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The geometry of Rayleigh dissipation Master’s Thesis Master in Theoretical Physics Autonomous University of Madrid and Institute of Theoretical Physics (CSIC-UAM)
Author: Asier López-Gordón Supervisor: Manuel de León (ICMAT-CSIC) Tutor: Germán Sierra
July 9, 2021
Contents
Contents iii
Abstractv
Resumen vii
Acknowledgements ix
1 Introduction1
2 Hamiltonian and Lagrangian mechanics3 2.1 Geometric foundations ...... 3 2.2 Symplectic geometry ...... 13 2.3 Hamiltonian mechanics ...... 15 2.4 The geometry of tangent bundles ...... 19 2.5 Lagrangian mechanics ...... 24 2.6 Momentum map and symplectic reduction ...... 34
3 Mechanical systems subjected to external forces 39 3.1 Mechanical systems subjected to external forces ...... 39 3.2 Symmetries and constants of the motion ...... 42 3.3 Momentum map and reduction ...... 48 3.4 Rayleigh systems ...... 51
4 Conclusions and future work 59
Bibliography 61
iii
Abstract
Geometric mechanics is a branch of mathematical physics that studies classical mechanics of particles and elds from the point of view of geometry. In a geometric language, symmetries can be expressed in a natural manner as vector elds that generate the corresponding symmetry group. Moreover, with the geometric machinery, the phase space of a mechanical system with symmetries can be reduced to a space with less dimensions. As it is well-known, non-conservative forces cannot be written as the gradient of a potential, so they cannot be absorbed in the Lagrangian or Hamiltonian. The description of a mechanical system subject to a non-conservative force requires an external force together with the Lagrangian or Hamiltonian function. In addition, external forces emerge for the description of certain systems with non-holonomic constraints. In this master’s thesis, autonomous Hamiltonian and Lagrangian systems are studied in the framework of symplectic geometry. After introducing the geometric tools that will be employed, several results regarding symmetries and constants of the motion are reviewed. The method of symplectic reduction for systems with symmetry is also presented. In a second part, non-conservative systems are presented in a geometric language. A Noether’s theorem for Lagrangian systems subject to external forces is obtained. Other results regarding symmetries and constants of the motion are derived as well. Furthermore, a theory for the reduction of forced Lagrangian systems invariant under the action of a Lie group is presented. These results are particularized for the so-called Rayleigh dissipation, that is, external forces that can be written as the derivative of a “potential” with respect to the velocities.
v
Resumen
La mecánica geométrica es la rama de la física matemática que estudia la mecánica clásica, tanto de partículas como de campos, desde el punto de vista de la geometría. En el lenguaje de la geometría, las simetrías pueden verse de manera natural como campos de vectores, los cuales son generadores del grupo de simetría correspondiente. Asimismo, con la maquinaria de la geometría, el espacio de fases de un sistema mecánico con simetrías puede ser reducido a un espacio de dimensión inferior. Como es bien sabido, las fuerzas no conservativas no pueden escribirse como el gradiente de un potencial, de modo que no se pueden absorber en un lagrangiano o hamil- toniano. La descripción de un sistema mecánico sometido a una fuerza no conservativa requiere considerar una fuerza externa junto al lagrangiano o hamiltoniano. Además, las fuerzas externas aparecen al estudiar ciertos sistemas con ligaduras no holónomas. En este Trabajo Fin de Máster se estudian sistemas autónomos, tanto hamiltoni- anos como lagrangianos, en el marco de la geometría simpléctica. Tras introducir las herramientas geométricas que se emplearán, se revisan los resultados en la bibliografía concernientes a las simetrías y constantes del movimiento. También se presenta el método de la reducción simpléctica. En una segunda parte, se presentan los sistemas no con- servativos en un lenguaje geométrico. Se obtiene un teorema de Noether para sistemas lagrangianos sometidos a fuerzas externas. Se obtienen además otros resultados acerca de las simetrías y constantes del movimiento de estos sistemas. Asimismo, se presenta una teoría para la reducción de sistemas lagrangianos forzados invariantes bajo la acción de un grupo de Lie. Estos resultados se particularizan para la llamada disipación de Rayleigh, esto es, fuerzas externas que se pueden escribir como la derivada de un «potencial» con respecto a las velocidades.
vii
Acknowledgements
I would like to thank my supervisor, Manuel de León, for introducing me to the re- search world and to the ICMAT’s Geometric Mechanics and Control Group. His careful corrections and comments on the original manuscript are deeply appreciated. I am grateful with Manuel Lainz for several enlightening discussions and explanations. I am also thankful with ICMAT’s administration sta , specially with Esther Ruiz, for helping me deal with several bureaucratic processes. Additionally, I want to thank Germán Sierra for having been my tutor. I am wholeheartedly thankful to my parents for having supported all my academic and personal projects. I want to acknowledge them for having made me a curious and hard-working person. I am grateful for the love and advice from all my friends, who have supported me in my best and worst moments. In particular, I must thank my friends and colleagues Laura Pérez and Irene Abril for encouraging me to apply for a JAE Intro grant, which eventually has lead to the existence of this text. This work has been supported by the CSIC’s JAE Intro Grant JAEINT_20_01494. A renewal of the grant by ICMAT is also acknowledged.
ix
Part 1
Introduction “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — E.P. Wigner, Comm. Pure Appl. Math., 13: 1-14 (1960)
Since the dawn of physics, mechanics has been one of its mostly studied branches. More- over, mechanics is the basis of astronomy, engineering and other disciplines. Analogously, geometry is one of the most ancient and fundamental branches of mathematics. As it is well-known, the rst mathematically precise formulation of mechanics is due to Newton. Unlike in the Newtonian framework, in the Lagrangian and Hamiltonian formalisms mechanical systems can be parametrized in arbitrary systems of coordinates without the need of incorporating ad hoc forces. The next logical step, accomplished by geometric mechanics, is to formulate laws of motion in an intrinsic way, that is, without the need of using coordinates. See reference [Leó17] for a historical review of geometric mechanics. In the language of geometry, the phase space of a system has the structure of a manifold endowed with a symplectic form, and its dynamics is given by the integral curves of a certain vector eld. Symmetries and constants of the motion are key concepts in modern mechanics. The knowledge of these properties gives a valuable information of the mechanical system to be studied, and it simplies the resolution of its equations of motion. Their relations have been deeply explored since the seminal work by Emmy Noether [Kos11; Nee99; Noe71]. The geometric formulation of mechanics, together with the theory of Lie groups and Lie algebras, allows to reduce the phase space of a system with symmetry, that is, to describe the dynamics in a space with less dimensions. The description of multiple physical systems requires considering an external force, together with the Lagrangian or the Hamiltonian, for instance, a dissipative force. More- over, non-holonomic systems have external forces caused by the constraints [ILD08; LD96]. These forces restrain the dynamics to the submanifold of constraints, which is the actual phase space of the system. Geometrically, an external force can be regarded as a semibasic 1-form. A particular type of external forces are the ones derived from the so-called Rayleigh dissipation function, that is, forces which are linear in the velocities. The concept of Rayleigh dissipation can be generalised for non-linear forces that can be written as the derivative of a “potential” with respect to the velocities. Symmetries and constants of the motion can also be considered for non-conservative systems (that is, systems with external forces), where they will depend on both the Lagrangian or Hamiltonian and on the external
1 2 PART 1. INTRODUCTION force. The method of reduction can also be generalised for non-conservative systems, requiring that the (sub)group of symmetries leaves the external force invariant as well as the Lagrangian or Hamiltonian. This master’s thesis is organized as follows:
• Part2 is a bibliographical revision of geometric mechanics for autonomous Hamil- tonian and Lagrangian systems depending on the positions and the velocities.
– Section 2.1 is an introduction to the basic mathematical tools that will be employed throughout the text: bre bundles, Cartan’s calculus of di erential forms and the actions of Lie groups and Lie algebras over manifolds. – Section 2.2 presents symplectic geometry, the geometric structure of phase spaces. – Section 2.3 is devoted to the Hamiltonian formalism presented in an intrinsic form. Symmetries and constants of the motion are presented in the language of di erential geometry. – Section 2.4 introduces the geometric structures in tangent bundles required for formulating the Lagrangian formalism. – Section 2.5 presents the Lagrangian formalism in its intrinsic form. The con- cepts introduced in the previous section are employed for studying symmetries and constants of the motion. – Section 2.6 is dedicated to symplectic reduction. If a Lie group of symmetries acts over the conguration space of a physical system, its dynamics can be described in a reduced space by means of this procedure.
• Part3 is devoted to mechanical systems with external forces. It covers the results from a paper of the author, coauthored by M. de León and M. Lainz, [LLL21b].
– Section 3.1 introduces mechanical systems subject to external forces in the language of geometry, both in the Lagrangian and Hamiltonian formalisms. – Section 3.2 studies symmetries and constants of the motion of forced La- grangian systems, generalizing the results from Section 2.5. – Section 3.3 extends symplectic reduction, explained in Section 2.6, for forced Lagrangian systems in which both the Lagrangian and the external force are invariant under a Lie (sub)group of symmetries. – Section 3.4 covers mechanical systems subject to Rayleigh external forces, particularizing the results from sections 3.2 and 3.3.
• Part4 summarizes the results obtained throughout this master’s thesis, and presents several lines for future research. Part 2
Hamiltonian and Lagrangian mechanics
2.1 Geometric foundations
This section is devoted to establish and recall the basic mathematical framework that will be employed throughout the text. The main references are [AM08; God69; LR89]. See also references [CCL99; DFN85; Lee12] for an introduction to di erential geometry of manifolds. Recall that a manifold is roughly a space that locally resembles R푛. More specically, a di erentiable manifold1 can be dened as follows. A di erentiable 푛-dimensional manifold is a set 푀 with the following properties:
i) The set 푀 is a nite or countably innite union of subsets 푈푞. 푈 푥훼 푈 훼 , . . . , 푛 ii) Each subset 푞 has dened on it local coordinates 푞 : 푞 → R, with = 1 .
iii) Each non-empty intersection of subsets, 푈푝 ∩ 푈푞, has dened on it at least two coordinate systems. Each of these two coordinate systems are required to be expressible in terms of the other as one-to-one di erentiable functions: 푥훼 = 푥훼 푥1, . . . , 푥푛 , 푝0 푝 푞 푞 (2.1) 푥훼 푥훼 푥1, . . . , 푥푛 . 푞 = 푞 푝 푝
휕푥훼 푥훼 Then the Jacobian det( 푝 / 푞 ) is non-zero in the region of intersection. 푈 , 휑 휑 푥1, . . . , 푥푛 푀 Each pair ( 푝 푈푝 ), with 푈푝 = ( 푝 푝 ), is called a coordinate chart of . A set 푈 , 휑 푀 푈 푀 of charts ( 푖 푈푖 ) which covers (that is, ∪푖 푖 = ) is called an atlas. The functions (2.1) are called transition functions. If, for each pair of charts in an atlas, the transition function between them is of class 퐶푟 , then the atlas is said to be of class 퐶푟 . The 퐶푟 di erentiable structure of 푀 is given by the maximal atlas of class 퐶푟 , that is, the union of every atlas of class 퐶푟 . It endows the set 푀 with the structure of a topological space.
1Unless otherwise stated, throughout this thesis every manifold will be assumed to be a di erentiable manifold. One can also dene the more general concept of topological manifold.
3 4 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
Let 푀 and 푁 be two manifolds with dim 푀 = 푚 and dim 푁 = 푛. Consider a continuous mapping 푓 : 푀 → 푁 . Suppose that there exists coordinate charts (푈 , 휑푈 ) and (푉,휓푉 ) at 푝 ∈ 푀, and 푓 (푝) ∈ 푁 , respectively, such that the mapping
휓 푓 휑−1 휑 푈 휓 푉 푉 ◦ ◦ 푈 : 푈 ( ) → 푉 ( )
∞ ∞ is of class 퐶 at the point 휑푈 (푝). Then the mapping 푓 is called 퐶 at 푝. A mapping 푓 : 푀 → 푁 that is 퐶∞ at every 푝 ∈ 푀 is called a di erentiable mapping (or a smooth mapping). Moreover, if 푚 ≥ 푛, a di erentiable mapping 푓 : 푀 → 푁 is said to be a submersion if rank 푓 = 푛 at every point of 푀. A bre bundle is a triple (퐸, 휋, 푀), where 퐸 is a manifold called the total space, 푀 is a manifold called the base space and 휋 : 퐸 → 푀 is a surjective submersion called the −1 projection of the bundle. For each 푥 in 푀, the submanifold 휋 (푥) = 퐸푥 is a surjective submersion called the bre over 푥. Moreover, 퐸 is called a bred manifold over 푀.A mapping 푠 : 푀 → 퐸 such that 휋 ◦ 푠 = id is called a section of 퐸. There are two types of bre bundles which are particularly relevant in physics: vector bundles and principal bundles. In the former each bre has the structure of a vector eld, whereas in the latter each bre has the structure of a Lie group. Throughout this thesis vector bundles, and particularly the tangent and cotangent bundle, will play the major role. However, it is also worth mentioning that principal bundles are closely related with gauge theories (see reference [CW06] and references therein). Recall that, given a vector space 푉 , GL(푉 ) denotes the general linear group of 푉 , that is, the set of all invertible linear transformations 푉 → 푉 , together with the composition 퐸 푀 푈 , 휑 of these transformations. Consider two manifolds and , and let ( 푖 푈푖 ) be an atlas of 푀.A (real) vector bundle 퐸 of rank 푞 over 푀, with typical bre 푉 = R푞, is a bre bundle (퐸, 휋, 푀) such that: 휑 푈 푞 휋−1 푈 i) Every map 푈푖 is a di eomorphism from 푖 × R to ( 푖), and 휋 휑 푝,푦 푝 ◦ 푈푖 ( ) =
for every 푝 ∈ 푈 and every 푦 ∈ R푞. ii) For any xed 푝 ∈ 푈 , let
휑 푦 휑 푝,푦 , 푦 푞. 푈푖,푝 ( ) = 푈푖 ( ) ∈ R 휑 푞 휋−1 푝 Then 푈푖,푝 : R → ( ) is an homeomorphism (that is, it is continuous and has a continuous inverse mapping).
iii) When 푈푖 ∩ 푈푗 ≠ for 푖 ≠ 푗, then, for any 푝 ∈ 푈푖 ∩ 푈푗 ,
푔 (푝) = 휑−1 ◦ 휑 : 푞 → 푞 푈푖푈푗 푈푗 ,푝 푈푖,푝 R R 푉 푞 푔 푝 푉 is a linear automorphism of = R , that is, 푈푖푈푗 ( ) ∈ GL( ). Moreover, the map 푔 푈 푈 푉 푈푖푈푗 : 푖 ∩ 푗 → GL( ) is di erentiable. Let 푀 be an 푚-dimensional di erentiable manifold. A curve at 푥 ∈ 푀 is a di eren- tiable mapping 휎 : 퐼 → 푀, with 퐼 3 0 an open interval of R and 휎 (0) = 푥. Two curves 2.1. GEOMETRIC FOUNDATIONS 5
휎 and 휏 are said to be tangent at 푥 if there exists a neighbourhood 푈 of 푥 with local coordinates (푥푖) such that
d(푥푖 ◦ 휎) d(푥푖 ◦ 휏) = , ≤ 푖 ≤ 푚. 푡 푡 1 d 푡=0 d 푡=0 This denition is independent of the local coordinate system and denes an equivalence class of curves. The equivalence class [휎] of 휎 is called the tangent vector of 휎 at 푥, also denoted by 휎¤ (0). The set of equivalence classes of curves at 푥 is called the tangent 푖 푖 space of 푀 at 푥, denoted by 푇푥 푀. The coordinates (푥 ) induce a basis 휕/휕푥 of 푇푥 푀. Let Ø 푇 푀 = 푇푥 푀. 푥∈푀
Let 휏푀 : 푇 푀 → 푀 be the canonical projection, given by 휏푚 (푣) = 푥 for each 푣 ∈ 푇푥 푀. The triple (푇 푀, 휏푀, 푀) is a vector bundle called the tangent bundle of 푀. The 2푚- dimensional manifold 푇 푀 is also sometimes called the tangent bundle of 푀. If 푓 : 푀 → 푁 is a mapping of class 퐶1 between the manifolds 푀 and 푁 , the tangent of 푓 is given by 푇 푓 : 푇 푀 → 푇 푁
푇 푓 ([휎]푥 ) = [푓 ◦ 휎] 푓 (푥) for each 푥 ∈ 푀. Consider a vector bundle (퐸, 휋, 푀). A di erentiable mapping 푠 : 푀 → 퐸 is called a section of the vector bundle if
휋 ◦ 푠 = id : 푀 → 푀.
A vector eld on 푀 is a section 푋 : 푀 → 푇 푀 assigning to each point 푥 ∈ 푀 a tangent 푋 (푥) 푀 푥 푀 (푥푖) (휕/휕푥푖) dim 푀 vector of at . If has local coordinates , then 푥 푖=1 is a basis for 2 푇푥 푀 for each 푥 ∈ 푀. A vector eld 푋 can thus be locally written as 휕 푋 = 푋 푖 (푥) . 휕푥푖
A curve 휎 : 퐼 → 푀 is called an integral curve of 푋 if, for every 푡 ∈ 퐼, 푋 (휎 (푡)) is the tangent vector to the curve at 휎 (푡). A vector eld acts as a di erential operator on a di erentiable function 푓 , locally 휕푓 푋 (푓 ) = 푋 푖 . 휕푥푖
Let 푀 be a 푚-dimensional di erentiable manifold. The cotangent space of 푀 at 푥, 푇 ∗푀 푇 푀 denoted by 푥 , is the dual vector space of 푥 (see [IR01] for dual vector space and 훼 푇 ∗푀 other concepts from linear algebra). An element of 푥 is called a tangent covector 푀 푥 푥푖 푥푖 푇 ∗푀 of at . The coordinates ( ) induce a basis d of 푥 . Set
푇 ∗푀 Ø 푇 ∗푀, = 푥 푥∈푀 2Throughout the text, the summation convention over repeated indices is employed. 6 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
휋 푇 ∗푀 푀 휋 훼 푥 훼 푇 ∗푀 and let 푀 : → be the canonical projection given by 푀 ( ) = for each ∈ 푥 . ∗ The triple (푇 푀, 휋푀, 푀) is a vector bundle called the cotangent bundle of 푀. The 2푚-dimensional manifold 푇 ∗푀 is also sometimes referred to as the cotangent bundle of 푀. A 1-form on 푀 is a section 훼 : 푀 → 푇 ∗푀 assigning to each point 푥 ∈ 푀 a tangent 훼 푥 푇 ∗푀 푓 푀 covector ( ) ∈ 푥 . Consider a di erentiable function : → R. The di erential of 푓 at 푥 ∈ 푀 is a linear mapping 푓 푥 푇 푀 푇 . d ( ) : 푥 → 푓 (푥)R R
Moreover, d푓 (푥) can be regarded as a tangent covector of 푀 at 푥. Let 푣 ∈ 푇푥 푀 and consider a curve 휎 on 푀 such that 휎 (0) = 푥 and 휎¤ (0) = 푣. Then
d푓 (푥)(푣) = d푓 (푥)[푣] = [푓 ◦ 휎], but [푓 ◦ 휎] is the tangent vector of R at 푓 (푥) given by the curve 푓 ◦ 휎, so if 푡 denotes the coordinate of R, d(푓 ◦ 휎) 휕 [푓 ◦ 휎] = . 푡 휕푡 d 푡=0 푓 (푥) On the other hand, d(푓 ◦ 휎) 푣 (푓 ) = , 푡 d 푡=0 and hence d푓 (푥)(푣) = 푣 (푓 ). 푀 (푥푖) 푥푖 dim 푀 In particular, if has local coordinates , then the set of 1-forms d 푖=1 satises 휕 휕 (d푥푖)(푥) = (푥푖) = 훿푖 , 휕푥 푗 휕푥 푗 푗 푥 푥
훿푖 ( 푥푖)(푥) dim 푀 푇 ∗푀 where 푗 is the well-known Kronecker delta. Therefore d 푖=1 is a basis for 푥 . (휕/휕푥푖) dim 푀 훼 In fact, it is the dual basis of 푥 푖=1 . Then a 1-form can be locally written as
푖 훼 = 훼푖 (푥)d푥 .
The di erential of 푓 at 푥 given by 휕푓 d푓 (푥) = (d푥푖)(푥), 휕푥푖 푥 which is the usual expression from elementary calculus. A 1-form 훼 is naturally paired to a vector eld 푋, yielding a function h훼, 푋i on 푀, locally given by 푖 h훼, 푋i ≡ 훼 (푋) = 훼푖 (푥)푋 (푥). From the point of view of geometric mechanics, if a manifold 푄 is the conguration space (that is, the space of positions), its tangent bundle 푇푄 can be regarded as the space of velocities, and its cotangent bundle푇 ∗푄 as the phase space. In fact, if 푄 has coordinates 2.1. GEOMETRIC FOUNDATIONS 7
푖 푖 푖 푖 ∗ (푞 ), it induces coordinates (푞 , 푞¤ ) on 푇푄, and coordinates (푞 , 푝푖) on 푇 푄. Physically (푞푖) are interpreted as the positions (generalized coordinates), (푞 ¤푖) as the (generalized) velocities, and (푝푖) as the conjugate momenta. Let 푉 and 푊 be vector spaces, and denote by 푉 ∗ and 푊 ∗ their respective dual vector spaces. The tensor product 푉 ∗ ⊗ 푊 ∗ is dened so that for each 푣∗ ∈ 푉 ∗ and 푤 ∗ ∈ 푊 ∗, there exists a 푣∗ ⊗ 푤 ∗ ∈ 푉 ∗ ⊗ 푊 ∗ such that
푣∗ ⊗ 푤 ∗(푥,푦) = 푣∗(푥) 푤 ∗(푦) for each 푥 ∈ 푉 , 푦 ∈ 푊 . Similarly, one can dene 푉 ⊗ 푊 . In addition, the condition
푣∗ ⊗ 푤 ∗ (푥 ⊗ 푦) = 푣∗(푥) 푤 ∗(푦) is required, so that 푉 ∗ ⊗ 푊 ∗ is the dual of 푉 ⊗ 푊 . The tensor space of type (푟, 푠) is given by 푇 푟 푉 푉 푉 푉 ∗ 푉 ∗. 푠 ( ) = ⊗ · · · ⊗ ⊗ ⊗ · · · ⊗ | {z } | {z } 푟-times 푠-times A tensor eld 퐾 of type (푟, 푠) on a manifold 푀 is a mapping assigning an element 퐾 푥 푇 푟 푇 푀 푥 푀 ( ) ∈ 푠 ( 푥 ) to each point ∈ , in other words, it is a section of the tensor bundle 푇 푟 푇 푀 푥푖 퐾 푠 ( ). Locally, if M has coordinates ( ), can be written as
··· 휕 휕 퐾 = 퐾푖1 푖푟 ⊗ · · · ⊗ ⊗ d푥 푗1 ⊗ · · · ⊗ d푥 푗푠 . 푗1···푗푠 휕푥푖1 휕푥푖푟
A skew-symmetric tensor eld of type (0, 푝) is called a 푝-form. A function 푓 : 푀 → R is sometimes called a 0-form. The set of 푝-forms on 푀 forms a vector bundle denoted by Ω푝 (푀). The full space of di erential forms of a manifold 푀 is
푚 Ê Ω(푀) B Ω푝 (푀). 푝=0 푝 푇 ∗푀 The exterior product (or wedge product) of basis elements of 푥 is given by 1 ∑︁ 푥푖1 ∧ 푥푖2 ∧ · · · ∧ 푥푖푝 = (휎) 푥휎(푖1) ⊗ 푥휎(푖2) ⊗ · · · ⊗ 푥휎 (푖푝 ), d d d 푝 sgn d d d ! 휎 where the sum is taken over all permutations 휎 of (1, 2, . . . , 푝). These products form a local basis of Ω푝 (푀). For instance, {d푥 ∧ d푦, d푥 ∧ d푧, d푦 ∧ d푧} is a basis of Ω2(R3). Then a 푝-form 훼 can be locally written as
훼 훼 푥푖1 푥푖푝 . = 푖1···푖푝 d ∧ · · · ∧ d
The exterior product of a 푝-form 훼 and a 푞-form 훽 is given by 1 훼 ∧ 훽 = 훼 훽 d푥푖1 ∧ · · · ∧ d푥푖푝 ∧ d푥 푗1 ∧ · · · ∧ d푥 푗푞 . 푝!푞! 푖1...푖푝 푗1...푗푞
An 푚-dimensional manifold 푀 is said to be orientable if there exists an 푚-form 휔 on 푀 such that 휔 (푥) ≠ 0 for all 푥 ∈ 푀; 휔 is called a volume form. If a volume form Ω has 8 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS compact support in a region contained in a neighbourhood 푈 ⊂ 푀, it can be integrated, yielding the volume of the corresponding orientable manifold. If (푥푖) are coordinates on 푈 , Ω can be written as
Ω = 푓 (푥1, . . . , 푥2푚) d푥1 ∧ · · · ∧ d푥2푚, and the integral of Ω on 푀 is given by ∫ ∫ Ω = 푓 (푥1, . . . , 푥2푚) d푥1 ··· d푥2푚, 푀 푈 where the integral on the right-hand side is the usual Riemann integral. The exterior derivative on 푀 is the unique linear operator d : Ω(푀) → Ω(푀) such that
i) d : Ω푝 (푀) → Ω푝+1(푀),
ii) d2 = 0,
iii) for each 푓 ∈ 퐶∞(푀), d푓 is the di erential of 푓 , locally given by 휕푓 d푓 = d푥푖; 휕푥푖
iv) if 훼 ∈ Ω푝 (푀) and 훽 ∈ Ω(푀), then
d(훼 ∧ 훽) = d훼 ∧ 훽 + (−1)푝 ∧ d훽.
A 푝-form 휔 is called closed if d휔 = 0, and exact if there exists a (푝 − 1)-form 훼 such that 휔 = d훼. Clearly, every exact form is closed. Furthermore, if 휔 is closed it is locally exact, that is, for each point in 푀 there exists a neighbourhood 푈 such that 휔|푈 is exact. The latter result is known as the Poincaré lemma. The interior product of a vector eld 푋 by a 푝-form 훼 is dened as follows:
i) 휄푋 훼 = 0 if 푝 = 0,
ii) 휄푋 훼 = 훼 (푋) if 푝 = 1,
iii) 휄푋 훼 is a (푝 − 1)-form such that
(휄푋 훼)(푌1, . . . , 푌푝−1) = 훼 (푋, 푌1, . . . , 푌푝−1)
for each 푌1, . . . , 푌푝−1 ∈ 푇 푀. Let 푈 be a open neighbourhood of the manifold 푀.A local one-parameter group (of di eomorphisms) is a smooth mapping Φ : (−휖, 휖) × 푈 → 푀, denoted by Φ푡 (푝) for each 푝 ∈ 푈 and each |푡 | < 휖, such that for any 푝 ∈ 푈 and any |푠| , |푡 | < 휖
i) Φ0(푝) = 푝;
ii) if |푡 + 푠| < 휖 and Φ푡 (푝) ∈ 푈 , then Φ푡+푠 (푝) = Φ푠 ◦ Φ푡 (푝). 2.1. GEOMETRIC FOUNDATIONS 9
A local one-parameter group induces a vector eld on 푈 . Indeed, if (푥푖) are coordinates at 푈 one can write 휕 푋 = 푋 푖 , 푝 푝 휕푥푖 푝 where d푥푖 (Φ (푝)) 푋 푖 = 푡 . 푝 푡 d 푡=0 Conversely, a vector eld 푋 on 푀 denes a local one-parameter group for each point 푝 ∈ 푀. Consider the system of ordinary di erential equations
푥푖 푡 d 푝 ( ) = 푋 푖 (푥 (푡)), 1 ≤ 푖 ≤ 푚, d푡 푝 푥 푥1 , . . . , 푥푚 푝 with the initial conditions 푝 (0) = ( 푝 (0) 푝 (0)) = . It can be easily shown that
Φ푡 (푝) = Φ(푡, 푝) = 푥푝 (푡) is a local one-parameter group. The local one-parameter group generated by 푋 is also known as the ow of 푋. As it will be detailed throughout the text, physically vector elds can represent innitesimal symmetries, and the one-parameter groups they generate represent their associated nite symmetries. For instance, 푋 = 휕/휕푥 generates translations along the 푥-direction. Let 휑 : 푀 → 푁 be a smooth mapping between the manifolds 푀 and 푁 . The di erential of 휑 at 푥 is a linear mapping d휑 such that, for any function 푓 on 푁 and every vector eld 푋 on 푀, d휑 (푋)(푓 ) = 푋 (푓 ◦ 휑). Let 훼 be a 푝-form on 푁 . The pullback of 훼 by 휑, denoted by 휑∗훼, is a 푝-form on 푀 given by ∗ 휑 훼 (푋1, . . . , 푋푝)(푥) = 훼 (d휑 (푋1),..., d휑 (푋푝))(휑 (푥)) 0 ∗ for each 푥 ∈ 푀 and 푋1, . . . , 푋푝 ∈ 푇푥 푀. In particular, for each 푓 ∈ Ω (푁 ), 휑 푓 = 푓 ◦ 휑. Let now 푋 be a vector eld on 푀. The pushforward of 푋 by 휑 is a vector eld 휑∗푋 on 푁 dened by (휑∗푋)(푔) = 푋 (푔 ◦ 휑). for each function 푔 : 푁 → R. Let 푋 a vector eld on 푀, and Φ푡 the local one-parameter group generated by 푋. Let 훼 be a 푝-form on 푀. The Lie derivative of 훼 with respect to 푋, denoted by L푋 훼, is a 푝-form on 푀 given by ∗ 훼 훼 Φ푡 ( ) − L푋 훼 = lim . (2.2) 푡→0 푡 In practice, the Lie derivative can be computed by means of the so-called Cartan’s formula: L푋 훼 = 휄푋 (d훼) + d(휄푋 훼). (2.3) In particular, L푋 푓 = 푋 (푓 ) 10 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS for every function 푓 on 푀. The denition of Lie derivative can be extended for a tensor eld of type (푟, 푠). Consider the mapping ˜ 푇 푟 푇 푀 푇 푟 푇 푀 Φ푡 : 푠 ( Φ푡 (푝) ) → 푠 ( 푝 ) given by ˜ 푣 푣 푣∗1 푣∗푠 −1 푣 −1 푣 ∗ 푣∗1 ∗ 푣∗푠 , Φ푡 1 ⊗ · · · ⊗ 푟 ⊗ ⊗ · · · ⊗ = Φ푡 ∗ ( 1)⊗· · ·⊗ Φ푡 ∗ ( 푟 )⊗Φ푡 ( )⊗· · ·⊗Φ푡 ( )
∗1 ∗푠 ∗ for any 푣1, . . . , 푣푟 ∈ 푇Φ (푝)푀 and any 푣 , . . . , 푣 ∈ 푇 푀. If 푇 is a (푟, 푠)-type tensor eld, 푡 Φ푡 (푝) its Lie derivative with respect to 푋 is dened by
Φ˜ 푡 (푇 ) − 푇 L푋푇 = lim , 푡→0 푡 and it is also an (푟, 푠)-type tensor eld. Given two vector elds 푋 and 푌 on 푀, consider a new vector eld [푋, 푌] on 푀 such that [푋, 푌](푓 ) = 푋 (푌 (푓 )) − 푌 (푋 (푓 )) for every function 푓 on 푀. This vector eld is called the Lie bracket (or the commutator) of 푋 and 푌. The commutator of 푋 and 푌 coincides with the Lie derivative of 푌 with respect to 푋, namely [푋, 푌] = L푋푌 = −L푌 푋 . In addition, one can easily check that the following properties hold: i) [푋, 푌] = −[푌, 푋], ii) [푓 푋,푔푌] = 푓 푋 (푔)푌 − 푔푌 (푓 )푋 + 푓 푔[푋, 푌], iii) [[푋, 푌], 푍] + [[푌, 푍], 푋] + [[푍, 푋], 푌] = 0 (Jacobi identity). Other relevant identities that will be employed are the following:
푝 휄푋 (훼 ∧ 훽) = 휄푋 훼 ∧ 훽 + (−1) 훼 ∧ 휄푋 훽, 휄푋 휄푋 훼 = 0, 푋 (푓 ) = 휄푋 d푓 , 푝 ∑︁ 푖 d훼 (푋0, . . . , 푋푝) = (−1) 푋푖 훼 (푋0, . . . , 푋푖−1, 푋푖+1, . . . , 푋푝) 푖=0 ∑︁ 푖+푗 + (−1) 훼 [푋푖, 푋푗 ], 푋0, . . . , 푋푖−1, 푋푖+1, . . . , 푋푗−1, 푋푗+1, . . . , 푋푝 푖<푗 (2.4) 휑∗ 휄 훼 휄 휑∗훼 , ( 푋 ) = 휑∗푋 ( ) 휑∗(d훼) = d(휑∗훼),
L푓 푋 훼 = 푓 L푋 훼 + d푓 ∧ 휄푋 훼, 휄[푋,푌]훼 = L푋 휄푌 훼 − 휄푌 L푋 훼, L푋 d훼 = dL푋 훼, L푋 휄푋 훼 = 휄푋 L푋 훼, 2.1. GEOMETRIC FOUNDATIONS 11
where 훼 and 훽 are a 푝-form and a 푞-form respectively, and 푋0, . . . , 푋푝 are vector elds. Finally, the concepts of Lie group and Lie algebras, as well as its actions on manifolds, will be important for the study of symmetries and reduction. See references [AM08; Hal15; LR89; OR04]. Recall that a Lie group 퐺 is a di erentiable manifold which is also an algebraic group whose operations are di erentiable, that is,
i) it is equipped with a smooth and associative multiplication map 휇 : 퐺 × 퐺 → 퐺,
ii) there exists an identity element 푒,
iii) for each 푔 ∈ 퐺, there exists an inverse 푔−1, so that, for any 푔 ∈ 퐺, 휇(푔, 푒) = 휇(푒,푔) = 푔, and 휇(푔,푔−1) = 휇(푔−1,푔) = 푒.
An action of a Lie group 퐺 on a manifold 푀 is a smooth mapping Φ : 퐺 × 푀 → 푀 such that
Φ(푒, 푥) ≡ Φ푒 (푥) = 푥 for any 푥 ∈ 푀. An action is said to be
i) free if, for every 푥 ∈ 푀, Φ푔 (푥) = 푥 if and only if 푔 = 푒,
ii) proper if and only if Φ is a proper mapping, that is, for any compact subset 퐾 ⊂ 푀, Φ−1(퐾) is also compact.
Remark 1. The action of a compact Lie group is always proper. The isotropy group of 푥 ∈ 푀 is the Lie subgroup 퐺푥 = 푔 ∈ 퐺 | 휙푔 (푥) = 푥 ⊂ 퐺
Let 퐹 : 푀 → 푁 be a mapping between the manifolds 푀 and 푁 , and let Φ : 퐺 × 푀 → 푀 and Ψ : 퐺 × 푁 → 푁 be group actions. The mapping 퐹 is said to be equivariant (with respect to these actions of 퐺) if
퐹 (Φ푔푥) = Ψ푔퐹 (푥) for any 푔 ∈ 퐺. An algebra is a vector space 푉 equipped with a bilinear multiplication. If this multiplication, denoted by [·, ·] : 푉 × 푉 → 푉 , additionally satises
i) anti-symmetry: [푋, 푌] = −[푌, 푋],
ii) Jacobi identity: [푋, [푌, 푍]] + [푌, [푍, 푋]] + [푍, [푋, 푌]] = 0, 12 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS for every 푋, 푌, 푍 ∈ 푉 , then (푉, [·, ·]) forms a Lie algebra. For each Lie group 퐺, there is a corresponding Lie algebra 픤 u 푇푒퐺. The left translation 퐿푔 : 퐺 → 퐺 dened by 푔 ∈ 퐺 is given by
퐿푔 (ℎ) = 푔ℎ for each ℎ ∈ 퐺. A vector eld 푋 on 퐺 is called left invariant if the pushforward of every left translation leaves it unchanged, namely
퐿푔∗푋 = 푋 for every 푔 ∈ 퐺. The vector subspace 픛퐿 (퐺) of left invariant vector elds on 퐺 is isomorphic to 푇푒퐺, the tangent space of 퐺 at the identity. More specically, for each 푋 ∈ 픛퐿 (퐺), it corresponds its value at the identity 푋 (푒) ∈ 푇푒퐺. The inverse of this map is given by
푇푒퐺 → 픛퐿 (퐺) 휉 ↦→ 휉퐿, with 휉퐿 such that 휉퐿 (푒) = 휉. Via this isomorphism, 푇푒퐺 becomes a Lie algebra, with Lie bracket
[휉퐿, 휂퐿] = [휉, 휂]퐿, where [·, ·] on the left-hand side denotes the Lie bracket of vector elds on 퐺. It is known as the Lie algebra of 퐺 and usually denoted by 픤. Consider a left invariant vector eld 푋 on 퐺. The exponential map is given by
exp : 픤 → 퐺
exp(휉) = 훾휉 (1),
휉 훾 퐺 휉 푒 where ∈ 픤, and 휉 : R → is the integral curve of through . The adjoint action is an action of 퐺 on its Lie algebra 픤 given by
Ad : 퐺 × 픤 → 픤 d (푔, 휉) ↦→ 휉 = 푔 (푡휉)푔−1 . Ad푔 푡 exp d 푡=0
The coadjoint action is an action Ad∗ : 퐺 × 픤∗ → 픤∗, with 픤∗ the dual 픤, such that
∗ 훼 휉 훼 휉 (Ad푔 )( ) = (Ad푔 ) for each 훼 ∈ 픤∗, each 휉 ∈ 픤 and each 푔 ∈ 퐺. Given a group 퐺 acting on a manifold 푀, one can dene an equivalence relation by setting 푥 ∼ 푦 (with 푥,푦 ∈ 푀) if there exists a 푔 ∈ 퐺 such that Φ푔 (푥) = 푦. The equivalence classes are called the orbits of 퐺 in 푀. If 퐺 is a Lie group acting smoothly, freely and properly on a di erentiable manifold 푀, it can be shown (see reference [Lee12, Theorem 21.10]) that the orbit space 푀/퐺 is a di erentiable manifold of dimension dim 푀 −dim퐺. 2.2. SYMPLECTIC GEOMETRY 13
2.2 Symplectic geometry
Symplectic geometry is the natural framework for Hamiltonian mechanics (see references [AM08; Arn78; God69; LR89; MR07]). As it will be seen, a phase space has the structure of a symplectic manifold. Moreover, Lagrangian systems can also be endowed with a symplectic structure when the Legendre transform is well-dened. Consider a manifold 푀 and a 2-form 휔 on 푀. For each point 푥 ∈ 푀 one can dene a mapping ♭ 푇 푀 푇 ∗푀 휔 (푥) : 푥 → 푥 푋 ↦→ 휄푋 휔 (푥).
The dimension of the image of ♭휔 (푥) is called the rank of 휔 at 푥:
rank 휔 (푥) = dim(Im ♭휔 (푥)).
It can be shown that the rank of every 2-form is an even number. The dimension of the kernel of ♭휔 (푥) is called the corank of 휔 at 푥:
corank 휔 (푥) = dim(ker ♭휔 (푥)), where ker ♭휔 (푥) = {푋 ∈ 푇푥 푀 | 휄푋 휔 (푥) = 0} . A 2-form is called non-degenerate if corank 휔 (푥) = 0 for every 푥 ∈ 푀.A symplectic form (or symplectic structure) on a manifold 푀 is a non-degenerate closed 2-form on 푀. A manifold 푀 endowed with a symplectic form 휔, denoted by (푀, 휔), is called a symplectic manifold. Every symplectic manifold is even-dimensional. By the Darboux theorem, if (푀, 휔) is a 2푚-dimensional symplectic manifold, there 1 푚 exists a neighbourhood of each point with local coordinates (푞 , . . . , 푞 , 푝1, . . . , 푝푚) such that the symplectic form can be written as
푖 휔 = d푞 ∧ d푝푖 . (2.5)
Consider two symplectic manifolds (푀, 휔) and (푁, 훼). A di erentiable mapping 휑 : 푀 → 푁 is called a symplectic transformation if 휑∗훼 = 휔. If a symplectic transformation is a global di eomorphism, it is called a symplectomorphism. In particular, a symplectic transformation 휑 : 푀 → 푀 which preserves the symplectic form, 휑∗휔 = 휔, is called a canonical transformation. A symplectic vector eld (or an innitesimal symplectic transformation) is a vector eld 푋 on 푀 which generates symplectic transformations. A vector eld 푋 is a symplectic vector eld if and only if
L푋 휔 = 0.
By Cartan’s formula (2.3), this is equivalent to 휄푋 휔 being a closed 1-form. Let Φ푡 be the ow of 푋. By the denition of Lie derivative (2.2),
d Φ∗휔 = Φ∗L 휔 = 0, d푡 푡 푡 푋 14 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
∗휔 푡 ∗휔 휔 so Φ푡 is constant in . On the other hand, Φ0 is the identity, and hence Φ푡 = . That is, the ow of a symplectic vector eld is a canonical transformation. Conversely, a canonical transformation induces a symplectic vector eld. A symplectic manifold (푀, 휔) has two natural isomorphisms between 푇 푀 and 푇 ∗푀:
∗ ♭휔 : 푇 푀 → 푇 푀 푋 ↦→ ♭휔 (푋) = 휄푋 휔,
♯ ♭−1 and 휔 B 휔 , both of which are called musical isomorphisms. To each function 푓 퐶1 푀, 푋 푀 ∈ ( R), one can associate a vector eld 푓 on , given by
푋푓 = ♯휔 (d푓 ), called a Hamiltonian vector eld. Clearly, every Hamiltonian vector eld is symplectic. The Poisson bracket is the bilinear operation
∞ ∞ ∞ {·, ·} : 퐶 (푀, R) × 퐶 (푀, R) → 퐶 (푀, R) {푓 ,푔} = 휔 (푋푓 , 푋푔), with 푋푓 , 푋푔 the Hamiltonian vector elds associated to 푓 and 푔, respectively. It satisfyes the following properties:
i) anti-symmetry: {푓 ,푔} = −{푔, 푓 },
ii) Leibniz product: {푓 ℎ,푔} = {푓 ,푔}ℎ + 푓 {ℎ,푔},
iii) Jacobi identity: {푓 , {푔, ℎ}} + {푔, {ℎ, 푓 }} + {ℎ, {푓 ,푔}} = 0,
∞ ∞ for each 푓 ,푔 ∈ 퐶 (푀, R). By the properties i) and iii), the vector space of 퐶 -functions on 푀 forms a Lie algebra, with Lie bracket being the Poisson bracket. In addition, 푓 ,푔 푔 푋 푔 푓 푋 푓 . { } = −L푋푓 = − 푓 ( ) = L푋푔 = 푔 ( )
In canonical coordinates (푞, 푝), 휕푓 휕푔 휕푓 휕푔 {푓 ,푔} = − , 푖 푖 휕푞 휕푝푖 휕푝푖 휕푞 which is the denition of Poisson bracket usually given in undergraduate Classical Mechanics courses [Gan70; Gol87]. The Poisson bracket of two 1-forms 훼 and 훽 on 푀 is a 1-form on 푀 given by {훼, 훽} = −♭휔 [푋훼, 푋훽 ] . (2.6) Hence the diagram
−[· ·] 푇 푀 × 푇 푀 , 푇 푀
♭휔 ×♭휔 ♭휔
{·,·} Ω1(푀) × Ω1(푀) Ω1(푀) 2.3. HAMILTONIAN MECHANICS 15 commutes (that is, all the directed paths in the diagram with the same start and endpoints lead to the same result). Then the vector space Ω1(푀) with the Poisson bracket dened in equation (2.6) is also a Lie algebra. The Poisson bracket of 1-forms satisfyes the following properties: 훼, 훽 퐿 훽 퐿 훼 푑 푖 푖 휔 i) { } = − 푋훼 + 푋훽 + 푋훼 푋훽 ,
ii) if 훼 and 훽 are closed 1-forms, then 훼, 훽 is exact;
iii) {d푓 , d푔} = d {푓 ,푔} for any 푓 ,푔 ∈ 퐶∞(푀).
A corollary of the latter property is that
푋{퐹,퐺} = −[푋푓 , 푋푔].
Every cotangent bundle 푇 ∗푄 is itself a symplectic manifold. In fact, the concept of symplectic manifolds emerged from cotangent bundles and Hamiltonian mechanics. Let 푄 be a manifold with coordinates (푞), and (푞, 푝) the induced coordinates on its cotangent bundle 푇 ∗푄. The 1-form on 푇 ∗푄 dened locally as
푖 휃 = 푝푖d푞 (2.7) is called the canonical 1-form (or tautological 1-form). The 2-form 휔 = −d휃 is precisely the symplectic form in the Darboux coordinates as in equation (2.5). This 2-form is known as the canonical symplectic form. The canonical 1-form on 푇 ∗푄 is the unique 1-form on 푇 ∗푄 such that
훼∗휃 = 훼 (2.8) for any 1-form 훼 on 푄, and thus
훼∗휔 = −훼∗d휃 = −d(훼∗휃) = −d훼.
If 푄 is an 푛-dimensional manifold, then 푇 ∗푄 is 2푛-dimensional and the 2푛-form Ω given by Ω = 휔 ∧ · · · ∧ 휔 ≡ 휔푛 | {z } 푛 times is a volume form.
2.3 Hamiltonian mechanics
The main references for this section are [AM08; LR89]; see also references [Arn78; JS98]. Geometrically, a Hamiltonian system is a triple (푀, 휔, 퐻), with (푀, 휔) a symplectic manifold and 퐻 : 푀 → R the Hamiltonian function. If a mechanical system has conguration space 푄, its phase space is the cotangent bundle 푇 ∗푄, with canonical symplectic structure (2.5). The trajectories of the system are the integral curves of the vector eld 푋퐻 on 푀 given by 휄 휔 퐻. 푋퐻 = d 16 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
Locally, 푋퐻 can be written as 휕 휕 푋 = 푋 푖 + 푋˜ , 퐻 푖 푖 휕푞 휕푝푖 so 휄 휔 푋 푖 푝 푋˜ 푞푖 . 푋퐻 = d 푖 − 푖d On the other hand, 휕퐻 휕퐻 퐻 = 푞푖 + 푝 , d 푖 d d 푖 휕푞 휕푝푖 푖 푖 so 푋 = 휕퐻/휕푝푖 and 푋˜푖 = 휕퐻/휕푞 , that is, 휕퐻 휕 휕퐻 휕 푋 = − . 퐻 푖 푖 (2.9) 휕푝푖 휕푞 휕푞 휕푝푖
The integral curves of 푋퐻 are then given by 휕퐻 푞¤푖 = , 휕푝푖 휕퐻 푝¤ = − , 푖 휕푞푖 which are the well-known Hamilton’s equations. They can also be written as
푞¤푖 = 푞푖, 퐻 , (2.10) 푝¤푖 = − {푝푖, 퐻 } , where {·, ·} is the Poisson bracket on (푀, 휔). Since 푋퐻 is, by denition, a Hamiltonian vector eld, it is also a symplectic vector ∗휔 휔 eld. Therefore its ow Φ푡 is a canonical transformation, that is, Φ푡 = . From here, one can immediately obtain the following well-known result in Hamiltonian mechanics.
푛 Proposition 1 (Liouville’s theorem). The ow of 푋퐻 preserves the volume form 휔 , that is, ∗휔푛 휔푛. Φ푡 = In other words, the volume of the phase space is preserved along the evolution of the system.
One can also see that a canonical transformation 휑 : 푇 ∗푄 → 푇 ∗푄 preserves the Hamilton’s equations. Indeed,
휑∗ {푓 ,푔} 휑∗(푋 (푓 )) 휑∗(휄 푓 ) 휄 (휑∗ 푓 ) 휄 (휑∗푓 ) 휄 휄 휔 −휄 휄 휔, = 푔 = 푋푔 d = 휑∗푋푔 d = 휑∗푋푔 d = 휑∗푋푔 푋휑∗푓 = 푋휑∗푓 휑∗푋푔 where the properties (2.4) have been used, but
휄 휔 휄 (휑∗휔) 휑∗(휄 휔) 휑∗( 푔) (휑 ∗ 푔) 휄 휔, 휑∗푋푔 = 휑∗푋푔 = 푋푔 = d = d = 푋휑∗푔 so ∗ ∗ ∗ 휑 {푓 ,푔} −휄 휄 휔 휔 (푋 ∗ , 푋 ∗ ) {휑 푓 , 휑 푔} , = 푋휑∗푓 푋휑∗푔 = 휑 푓 휑 푔 = 2.3. HAMILTONIAN MECHANICS 17 that is, {푓 ,푔} ◦ 휑 = {푓 ◦ 휑,푔 ◦ 휑} . In particular, if 휑 : (푞, 푝) ↦→ (푞,˜ 푝˜), with (푞, 푝) and (푞,˜ 푝˜) canonical coordinates, d푞˜푖 푞푖, 퐻 ◦ 휑 = 푞푖 ◦ 휑, 퐻 ◦ 휑 = 푞˜푖, 퐾 = , d푡 d푝˜ {푝 , 퐻 } ◦ 휑 = {푝 ◦ 휑, 퐻 ◦ 휑} = {푝˜ , 퐾} = − 푖 , 푖 푖 푖 d푡 with 퐾 = 퐻 ◦ 휑 the Hamiltonian in the new coordinates. The right-hand side equalities take the same form as the original Hamilton’s equations (2.10). Consider a vector eld 푋 on a manifold 푀. A 1-form 훼 on 푀 is called a rst integral of 푋 if 훼 (푋) = 0. Similarly, a function 푓 on 푀 such that 푋 (푓 ) = 0 is also called a rst integral of 푋. Clearly, ♭휔 (푋) is a rst integral of 푋, for any 푋 on a symplectic manifold (푀, 휔). Any function 푓 is a rst integral of its corresponding Hamiltonian vector eld 푋푓 . In particular, a rst integral 푓 of 푋퐻 is called a constant of the motion. Indeed, if 푓 is a constant of the motion and 휎(푡) is an integral curve of 푋퐻 , d 0 = 푋 (푓 )(휎 (푡)) = 푓 (휎 (푡)), 퐻 d푡 so 푓 is constant along the evolution of the system. In particular, the Hamiltonian function 퐻 is conserved along the evolution (conservation of the energy). As it is well-known, there are multiple physical systems for which the energy is not conserved. In order to describe some of these systems, one can consider non-autonomous Hamiltonians, that is, 퐻 depending explicitly on time. These systems are naturally described in cosymplectic geometry3. Other non-conservative systems are characterized by an external force (described by a 1-form) together with the Hamiltonian function. Part 3 of this thesis is devoted to mechanical systems with external forces. Other physical systems for which the energy is not conserved can be described in the framework of contact geometry, or in other geometric structures, rather than symplectic geometry. See references [LL19; LL21] and references therein for contact geometry and its applications in physics. In terms of Poisson brackets, one can write 푋퐻 (푓 ) = {푓 , 퐻 }. If {푓 , 퐻 } = 0, obviously {d푓 , d퐻 } = d {푓 , 퐻 } = 0. On the other hand, { 푓 , 퐻 } { 퐻, 푓 } ♭ 푋 , 푋 휄 휔 휄 휔 휄 휔. d d = − d d = 휔 ([ 퐻 푓 ]) = [푋퐻 , 푋푓 ] = L푋퐻 푋푓 − 푋푓 L푋퐻 푋 휔 Now, since 퐻 is a Hamiltonian vector eld, L푋퐻 = 0. Then 푓 , 퐻 휄 휔 푓 푓 푋 푓 , {d d } = L푋퐻 푋푓 = L푋퐻 (d ) = d(L푋퐻 ) = d( 퐻 ( )) so {d푓 , d퐻 } vanishes if and only if 푋퐻 (푓 ) is a constant function. Without loss of generality, this constant can be taken as zero (clearly, 푓 is a constant of the motion if and only if 푓 + 퐾 is a constant of the motion for any 퐾 ∈ R). In addition, since 휔 is non-degenerate, 휄 휔 푋 , 푋 푋 푓 푋 퐻 [푋퐻 , 푋푓 ] vanishes if and only if [ 퐻 푓 ] does. Moreover, observe that 퐻 ( ) = − 푓 ( ). This results can be summed up as follows. 3A cosymplectic manifold [LS17] is an (2푛 + 1)-dimensional manifold equipped with a closed 1-form 휂 푛 ∗ 푖 and a closed 2-form 휔 such that 휂 ∧ 휔 ≠ 0. In 푇 푄 × R with Darboux coordinates 푡, 푞 , 푝푖 , 휂 = d푡 and 휔 takes the form in equation (2.5), which is basically the framework for autonomous Hamiltonian systems with the addition of a time dimension. 18 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
Proposition 2. The following statements are equivalent:
i) 푓 is a constant of the motion,
ii) {푓 , 퐻 } = 0,
iii) [푋퐻, 푋푓 ] = 0,
iv) 퐻 is a rst integral of 푋푓 .
Example 1 (Conservation of momentum). Consider a Hamiltonian system (푀, 휔, 퐻) with canonical coordinates (푞, 푝). Each momentum 푝푖 is itself a function on 푀. By equation (2.9), 휕퐻 푋 (푝 ) = − , 퐻 푖 휕푞푖
푖 so 푝푖 is a constant of the motion if and only if 휕퐻/휕푞 = 0, that is, if and only if 퐻 is invariant under translations along the 푞푖-direction. ♦
Example 2 (Conservation of angular momentum). Consider a Hamiltonian system ∗ (푇 (R푛), 휔, 퐻), with Hamiltonian function of the form
p2 퐻 = + 푉 (kxk), 2푚
푥1, . . . , 푥푛 푝 , . . . , 푝 푇 ∗ 푛 2푛 with x = ( ) and p = ( 1 푛) the canonical coordinates on (R ) u R . Then 푝 휕 푉 0(kxk) 휕 푋 = 푖 − 푥푖 . 퐻 푖 (2.11) 푚 휕푥 kxk 휕푝푖
1 2 Let 푓 = 푥 푝2 − 푥 푝1. Then
푝 푝 푉 0(kxk) 푉 0(kxk) 푋 (푓 ) = 1 푝 − 2 푝 − 푥1(−푥2) − 푥2푥1 = 0, 퐻 푚 2 푚 1 kxk kxk so 푓 is a constant of the motion. The Hamiltonian vector eld associated to 푓 is given by
휕 휕 휕 휕 푋 = −푥2 + 푥1 − 푝 + 푝 . 푓 1 2 2 1 휕푥 휕푥 휕푝1 휕푝2
The reader can easily check that each of the statements in Proposition2 is veried. 푗 Analogously, one can show that 퐿푖 = 휀푖 푗푘푥 푝푘 , for 푖 = 1, . . . , 푛, are constants of the motion, where 휀푖 푗푘 is the totally antisymmetric Levi-Civita symbol. In particular, for a mechanical system on R3 with a Hamiltonian function of the form (2.11), the angular momentum L = x × p is conserved. ♦ 2.4. THE GEOMETRY OF TANGENT BUNDLES 19
2.4 The geometry of tangent bundles
As it has been explained in Section 2.3, the Hamiltonian description of a mechanical system is settled in the cotangent bundle of its conguration space. Similarly, the Lagrangian formalism is settled in the tangent bundle of its conguration space. Unlike the cotangent bundle, the tangent bundle does not have a canonical symplectic structure. However, as it will be explained in Section 2.5, when the Lagrangian function 퐿 is “well-behaving”, the symplectic structure on 푇 ∗푄 induces a symplectic structure on 푇푄 that depends on 퐿. Before presenting the Lagrangian formalism in its intrinsic form, some further ge- ometric structures need to be introduced. See references [Cra83; God69; LR89; MR07; YI67; YI73]. Let 푄 be an 푛-dimensional manifold. If 푓 : 푄 → R is a function on 푄, its vertical lift is a function 푣 푓 = 푓 ◦ 휏푄 : 푇푄 → R, on 푇푄, with 휏푄 : 푇푄 → 푄 the canonical projection. If a point 푃 ∈ 푇푄 has induced coordinates (푞, 푞¤) = (푞1, . . . , 푞푛, 푞¤1,..., 푞¤푛), then
푣 푣 푓 (푃) = 푓 (푞, 푞¤) = 푓 ◦ 휏푄 (푞, 푞¤) = 푓 (푞). If 훼 is a 1-form on 푄, it can be naturally regarded as a function on 푇푄, denoted by 횤훼. Locally, if 푄 has coordinates (푞푖), 푇푄 has induced coordinates (푞푖, 푞¤푖) and
푖 훼 = 훼푖d푞 , then 푖 횤훼 = 훼푖푞¤ . A vertical vector eld 푋˜ is a vector eld on 푇푄 such that 푋˜ (푓 푣 ) = 0 for every function 푓 on 푄. Locally, if 휕 휕 푋˜ = 푋 푖 + 푌 푖 , 휕푞푖 휕푞¤푖 then 휕푓 푋˜ (푓 푣 ) = 푋 푖 , 휕푞푖 so 푋˜ is a vertical vector eld if and only if it is of the form 휕 푋˜ = 푋˜ 푖 . 휕푞¤푖 If 푋 is a vector eld on 푄, its vertical lift is the unique vector eld 푋 푣 on 푇푄 such that 푋 푣 (횤훼) = (훼 (푋))푣 for every 1-form 훼 on 푄. Clearly, every vertical lift is a vertical vector eld by construction. Locally, if 휕 푋 = 푋 푖 , 휕푞푖 20 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS then its vertical lift is given by 휕 푋 푣 = 푋 푖 . 휕푞¤푖 It can be easily proven that [푋 푣, 푌 푣 ] = 0 holds for every pair of vector elds 푋 and 푌 on 푄. If 푓 is a function on 푄, its complete lift is a function 푓 푐 on 푇푄 dened by
푓 푐 = 휄(d푓 ).
Locally, 휕푓 푓 푐 = 푞¤푖 . 휕푞푖 If 푋 is a vector eld on 푄, its complete lift is the unique vector 푋푐 on 푇푄 such that
푋푐 (푓 푐) = (푋 (푓 ))푐 for every function 푓 on 푄. Locally, if 푋 is given by 휕 푋 = 푋 푖 , 휕푞푖 then its complete lift is 휕 휕푋푖 휕 푋푐 = 푋 푖 +푞 ¤푗 . 휕푞푖 휕푞 푗 휕푞¤푖 Two relevant relations that hold are
[푋 푣, 푌 푐] = [푋, 푌]푣, and [푋푐, 푌 푐] = [푋, 푌]푐 for every pair of vector elds 푋 and 푌 on 푄. Let 푀 be a 2푚-dimensional manifold. An almost tangent structure 푆 on 푀 is a tensor eld of type (1, 1) such that
i) 푆 has constant rank 푛,
ii) 푆2 = 0.
Then 푀 is called an almost tangent manifold. For each 푥 ∈ 푀,
푆푥 : 푇푥 푀 → 푇푥 푀 is a linear endomorphism. One can dene a tensor eld 푆 of type (1, 1) on 푇푄 such that
푣 푆푦 (푋˜ ) = (휏푄∗푋˜ ) 2.4. THE GEOMETRY OF TANGENT BUNDLES 21
for each 푦 ∈ 푇푄 and each 푋˜ ∈ 푇푦 (푇푄). Locally 푆 is given by 휕 휕 휕 푆 = , 푆 = 0, 휕푞푖 휕푞¤푖 휕푞¤푖 or equivalently, 휕 푆 = ⊗ d푞푖 . 휕푞¤푖 Clearly, 푆 has constant rank 푛 and 푆2 = 0, so it is an almost tangent structure on 푇푄 called the canonical almost tangent structure. It is also known as the vertical endomorphism. Observe that a vector eld 푋˜ on 푇푄 is a vertical vector eld if and only if 푆 (푋˜ ) = 0. In particular, 푆 (푋 푣 ) = 0 for every 푋 on 푄. Moreover, 푆 (푋푐) = 푋 푣 . The adjoint operator 푆∗ of the vertical endomorphism 푆 on 푇푄 is dened by
푆∗(푓 ) = 푓 , ∗ (푆 휔)(푋1, . . . , 푋푝) = 휔 (푆 (푋1), . . . 푆 (푋푝)), for every function 푓 , every 푝-form 휔 and every vector elds 푋1, . . . , 푋푝 on 푇푄. Locally,
푆∗(d푞푖) = 0, and 푆∗(d푞¤푖) = d푞푖, with (푞푖, 푞¤푖) the induced coordinates on 푇푄. It satises
∗ ∗ 휄푋 푆 = 푆 ◦ 휄푆푋 for any vector eld 푋 on 푇푄. Let 푞 ∈ 푄 and (푞, 푞¤) ∈ 푇푄. Consider the vertical lift of 푞¤ ∈ 푇푥푄, namely 휕 휕 푞¤ = 푞¤푖 ↦→푞 ¤푣 = 푞¤푖 . 휕푞푖 휕푞¤푖
This denes a vector eld on 푇푄:
Δ : 푇푄 → 푇푇푄 (푞, 푞¤) ↦→ ((푞, 푞¤), (푞, 푞¤)푣 ), called the Liouville vector eld, locally given by 휕 Δ = 푞¤푖 . 휕푞¤푖 22 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
A second order di erential equation (in what follows abbreviated as SODE; also known as semispray) on 푄 is a vector eld on 푇푄 that is a section of both 휏푇푄 : 푇푇푄 → 푇푄 and 푇휏푄 : 푇푇푄 → 푇푄. Locally, if 휕 휕 휉 = 휉˜푖 + 휉푖 , 휕푞푖 휕푞¤푖 then 휕 푇휏 (휉) = 휉˜푖 , 푄 휕푞푖 since 휏푄 (푞, 푞¤) = 푞, that is,
푖 푖 푖 푖 푖 푖 푖 푖 푇휏푄 (휉 (푞 , 푞¤ )) = 푇휏푄 푞 , 푞¤ , 휉˜ , 휉 = 푞 , 휉˜ , but 푖 푖 푖 푖 휏푇푄 (휉 (푞 , 푞¤ )) = (푞 , 푞¤ ), so a SODE 휉 is locally given by 휕 휕 휉 = 푞¤푖 + 휉푖 . 휕푞푖 휕푞¤푖
From the local expressions, it is clear that a vector eld 휉 on 푇푄 is a SODE if and only if
푆 (휉) = Δ, (2.12) with 푆 the vertical endomorphism on 푇푄. 훾 퐼 푄 푄 훾 푡 Let : ⊂ R → be a curve on , and let ¤( 0) be the tangent vector of the curve at 훾 (푡0). Then the canonical lift of 훾 to 푇푄 is the curve
훾˜ : 퐼 → 푇푄 푡 ↦→ (훾 (푡),훾¤(푡)) .
Clearly, 휏푄 ◦ 훾˜ = 훾. To avoid ambiguity in the following paragraph, let (푞푖, 푣푖) be the bundle coordinates on 푇푄 (which are denoted as (푞푖, 푞¤푖) along the rest of the text). Consider a SODE 휉 on 푇푄 locally given by 휕 휕 휉 = 푣푖 + 휉푖 . 휕푞푖 휕푣푖
A curve 휎 : 퐼 → 푇푄 on 푇푄, with 휎(푡) = 푞푖 (푡), 푣푖 (푡), is an integral curve of 휉 if and only if 휎¤ (푡) = 휉 (휎 (푡)), that is, d푞푖 = (푣푖 ◦ 휎)(푡) = 푣푖 (푡), d푡 d푣푖 = (휉푖 ◦ 휎)(푡) = 휉푖 푞 푗 (푡), 푣 푗 (푡) , d푡 2.4. THE GEOMETRY OF TANGENT BUNDLES 23 so 휎 (푡) = 푞푖 (푡), d푞푖 (푡)/d푡, and then
d2푞푖 d푞푖 = 휉푖 푞푖, , 1 ≤ 푖 ≤ 푚 = dim 푀. (2.13) d푡2 d푡 A solution of a SODE 휉 is a curve 훾 : 퐼 → 푄 on 푄 such that its canonical lift to 푇푄 is an integral curve of 휉. As a matter of fact, if 훾 (푡) = (푞푖 (푡)), then 휎 (푡) = 훾˜(푡) = (푞푖 (푡), 푞¤푖 (푡)), so 훾 is given by the system of ordinary di erential equations (2.13). This motivates calling these vector elds second order di erential equations. A 1-form 훼 on 푇푄 is called semibasic if
훼 (푋˜ ) = 0 for any vertical vector eld 푋˜ on 푇푄. In particular, if a 1-form 훼 on 푇푄 can be written as
훼 = 푆∗(d푓 ) for some function 푓 on 푇푄, then is semibasic. Given a semibasic 1-form 훼 on 푇푄, one can dene the following morphism of bre bundles
∗ 퐷훼 : 푇푄 → 푇 푄, 퐷 푣 ,푤 훼 푣 푢 , 훼 ( 푞) 푞 = ( 푞)( 푤푞 ) 푣 ,푤 푇 푄, 푢 푇 푇푄 푇휏 푢 푤 for every 푞 푞 ∈ 푞 푤푞 ∈ 푤푞 ( ), with 푄 ( 푤푞 ) = 푞. In local coordinates, if
푖 훼 = 훼푖 (푞, 푞¤)d푞 , then 푖 푖 푖 푖 푖 퐷훼 (푞 , 푞¤ ) = 푞 , 훼푖 (푞 , 푞¤ ) . Conversely, given a morphism of bre bundles
퐷 : 푇푄 푇 ∗푄 , 휏푞 휋푄 푄 a semibasic 1-form 훼 on 푇푄 can be dened by 훼 푣 푢 퐷 푣 ,푇휏 푢 , 퐷 ( 푞)( 푣푞 ) = ( 푞) 푄 ( 푣푞 ) 푣 푇 푄, 푢 푇 푇푄 퐷 where 푞 ∈ 푞 푣푞 ∈ 푣푞 ( ). If locally is given by
푖 푖 푖 퐷(푞 , 푞¤ ) = (푞 , 퐷푖 (푞, 푞¤)), then 푖 훼퐷 = 퐷푖 (푞, 푞¤)d푞 . So there exists a one-to-one correspondence between semibasic 1-forms and bred morphisms from 푇푄 to 푇 ∗푄. If 휃 is the canonical 1-form on 푇 ∗푄 and 퐷 : 푇푄 → 푇 ∗푄 a morphism of bre bundles, then ∗ 훼퐷 = 퐷 휃. 24 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
2.5 Lagrangian mechanics
This section covers a geometric formulation of Lagrangian mechanics developed by J. Klein [Kle62; Kle63a; Kle63b]. See also references [AM08; Cra83; God69; LR89; MR07]. Consider a mechanical system with conguration space 푄, with 푄 an 푛-dimensional manifold. Then its space of velocities is the tangent bundle 푇푄, and its Lagrangian function is a di erentiable function 퐿 : 푇푄 → R on 푇푄. The Poincaré-Cartan 1-form is given by ∗ 훼퐿 = 푆 (d퐿), where 푆∗ is the adjoint operator of the vertical endomorphism. The Poincaré-Cartan 2-form is 휔퐿 = −d훼퐿. Locally, if 푇푄 has induced coordinates (푞푖, 푞¤푖), then 휕퐿 휕퐿 d퐿 = d푞푖 + d푞¤푖, 휕푞푖 휕푞¤푖 so 휕퐿 훼 = d푞푖, 퐿 휕푞¤푖 and hence 휕2퐿 휕2퐿 휔 = d푞푖 ∧ d푞 푗 + d푞푖 ∧ d푞¤푗 . (2.14) 퐿 휕푞 푗 휕푞¤푖 휕푞¤푖 휕푞¤푗 One can write 푖 훼퐿 = 푝푖d푞 , 푖 where 푝푖 = 휕퐿/휕푞¤ is the canonical momenta. Notice that the Poincaré-Cartan 1-form is formally identical to the canonical 1-form 휃 on 푇 ∗푄 given by equation (2.7). In fact, as it will be seen latter, 훼퐿 is induced by 휃 via the Legendre transform. 푛 휔푛 The 2 -form 퐿 is locally given by 휕2퐿 휕2퐿 휕2퐿 휔푛 = d푞푖1 ∧ d푞¤푗1 ∧ d푞푖2 ∧ d푞¤푗2 ∧ · · · ∧ d푞푖푛 ∧ d푞¤푗푛 퐿 휕푞¤푖1 휕푞¤푗1 휕푞¤푖2 휕푞¤푗2 휕푞¤푖푛 휕푞¤푗푛 휕2퐿 = ± det d푞1 ∧ · · · ∧ d푞푛 ∧ d푞¤1 ∧ · · · ∧ d푞¤푛. 휕푞¤푖 휕푞¤푗
휔 휔푛 If 퐿 is symplectic, obviously 퐿 is a volume form. On the other hand, if the Hessian matrix 휕2퐿 (푊 ) = 푖 푗 휕푞¤푖 휕푞¤푗 is non-singular, then 휔퐿 is non-degenerate. Thus 휔퐿 is a symplectic form if and only if the 푖 푖 Hessian matrix (푊푖 푗 ) is invertible for any coordinate system (푞 , 푞¤ ). If 휔퐿 is a symplectic form on 푇푄, the Lagrangian function 퐿 on 푇푄 is said to be regular. Otherwise 퐿 is said to be singular, irregular or degenerate. The energy associated to the Lagrangian 퐿 on 푇푄 is the function 퐸퐿 : 푇푄 → R given by 퐸퐿 = Δ(퐿) − 퐿, 2.5. LAGRANGIAN MECHANICS 25 with Δ the Liouville vector eld. Suppose that 퐿 is regular, then
휄 휔 퐸 휉퐿 퐿 = d 퐿 (2.15) is the intrinsic expression of the Euler-Lagrange equations. In other words, the trajectories of the Lagrangian system are given by the integral curves of 휉퐿. The vector eld 휉퐿 on 푇푄 is called the Euler-Lagrange vector eld. Since 퐿 is regular, 휔퐿 is symplectic and equation (2.15) admits a unique solution 휉퐿. Moreover, this vector eld 휉퐿 on 푇푄 is a SODE [LR89]. Therefore it is locally given by 휕 휕 휉 = 푞¤푖 + 휉푖 , 퐿 휕푞푖 퐿 휕푞¤푖 which combined with equation (2.14) yields
휕2퐿 휕2퐿 휕2퐿 휕2퐿 휄 휔 = 푞¤푖 − 푞¤푖 − 휉푖 d푞 푗 + 푞¤푖d푞¤푗 . 휉퐿 퐿 휕푞 푗 휕푞¤푖 휕푞푖 휕푞¤푗 휕푞¤푖 휕푞¤푗 휕푞¤푖 휕푞¤푗
On the other hand, 휕퐿 퐸 = 푞¤푖 − 퐿, 퐿 휕푞푖 so 휕2퐿 휕퐿 휕2퐿 d퐸 = 푞¤푖 − d푞 푗 + 푞¤푖d푞¤푗 . 퐿 휕푞 푗푞¤푖 휕푞 푗 휕푞¤푖푞¤푗 Then equation (2.15) holds if and only if
휕2퐿 휕2퐿 휕퐿 푞¤푖 + 휉푖 = , 1 ≤ 푗 ≤ 푛 = dim푄. 휕푞푖 휕푞¤푗 휕푞¤푖 휕푞¤푗 휕푞 푗
푖 Let 휎 : 퐼 ⊂ R → 푄 be a solution of 휉퐿, with 휎(푡) = (푞 (푡)) . Then 휕2퐿 휕2퐿 휕퐿 푞¤푖 + 푞¥푖 = , 1 ≤ 푗 ≤ 푚, 휕푞푖 휕푞¤푗 휕푞¤푖 휕푞¤푗 휕푞 푗 that is, d 휕퐿 휕퐿 − = 0, 1 ≤ 푖 ≤ 푛, d푡 휕푞¤푖 휕푞푖 which are the well-known Euler-Lagrange equations. Observe that 휉 퐸 휄 퐸 휄 휄 휔 , 퐿 ( 퐿) = 휉퐿 d 퐿 = 휉퐿 휉퐿 퐿 = 0 in other words, the energy is conserved along the evolution of the Lagrangian system. As it was discussed in Section 2.3, non-conservative systems can be described by non- autonomous Lagrangians, external forces or non-symplectic geometries.
Example 3. Consider a Lagrangian system on 푇푄 with Lagrangian function of the usual form 1 퐿 = 푔 푞¤푖푞¤푗 − 푉 (푞), 2 푖 푗 26 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
1 푛 where 푞 = (푞 , . . . , 푞 ) and (푔푖 푗 ) a positive-denite symmetric matrix. Then
1 Δ(퐿) = Δ 푔 푞¤푖푞¤푗 = 푔 푞¤푖푞¤푗, 2 푖 푗 푖 푗 so the energy of the system is
1 퐸 = 푔 푞¤푖푞¤푗 + 푉 (푞). 퐿 2 푖 푗 The Poincaré-Cartan 1-form is 푖 푗 훼퐿 = 푔푖 푗푞¤ d푞 , and the Poincaré-Cartan 2-form is
푖 푗 휔퐿 = 푔푖 푗 d푞 ∧ d푞¤ .
♦
Consider a mechanical system with conguration space 푄, space of velocities 푇푄 and phase space 푇 ∗푄. The Legendre transform is a mapping Leg : 푇푄 → 푇 ∗푄 such that the diagram
푇푄 Leg 푇 ∗푄
휏푞 휋푄 푄
commutes. Here 휏푞 and 휋푄 are the canonical projections on 푄. Locally,
푖 푖 푖 Leg : (푞 , 푞¤ ) ↦→ (푞 , 푝푖),
푖 with 푝푖 = 휕퐿/휕푞¤ . Moreover, ∗ Leg 휃 = 훼퐿, ∗ where 휃 and 훼퐿 is the canonical 1-form on 푇 푄 and the Poincaré-Cartan 1-form on 푇푄, respectively. Thus ∗ Leg 휔 = 휔퐿.
Clearly, 휔퐿 is symplectic if and only if Leg is a symplectic map. Then the following result holds.
Theorem 3. The following assertions are equivalent:
i) 휔퐿 is a symplectic form on 푇푄,
ii) 퐿 is a regular Lagrangian,
iii) Leg : 푇푄 → 푇 ∗푄 is a local di eomorphism. 2.5. LAGRANGIAN MECHANICS 27
In general, 퐿 being regular does not imply Leg to be a global di eomorphism. If Leg is globally a di eomorphism, then 퐿 is said to be hyperregular. In what follows, let us assume Leg to be hyperregular. Consider the vector eld 휉¯ on 푇 ∗푄 given by 휉¯ 휉 = Leg∗ 퐿 with 휉퐿 the Euler-Lagrange vector eld on 푇푄. Then
−1∗ −1∗ −1∗ 휄 ¯휔 = 휄 ¯ Leg 휔 = 휄 −1 Leg 휔 = Leg 휄휉 휔퐿 휉 휉 (Leg )∗휉 퐿 −1∗ −1 = Leg (d퐸퐿) = d 퐸퐿 ◦ Leg .
Therefore 휎 is an integral curve of 휉퐿 if and only if 훾 = Leg ◦휎 is an integral curve of 휉¯. In addition, 휋푄 ◦ 훾 = 휏푄 ◦ 휎. The vector eld 휉¯ is the Hamiltonian vector eld for the Hamiltonian function 퐻 = −1 ∗ 퐸퐿 ◦ Leg on 푇 푄. Example 4. Consider the Lagrangian system from Example3. The corresponding Leg- endre transform is given by
푖 푖 푖 푗 Leg : 푞 , 푞¤ ↦→ 푞 ,푔푖 푗푞¤ , so its inverse is −1 푖 푖 푖 푗 Leg 푞 , 푝푖 ↦→ 푞 ,푔 푝 푗 , 푖 푗 with (푔 ) the inverse matrix of (푔푖 푗 ). Then the Hamiltonian function of the system is given by 1 1 퐻 = 푔 푔푖푘푔푗푙푝 푝 + 푉 (푞) = 푔푘푙푝 푝 + 푉 (푞). 2 푖 푗 푘 푙 2 푘 푙 ♦
Symmetries and constants of the motion In this subsection the di erent types of symmetries in the Lagrangian formalism are discussed, as well as their corresponding constants of the motion. The main references are [Arn78; Cra83; LR89]. See also references [CLM89; Kos11; LM94; LM95; Lun90; Nee99; Noe71; Pri83; Pri85; Rom20; Sar83; SC81]. Let 퐿 be a Lagrangian function on 푇푄, with 훼퐿 and 휔퐿 the corresponding Poincaré- Cartan 1-form and 2-form, respectively. Let 휉퐿 be the corresponding Euler-Lagrange vector eld. Consider a function 퐹 on 푇푄. Observe that 휉 퐹 휄 퐹 휄 휄 휔 휄 휄 휔 휄 퐸 푋 퐸 , 퐿 ( ) = 휉퐿 d = 휉퐿 푋퐹 퐿 = − 푋퐹 휉퐿 퐿 = − 푋퐹 d 퐿 = − 퐹 ( 퐿) with 푋퐹 the Hamiltonian vector eld on 푇푄 associated to 퐹, given by 휄 휔 퐹 . 푋퐹 퐿 = d Recalling the discussion about constants of the motion in Section 2.3, one can see that the following statements are equivalent: 28 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
i) 퐹 is a constant of the motion,
ii) 휉퐿 (퐹) = 0,
iii) 푋퐹 (퐸퐿) = 0.
Consider a vector eld 푋 on 푄. Then
휄 휔 푋푐 휔 휉 , 푋푐 훼 휉 , 푋푐 휉 훼 푋푐 푋푐 훼 휉 훼 휉 , 푋푐 . 휉퐿 퐿 ( ) = 퐿 ( 퐿 ) = −d 퐿 ( 퐿 ) = − 퐿 ( 퐿 ( )) + ( 퐿 ( 퐿)) + 퐿 ([ 퐿 ])
Now, 훼 휉 푆∗ 퐿 휉 휄 푆∗ 퐿 휄 퐿 휄 퐿 퐿 , 퐿 ( 퐿) = ( d )( 퐿) = 휉퐿 ( d ) = 푆휉퐿 d = Δd = Δ( ) using equation (2.12) and the fact 휉퐿 is a SODE. Similarly,
푐 푣 훼퐿 (푋 ) = 휄푆푋푐 d퐿 = 휄푋 푣 d퐿 = 푋 (퐿).
푐 In addition, [휉퐿, 푋 ] is a vertical vector eld (which can be easily checked, for instance, 푐 using the local expressions), and hence 푆 [휉퐿, 푋 ] vanishes. Therefore 휄 휔 푋푐 휉 푋 푣 퐿 푋푐 퐿 . 휉퐿 퐿 ( ) = − 퐿 ( ( )) + (Δ( ))
On the other hand,
휄 휔 푋푐 퐸 푋푐 푋푐 퐸 푋푐 퐿 퐿 , 휉퐿 퐿 ( ) = d 퐿 ( ) = ( 퐿) = (Δ( ) − ) and thus 푣 푐 휉퐿 (푋 (퐿)) = 푋 (퐿). In particular, the left-hand side vanishes if and only if the right-hand side does. In other words, one has the following result.
Theorem 4 (Noether’s theorem). Let 푋 be a vector eld on 푄. Then 푋 푣 (퐿) is a constant of the motion if and only if 푋푐 (퐿) vanishes.
A vector eld 푋 on 푄 is called a symmetry of the Lagrangian 퐿 if 푋푐 (퐿) = 0. In 푐 fact, if 푋 generates a local one-parameter group Φ푡 of transformations on 푀, then 푋 generates 푇 Φ푡 . Therefore 푋 is a symmetry of 퐿 if and only if
∗ (푇 Φ푡 ) 퐿 = 퐿.
Example 5 (Conservation of momentum). Consider a Lagrangian system on 푇 R3 with Lagrangian function 퐿. Let (푥,푦, 푧) be the Cartesian coordinates on R3 and (푥,푦, 푧, 푥,¤ 푦,¤ 푧¤) the induced coordinates on 푇 R3. Translations along the 푧-direction are generated by the vector eld 휕 푋 = 휕푧 on R3. Then the vertical and complete lifts of 푋 are 휕 푋 푣 = , 휕푧¤ 2.5. LAGRANGIAN MECHANICS 29 and 휕 푋푐 = , 휕푧 respectively. As it is well-known, the 푧-th component of momentum is conserved if and only if 퐿 is invariant under translations along the 푧-direction. Indeed, 휕퐿 푋푐 (퐿) = 휕푧 vanishes if and only if 휕퐿 푋 푣 (퐿) = = 푝 휕푧¤ 푧 is a constant of the motion. ♦
Example 6 (Conservation of angular momentum). Consider a Lagrangian system on 푇 R3 with Lagrangian function 퐿. Let (푥,푦, 푧) be the Cartesian coordinates on R3 and (푥,푦, 푧, 푥,¤ 푦,¤ 푧¤) the induced coordinates on 푇 R3. Rotations around the 푧-direction are generated by the vector eld 휕 휕 푋 = 푥 − 푦 휕푦 휕푥 on 3. Then R 휕 휕 푋 푣 = 푥 − 푦 , 휕푦¤ 휕푥¤ and 휕 휕 휕 휕 푋푐 = 푥 − 푦 +푥 ¤ −푦 ¤ , 휕푦 휕푥 휕푦¤ 휕푥¤ so 휕퐿 휕퐿 푋 푣 (퐿) = 푥 − 푦 = 푥푝 − 푦푝 = 퐿 휕푦¤ 휕푥¤ 푦 푥 푧 is conserved if and only if 푋푐 (퐿) = 0. ♦
Example 7 (Lagrangian for a relativistic particle and Lorentz invariance). Consider a free particle in Minkowski spacetime, that is, 푄 = R4 endowed with the metric
휇 휈 휂 = 휂휇휈 d푥 ⊗ d푥 , 휂휇휈 = diag (−1, 1, 1, 1) , in Cartesian coordinates (푥 휇) = (푡, 푥,푦, 푧). The Lagrangian is then
√︁ 휇 휈 퐿 = 휂휇휈푥¤ 푥¤ , so 1 훼 휂 푥¤휇 푥휈, 퐿 = √︃ 휇휈 d 훼 훽 휂훼훽푥¤ 푥¤ and 1 휔 휂 푥 휇 ∧ 푥¤휈 . 퐿 = √︃ 휇휈 d d 훼 훽 휂훼훽푥¤ 푥¤ 30 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
Consider the following vector elds on 푄: 휕 휕 휕 휕 휕 휕 푋 = 푡 + 푥 , 푋 = 푡 + 푦 , 푋 = 푡 + 푧 , 1 휕푥 휕푡 2 휕푦 휕푡 3 휕푧 휕푡 which generate the boosts along the 푥-, 푦- and 푧-directions, respectively. These vector elds are symmetries of the Lagrangian, and their associated conserved quantities are 1 1 1 푋 푣 (퐿) = (푡푥¤−푥푡¤), 푋 푣 (퐿) = (푡푦¤−푦푡¤), 푋 푣 (퐿) = (푡푧¤−푧푡¤). 1 √︃ 2 √︃ 3 √︃ 훼 훽 훼 훽 훼 훽 휂훼훽푥¤ 푥¤ 휂훼훽푥¤ 푥¤ 휂훼훽푥¤ 푥¤ ♦
There are other types of innitesimal symmetries on 푄, that is, vector elds on 푄 which generate symmetries of the dynamics. A Lie symmetry is a vector eld 푋 on 푄 such that 푐 [푋 , 휉퐿] = 0.
Since 휔퐿 is non-degenerate, 푋 is a Lie symmetry if and only if
휄 푐 휔 푐 휄 휔 휄 푐 휔 푐 휄 휔 휄 푐 훼 0 = [푋 ,휉퐿] 퐿 = L푋 ( 휉퐿 퐿) − 휉퐿 (L푋 퐿) = L푋 ( 휉퐿 퐿) + 휉퐿 (L푋 d 퐿) 푐 푐 퐸 휄 푐 훼 푐 퐸 휄 푐 훼 푋 퐸 휄 푐 훼 . = L푋 (d 퐿) + 휉퐿 d(L푋 퐿) = d(L푋 퐿) + 휉퐿 d(L푋 퐿) = d( ( 퐿)) + 휉퐿 d(L푋 퐿)
In particular, 푋 is a Lie symmetry if
d(L푋푐 훼퐿) = 0, and 푐 d(푋 (퐸퐿)) = 0. This can be restated as follows.
Proposition 5. Let 푋 be a vector eld on 푄 such that L푋푐 훼퐿 is a closed 1-form. Then 푋 is 푐 a Lie symmetry if and only if 푋 (퐸퐿) is a constant function.
In particular, if L푋푐 훼퐿 is exact (that is, there exists a function 푓 on 푇푄 such that 푐 L푋푐 훼퐿 = d푓 ) and 푋 (퐸퐿) = 0, then 푋 is said to be a Noether symmetry. Obviously, every Noether symmetry is also a Lie symmetry. Observe that
∗ L푋푐 훼퐿 = 휄푋푐 (d훼퐿) + d(휄푋푐 훼퐿) = 휄푋푐 (d훼퐿) + d(휄푋푐 푆 d퐿) 푣 = 휄푋푐 (d훼퐿) + d(휄푆푋푐 d퐿) = 휄푋푐 (d훼퐿) + d(푋 (퐿)), so 푣 휄푋푐 (d훼퐿) = d푓 − d(푋 (퐿)), and thus 푣 푣 휄 휄 푐 훼 휄 푓 푋 퐿 휉 푓 푋 퐿 , 휉퐿 푋 (d 퐿) = 휉퐿 (d( − )) = 퐿 ( − ( )) but 푐 휄 휄 푐 훼 휄 푐 휄 훼 휄 푐 휄 휔 휄 푐 퐸 푋 퐸 . 휉퐿 푋 (d 퐿) = − 푋 휉퐿 (d 퐿) = 푋 휉퐿 퐿 = 푋 (d 퐿) = ( 퐿) This leads to the following result. 2.5. LAGRANGIAN MECHANICS 31
Proposition 6. Let 푋 be a vector eld on 푄 such that
L푋푐 훼퐿 = d푓 , then 푋 is a Noether symmetry if and only if 푓 − 푋 푣 (퐿) is a constant of the motion.
Observe that Noether symmetries are a generalization of symmetries of the La- grangian. Indeed, if 푋 is a Noether symmetry and 푓 is a constant function, clearly 푋 푣 (퐿) is a constant of the motion . In addition, L푋푐 훼퐿 vanishes so
푐 푐 0 = (L푋푐 훼퐿)(휉퐿) = 푋 (훼퐿 (휉퐿)) − 훼퐿 ([푋 , 휉퐿]) 푐 푐 푐 = 푋 (Δ(퐿)) − 푆 ([푋 , 휉퐿])(퐿) = 푋 (Δ(퐿)), and thus, 푐 푐 푐 푐 0 = 푋 (퐸퐿) = 푋 (Δ(퐿)) − 푋 (퐿) = −푋 (퐿), so a symmetry of 퐿 is also a Noether symmetry (and a Lie symmetry). The innitesimal symmetries just discussed are vector elds on 푄, sometimes called point-like symmetries [LM94; LM95]. One can also consider innitesimal symmetries on 푇푄, in other words, symmetries that depend on the velocities as well as the positions. A vector eld 푋˜ on 푇푄 is called a dynamical symmetry if
[푋,˜ 휉퐿] = 0.
A Cartan symmetry is a vector eld 푋˜ on 푇푄 such that 훼 푓 , L푋˜ 퐿 = d and 푋˜ (퐸퐿) = 0. Recalling the deductions above for point-like symmetries, the reader can analogously derive the following results. 푋˜ 푇푄 훼 푋˜ Proposition 7. Let be a vector eld on such that L푋˜ 퐿 is a closed 1-form. Then is a dynamical symmetry if and only if 푋˜ (퐸퐿) is a constant function. Proposition 8. Let 푋˜ be a vector eld on 푇푄 such that 훼 푓 , L푋˜ 퐿 = d then 푋˜ is a Cartan symmetry if and only if 푓 − (푆푋˜ )(퐿) is a constant of the motion.
Notice that there exists no analogous for Lagrangian symmetries on 푇푄, since [휉퐿, 푋˜ ] is not a vertical vector eld for a general vector eld 푋˜ on 푇푄. Furthermore, it is clear that
i) 푋 is a Lie symmetry if and only if 푋푐 is a dynamical symmetry,
ii) 푋 is a Noether symmetry if and only if 푋푐 is a Cartan symmetry,
iii) every Cartan symmetry is also a dynamical symmetry, 32 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS with 푋 a vector eld on 푄.
Example 8 (2-dimensional harmonic oscillator). In this case the conguration space is 푄 = R2, so 푇푄 ' R2 × R2. The Lagrangian function of the harmonic oscillator is 1 퐿 = (푞 ¤1)2 + (푞 ¤2)2 − (Ω 푞1)2 − (Ω 푞2)2 , 2 1 2 in units where the mass is equal to 1 for the sake of simplicity. Then the Poincaré-Cartan 1-form is 1 1 2 2 훼퐿 = 푞¤ d푞 +푞 ¤ d푞 , so the corresponding 2-form is
1 1 2 2 휔퐿 = d푞 ∧ d푞¤ + d푞 ∧ d푞¤ .
Moreover, the energy of the system is 1 퐸 = (푞 ¤1)2 + (푞 ¤2)2 + (Ω 푞1)2 + (Ω 푞2)2 . 퐿 2 1 2 Consider the following vector elds on 푇푄: 휕 휕 Ω1 1 1 푋1 = 푞 +푞 ¤ , 2 1 2 1 2 휕푞1 휕푞1 (Ω1) (푞 ) + (푞¤ ) ¤ 휕 휕 Ω2 2 2 푋2 = 푞 +푞 ¤ . 2 2 2 2 2 휕푞2 휕푞2 (Ω2) (푞 ) + (푞¤ ) ¤ One can check that 휔 , L푋푎 퐿 = 0 and 푋푎 (퐸퐿) = Ω푎, where 푎 = 1, 2, so 푋1 and 푋2 are dynamical symmetries. See reference [Rom20, Example 1] for the Hamiltonian counterpart of this example. ♦
Example 9 (Conservation of energy). Consider a Lagrangian system on 푇푄 with La- grangian function 퐿 and Euler-Lagrange vector eld 휉퐿. Observe that 훼 휄 훼 휄 훼 휄 휔 휄 푆∗ 퐿 퐸 휄 퐿 L휉퐿 퐿 = 휉퐿 d 퐿 + d 휉퐿 퐿 = − 휉퐿 퐿 + d 휉퐿 (d ) = −d 퐿 + d 푆휉퐿 (d ) = −d퐸퐿 + d ((푆휉퐿)(퐿)) = −d(퐸퐿 − Δ(퐿)) = d퐿, and 휉 퐸 휄 퐸 휄 휄 휔 , 퐿 ( 퐿) = 휉퐿 d 퐿 = 휉퐿 휉퐿 퐿 = 0 so 휉퐿 is a Cartan symmetry and
퐿 − (푆휉퐿)(퐿) = 퐿 − Δ(퐿) = 퐸퐿 is the associated conserved quantity. ♦ 2.5. LAGRANGIAN MECHANICS 33
Figure 2.1: Types of symmetries on 푄 and on 푇푄. The complete lifts of Lie symmetries and Noether symmetries correspond to dynamical symmetries and Cartan symmetries, respectively. Noether symmetries are the subset of Lie symmetries such that L푋푐 훼퐿 = d푓 and 푓 − 푋 푣 (퐿) is a constant of the motion. Cartan symmetries are the analogous subset of dynamical symmetries. The symmetries of the Lagrangian system are the subset of Noether symmetries for which L푋푐 훼퐿 vanishes, so 푓 is a constant function that can be taken as 0 without loss of generality.
The types of symmetries for Lagrangian systems on 푇푄 with Lagrangian function 퐿 are summarized in Figure 2.1. Finally, symmetries and constants of the motion in the Lagrangian and Hamiltonian formalisms can be related. Recall that 퐿 is assumed to be hyperregular. Let 퐻 ◦ Leg = 퐸퐿 ∗ be a Hamiltonian on 푇 푄 and 푋퐻 its corresponding Hamiltonian vector eld. Let 푋˜ be a vector eld on 푇푄 and 푋ˆ the Leg-related vector eld on 푇 ∗푄, that is,
푋 푋 . ˆ = Leg∗ ˜
It can be easily seen that the following statements hold:
i) 푋ˆ commutes with 푋퐻 if and only if 푋˜ is a dynamical symmetry of 퐿.
휃 푓 훼 푔 푔 푓 ii) L푋ˆ = d if and only if L푋˜ 퐿 = d , where = ◦ Leg. 휃 푓 iii) Suppose that L푋ˆ = d . Then the following assertions are equivalent:
a) 푋ˆ (퐻) = 0, b) 푓 − 휃 (푋ˆ) is a conserved quantity,
c) 푋˜ (퐸퐿) = 0,
d) 푓 ◦ Leg −훼퐿 (푋˜ ) is a conserved quantity. 34 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
2.6 Momentum map and symplectic reduction
In the previous section, the relation between innitesimal symmetries (and hence, of one-parameter groups of symmetries) with constants of the motion has been studied. These conserved quantities are frequently used to simplify the equations of motion. As it is well-known, many physical systems have a group of symmetries (which is not in general a one-parameter group). For instance, special-relativistic systems are invariant under the Poincaré group. One could then think of simplifying the equations of the motion by acting with the whole symmetry group. In fact, if a symmetry group acts over a system, then its number of independent degrees of freedom is reduced. In other words, the dimensions of 푄, and hence of 푇푄 and 푇 ∗푄, are reduced. A systematic procedure to explore this fact is known as reduction, which is due to Marsden and Weinstein. See references [AM08; MMR90; MW74; OR04]. See also the seminal works by Lie, Kostant, Souriau and Smale [Kos09; Mer10; Sma70; Sou67; Sou08]. Consider a Lie group 퐺 with Lie algebra 픤, and let 픤∗ be the dual of 픤. Let (푀, 휔) be a connected symplectic manifold. The action Φ : 퐺 × 푀 → 푀 of 퐺 on 푀 is called symplectic if the map Φ푔 : 푀 → 푀 푥 ↦→ Φ(푔, 푥) = 푔푥 is symplectic for each 푔 ∈ 퐺. In what follows, every group action is assumed to be free and proper. For each 휉 ∈ 픤, there is a vector eld 휉푀 on 푀 which is the innitesimal generator of the action corresponding to 휉. Consider a mapping 퐽 : 푀 → 픤∗, and a function 퐽 휉 : 푀 → R such that 퐽 휉 (푥) = h퐽 (푥), 휉i for each 휉 ∈ 픤 and each 푥 ∈ 푀. Then 퐽 is called a momentum map for the action if 퐽 휉 휄 휔 d = 휉푀 휉 ∈ 퐽 휉 ∈ 휉푐 for every 픤. In other words, is a momentum map provided that, for each 픤, 푄 is the Hamiltonian vector eld on 푀 associated to 퐽 휉 . The a ne action of 퐺 on 픤∗ associated with the momentum map 퐽 : 푀 → 픤∗ is given by 퐺 × 픤∗ → 픤∗ (푔, 휇) ↦→ ∗ 휇 + 퐽 (푥) − ∗ (퐽 (푥)) . Ad푔−1 Φ푔 Ad푔−1 This denition is independent of the choice of 푥 ∈ 푀. The isotropy group of 휇 ∈ 픤 under the a ne action is denoted by 퐺휇. Consider a function 퐹 on (푀, 휔) which is invariant under the action of 퐺, that is, 퐹 (푥) = 퐹 Φ푔 (푥) for every 푔 ∈ 퐺 and every 푥 ∈ 푀. Equivalently, 퐹 , L휉푃 = 0 2.6. MOMENTUM MAP AND SYMPLECTIC REDUCTION 35 for each 휉 ∈ 픤, that is, n o 퐹, 퐽 휉 = 0,
휉 where {·, ·} is the Poisson bracket dened by 휔, so 퐽 is a rst integral of 푋퐹 . In practice, for a Hamiltonian or Lagrangian invariant under 퐺, there is a constant of the motion 퐽 휉 for each 휉 ∈ 픤. A momentum map 퐽 is called Ad∗-equivariant if 퐽 (푥) ∗ 퐽 (푥) Φ푔 = Ad푔−1 for each 푔 ∈ 퐺 and each 푥 ∈ 푀. In other words, the following diagram commutes:
푀 Φ푔 푀
퐽 퐽 픤∗ 픤∗ Ad∗ 푔−1 An Ad∗-equivariant momentum map denes an homomorphism between the Lie alge- bra 픤 and the Lie algebra of functions on 푀 under the Poisson bracket. More specically, the following property holds: n o 퐽 휉, 퐽 휂 = 퐽 [휉,휂] for each 휉, 휂 ∈ 픤. If the symplectic form can be written as 휔 = −d휃, and the action leaves 휃 invariant ∗휃 휃 푔 퐺 (that is, Φ푔 = for each ∈ ), there is a natural momentum map given by 퐽 푥 휉 휄 휃 푥 ( )( ) = 휉푀 ( ) for each 푥 ∈ 푀. It can be shown that it is Ad∗-equivariant. In particular, if 휃 is the canonical 1-form on 푇 ∗푄, every group action leaves it invariant by the property (2.8), so the natural momentum map can always be dened in the Hamiltonian formalism. In the Lagrangian description, if the Lagrangian function 퐿 is 퐺-invariant, so is 훼퐿 and the natural momentum map can be used as well. The following theorem is the basis of symplectic reduction. Theorem 9 (Symplectic point reduction). Consider a Lie group 퐺 with symplectic action Φ on the connected manifold (푀, 휔). Then the following statements hold:
−1 i) The quotient space 푀휇 B 퐽 (휇)/퐺휇 is a symplectic manifold, with symplectic form 휔휇 uniquely characterized by the relation 휋∗휔 푖∗휔. 휇 휇 = 휇 −1 −1 −1 Here the maps 푖휇 : 퐽 (휇) ↩→ 푀 and 휋휇 : 퐽 (휇) → 퐽 (휇)/퐺휇 denote the inclusion and the projection, respectively. The pair (푀휇, 휔휇) is called the symplectic point reduced space.
ii) Let 푓 be a 퐺-invariant function on 푀, and 푋푓 the corresponding Hamiltonian vector 푀 퐹 푋 퐹 휇 푀 eld on . The ow 푡 of 푓 induces a ow 푡 on 휇 given by 휋 퐹 푖 퐹 휇 휋 . 휇 ◦ 푡 ◦ 휇 = 푡 ◦ 휇 36 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS
퐹 휇 푀 , 휔 iii) The vector eld generated by the ow 푡 on ( 휇 휇) is Hamiltonian with associated reduced function 푓휇 on 푀휇 dened by
푓휇 ◦ 휋휇 = 푓 ◦ 푖휇 . 푋 푋 휋 The vector elds 푓 and 푓휇 are 휇-related, that is, 휋 푋 푖 푋 . 휇∗ 푓휇 = 휇∗ 푓
iv) Let 푘 be another 퐺-invariant function. Then {푓 , 푘} is also 퐺-invariant and its associ- ated reduced function is given by {푓 , 푘} = 푓휇, 푘휇 , 휇 푀휇 {·, ·} 휔 푀 where 푀휇 denotes the Poisson bracket associated to 휇 on 휇. The proof is involved and can be found in reference [OR04]. The reduced space has dimensions
−1 dim 푀휇 = dim 퐽 (휇) − dim퐺휇 = dim 푀 − dim퐺 − dim퐺휇 .
Consider a Lie group 퐺 with action Φ : 퐺 × 푄 → 푄 on 푄. If Φ푔 : 푄 → 푄 is the di eomorphism given by Φ푔 (푞) = 푔푞 for each 푔 ∈ 퐺 and each 푞 ∈ 푄, it induces an action on 푇푄 given by 푇 Φ푔 : 푇푄 → 푇푄, known as the lifted action, which is symplectic. For each 휉 ∈ 픤, there is a vector eld 휉푄 on 푄 generating the corresponding innitesimal action on 푄. The generator of the lifted 휉푐 푄 action is then 푄 . Consider a Lagrangian system on with hyperregular Lagrangian function 퐿, and let 휔퐿 be the symplectic structure induced by 퐿 on 푇푄. As mentioned above, the natural momentum map on 푇푄 is given by 퐽 휉 훼 휉푐 ( ) = 퐿 푄 for each 휉 ∈ 픤. To sum up, if a Lagrangian 퐿 on 푇푄 is invariant under the action of a symmetry group 퐺 with Lie algebra 픤, then: 휉 ∈ 퐽 휉 훼 (휉푐 ) i) for each 픤, there is a conserved quantity = 퐿 푄 , ii) the symplectic point reduced space is (푇푄)휇, 휔휇 , with 휋∗휔 푖∗휔 휇 휇 = 휇 퐿;
iii) the reduced Lagrangian function 퐿휇 is given by
퐿휇 ◦ 휋휇 = 퐿 ◦ 푖휇,
휉 iv) the reduced Euler-Lagrange vector eld 퐿휇 is given by 휋 휉 푖 휉 . 휇∗ 퐿휇 = 휇∗ 퐿 2.6. MOMENTUM MAP AND SYMPLECTIC REDUCTION 37
It is worth remarking that (푇푄)휇 is denoted this way since it is the reduced space associated to 푇푄 but, in general, it is not a tangent bundle. One can also introduce a lifted action on 푇 ∗푄 and obtain the same result in the Hamiltonian formalism. Remark 2 (Reconstruction of the dynamics). Given the dynamics on 퐽 −1(휇) ⊂ 푇푄, one can obtain the dynamics on the reduced space (푇푄)휇 by the procedure above. From a practical point of view, one poses the inverse question, that is, how to determine the integral curves 푐 푡 휉 푐 푡 휉 푚 퐽 −1 휇 ( ) of 퐿 knowing the integral curves 휇 ( ) of 퐿휇 and an initial condition 0 ∈ ( ). −1 Take a smooth curve 푑(푡) in 퐽 (휇) such that 푑(0) = 푚0 and 휋휇 (푑(푡)) = 푐휇 (푡). If 푐(푡) denotes the integral curve of 휉퐿 with 푐(0) = 푚0, there exists a smooth curve 푔(푡) in 퐺휇 ⊂ 퐺 such that 푐(푡) = Φ푔(푡) (푑(푡)). (2.16) The problem is now to determine 푔(푡) and thus 푐(푡). By denition of integral curve,
휉퐿 (푐(푡)) = 푐¤(푡) 푇 푑 푡 푑¤ 푡 푇 푑 푡 푇퐿 푔 푡 푐 푑 푡 , = Φ푔(푡) ( ( ))( ( )) + Φ푔(푡) ( ( ))( 푔(푡)−1 ( ¤( )))푄 ( ( )) and using the Φ푔-invariance of 휉퐿 one has
휉 푑 푡 푑¤ 푡 푇퐿 푔 푡 푐 푑 푡 . 퐿 ( ( )) = ( ) + ( 푔(푡)−1 ( ¤( )))푄 ( ( ))
In order to solve this equation, one rst has to solve the algebraic problem
휉푐 푑 푡 휉 푑 푡 푑¤ 푡 , 푄 ( ( )) = 퐿 ( ( )) − ( ) for 휉 (푡) ∈ 픤, and then solve 푔¤(푡) = 푇퐿푔(푡)휉 (푡) for 푔(푡). The integral curve sought is given by equation (2.16).
Example 10 (Angular momentum). Consider 푄 = R푛 \{0}, and a Lagrangian 퐿 on 푇푄 that is spherically symmetric, say 퐿(푞, 푞¤) = 퐿(k푞k, k푞 ¤k). Consider the Lie group × 퐺 = SO(푛) = {푂 ∈ R푛 푛 | 푂푡푂 = Id, det(푂) = 1} acting by rotations on 푄. The action can be lifted to 푇푄 via the tangent lift. Explicitly, for 푂 ∈ SO(3) let
푔푂 : 푄 → 푄, (푞) ↦→ (푂 · 푞)
푇푔푂 : 푇푄 → 푇푄, (푞, 푞¤) ↦→ (푂 · 푞, 푂 ·푞 ¤).
The group SO(푛) acts freely and properly. The Lie algebra of the group is given by × 픤 = 픰픬(푛) = {표 ∈ R푛 푛 | 표푡 +표 = 0}. In particular, for 푛 = 3 the algebra 픤 can be identied with the algebra of 3-dimensional vectors with the cross product by taking
0 −휉3 휉2 휉1 © 휉 −휉 ª ↦→ ©휉 ª , 3 0 1® 2® −휉 휉 0 휉 « 2 1 ¬ « 3¬ 38 PART 2. HAMILTONIAN AND LAGRANGIAN MECHANICS so, for each 휉 ∈ 픤, one has
휉푄 (푞) = (휉 × 푞), 휉푐 푞, 푞 휉 푞, 휉 푞 , 푄 ( ¤) = ( × × ¤) 휉푣 푞, 푞 , 휉 푞 . 푄 ( ¤) = (0 × ¤)
∗ One can identify 픤 with 픤 by using the inner product on R3. The moment map is then given by [AM08, Example 4.2.15]
퐽 (푞, 푞¤) = 푞 ×푞. ¤
∗ Identifying 픤 ' R3 one sees that the a ne action of 퐺 is the usual one (by rotations). ∗ 1 Let 휇 ∈ 픤 , 휇 ≠ 0 then the isotropy group 퐺휇 ' 푆 of 휇 under the a ne action, which are the rotations around the axis 휇. −1 Without loss of generality, one can take 휇 = (0, 0, 휇0). Hence, if (푞, 푞¤) ∈ 퐽 (휇), both 1 1 2 푞 and 푞¤ lie on the 푥푦-plane. Moreover, they must satisfy the equation 푞¤ 푝2 −푞 ¤ 푞 = 휇0. Finally, applying the reduction method above to the system, one can nd out ((푇푄)휇, 퐿휇), which is a Lagrangian system over a 2-dimensional manifold [AM08, Example 4.3.4]. ♦
See reference [MMR90] for explicit expressions for 푐(푡) in some particular cases and their derivation. Part 3
Mechanical systems subjected to external forces
A well-known result from elementary calculus and classical mechanics is that a conser- vative force (that is, a force such that the total work it does moving a particle between two given points is independent of the path taken) can be written as the gradient of a potential. This potential can then be incorporated to the Lagrangian or Hamiltonian function of the system. However, this is not always the case. As a matter of fact, when non-conservative forces act on a mechanical system, its equations of motion depend on both the Lagrangian or the Hamiltonian and on the external forces [Gol87]. This is the case of many physical systems of interests [Can82; Can84; Can+02; EGG21]. Furthermore, external forces can arise in a more sophisticated manner, for instance, after a process of reduction of a nonholonomic system with symmetries [Can+02; Can+99; CL99]. In this part of the thesis, the concepts studied in Part2 are generalized for systems with external forces. The main reference is our original work [LLL21b].
3.1 Mechanical systems subjected to external forces
Geometrically, an external force is expressed as a semibasic 1-form on a symplectic manifold [AM08; God69; LR89]. Consider a forced Hamiltonian system on (푇 ∗푄, 휔), with Hamiltonian function 퐻 : 푀 → R and external force 훾. If (푞푖) are local coordinates on 푄, 푖 ∗ and (푞 , 푝푖) the induced coordinates on 푇 푄, the external force can be locally written as
푖 훾 = 훾푖 (푞, 푝)d푞 .
∗ The trajectories of the system are now the integral curves of the vector eld 푋퐻,훾 on 푇 푄 given by 휄 휔 퐻 훾. 푋퐻,훾 = d +
Let 푍훾 be the vector eld dened by 휄 휔 훾. 푍훾 =
Then 푋퐻,훾 can be written as 푋퐻,훾 = 푋퐻 + 푍훾,
39 40 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
with 푋퐻 the Hamiltonian vector eld of 퐻. Locally, 휕 푍훾 = −훾푖 , 휕푝푖 and recall that 휕퐻 휕 휕퐻 휕 푋 = − , 퐻 푖 푖 휕푝푖 휕푞 휕푞 휕푝푖 so 휕퐻 휕 휕퐻 휕 푋 = − + 훾 . 퐻,훾 푖 푖 푖 휕푝푖 휕푞 휕푞 휕푝푖 The equations of motion are then
d푞푖 휕퐻 = , d푡 휕푝푖 d푝 휕퐻 푖 = − + 훾 . d푡 휕푞푖 푖
Suppose that the external force is exact, that is, 훾 = d푓 for some function 푓 on 푄. Then 푍훾 is the Hamiltonian vector eld for 푓 , so one can dene a new Hamiltonian function 0 퐻 = 퐻 + 푓 , such that 푋퐻,훾 = 푋퐻 0 is its Hamiltonian vector eld. In that case, the problem is reduced to the conservative one (that is, without external forces). Therefore, stricto sensu external forces are those which are non-exact. The Lagrangian formalism is analogous. An external force is now given by a semibasic 1-form 훽 on 푇푄. In bundle coordinates,
푖 훽 = 훽푖 (푞, 푞¤)d푞 .
Hereinafter, a Lagrangian system on 푇푄 with Lagrangian function 퐿 and subject to an external force 훽 will be called a forced Lagrangian system (퐿, 훽) on 푇푄. The dynamics of (퐿, 훽) is given by the forced Euler-Lagrange vector eld 휉퐿,훽 via 휄 휔 퐸 훽. 휉퐿,훽 퐿 = d 퐿 + (3.1)
Recall that 휔퐿 and 퐸퐿 are the Poincaré-Cartan 2-form and the energy function associated to 퐿, respectively (see Section 2.5). Here L is assumed to be hyperregular. Let 휉훽 be the vector eld given by 휄 휔 훽, 휉훽 퐿 = so that 휉퐿,훽 = 휉퐿 + 휉훽, with 휉퐿 the Hamiltonian vector eld on (푇푄, 휔퐿) associated to 퐸퐿. Recall that locally 휕2퐿 휔 = d푞푖 ∧ d푞 푗 + 푊 d푞푖 ∧ d푞¤푗, 퐿 휕푞¤푖 휕푞 푗 푖 푗 where the Hessian matrix 휕2퐿 푊 = 푖 푗 휕푞¤푖 휕푞¤푗 3.1. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES 41 is invertible since L is regular. Let
휕 휕 휉 = 퐴푖 + 퐵푖 , 훽 휕푞푖 휕푞¤푖 then 휕2퐿 휕2퐿 휄 휔 = 퐴푖 − d푞 푗 + 퐴푖푊 d푞¤푗 − 퐵 푗푊 d푞푖, 휉훽 퐿 휕푞¤푖 휕푞 푗 휕푞¤푗 휕푞푖 푖 푗 푖 푗
푖 푗 so 퐴 = 0 and 퐵 푊푖 푗 = −훽푖, and thus
휕 휉 = −훽 푊 푖 푗 . 훽 푖 휕푞¤푖
Then 휕 휕 휉 = 푞¤푖 + 휉푖 − 훽 푊 푖 푗 . (3.2) 퐿,훽 휕푞푖 푗 휕푞¤푖
푖 Therefore, if (푞 (푡)) is a solution of the SODE 휉퐿,훽, it satises
푖 푖 푗푖 푞¥ = 휉 − 훽푗푊 .
Recall that 휉푖 is given by 휕2퐿 휕퐿 푞¤푖 + 푊 휉푖 = , (3.3) 휕푞푖 휕푞¤푗 푖 푗 휕푞 푗 so 휕2퐿 휕2퐿 휕퐿 푞¥푖 +푞 ¤푖 − + 훽 푊 푘푖푊 = 0, 휕푞¤푖푞¤푗 휕푞푖푞¤푗 휕푞 푗 푘 푖 푗 which nally yields d 휕퐿 휕퐿 − = −훽 . d푡 휕푞¤푖 휕푞푖 푖
These are the forced Euler-Lagrange equations. They can also be derived from the Lagrange-d’Alembert’s principle [Gol87; Lew+04; MW01]. Notice that
푆 (휉퐿,훽 ) = 푆 (휉훽 ) = Δ, where 푆 and Δ are the vertical endomorphism and the Liouville vector eld, respectively. Hence, 휉퐿,훽 is a SODE. Since 퐿 is assumed to be hyperregular, the Legendre transform Leg : 푇푄 → 푇 ∗푄 is a di eomorphism. Therefore the external force 훽 on 푇푄 is related with the external force 훾 on 푇 ∗푄 by 훽 훾. = Leg∗ 42 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
3.2 Symmetries and constants of the motion in the Lagrangian description
In this section, the results regarding symmetries and constants of the motion studied in section 2.5 are extended for forced Lagrangian systems. There are other approaches that can be found in the previous literature and have some relation with the one followed here. For instance, Cantrijn [Can82] considers Lagrangian systems that depend explicitly on time, and denes a 2-form on R × 푇푄 that depends on the Poincaré-Cartan 2-form of the Lagrangian and the semibasic 1-form representing the external force. Alternatively, van der Schaft [Sch81; Sch83] considers a framework steming from system theory, in which an “observation” manifold appears together with the usual state space, and obtains a Noether’s theorem for Hamiltonian systems in this frame. Other approaches using variational tools can be found in reference [BK87]. However, in the approach followed here [LLL21b] no additional structure or objects are introduced besides the proper external force. Consider a forced Lagrangian system (퐿, 훽) with forced Euler-Lagrange vector eld 휉퐿,훽 . If 푓 is a function on 푇푄, then
d 푓 (휎 (푡)) = 휉 (푓 )(휎 (푡)), d푡 퐿,훽 for any integral curve 휎 (푡) of 휉퐿,훽. Thus 푓 is a constant of the motion (or a conserved quantity) if it is a rst integral of 휉퐿,훽 (that is, if 휉퐿,훽 (푓 ) = 0). Observe that now a constant of the motion will not be, in general, a rst integral of 휉퐿. 푖 푖 Suppose that, for a given coordinate 푞 , 휕퐿/휕푞 = 훽푖. Then
d 휕퐿 = 0, d푡 휕푞¤푖
푖 so 푝푖 = 휕퐿/휕푞 is a constant of the motion. This motivates generalising Noether’s theorem for systems with external forces. Consider a vector eld 푋 on 푄. Then, by equation (3.1), 퐸 훽 푋푐 휄 휔 푋푐 휔 휉 , 푋푐 훼 휉 , 푋푐 (d 퐿 + )( ) =( 휉퐿,훽 퐿)( ) = 퐿 ( 퐿,훽 ) = −d 퐿 ( 퐿,훽 ) 푐 푐 푐 = − 휉퐿,훽 (훼퐿 (푋 )) + 푋 (훼퐿 (휉퐿,훽)) + 훼퐿 ([휉퐿,훽, 푋 ]).
Now, since 휉퐿,훽 is a SODE, 훼 휉 휄 푆∗ 퐿 푆휉 퐿 퐿. 퐿 ( 퐿,훽 ) = 휉퐿,훽 ( d ) = ( 퐿,훽) = Δ
푐 푣 푐 Moreover, recall that 푆푋 = 푋 . In addition, [휉퐿,훽, 푋 ] is a vertical vector eld, and thus 푐 푆 [휉퐿,훽, 푋 ] = 0. Then
푐 푣 푐 (d퐸퐿 + 훽)(푋 ) = −휉퐿,훽 (푋 퐿) + 푋 (Δ퐿).
On the other hand, one can write
푐 푐 푐 푐 d퐸퐿 (푋 ) = 푋 (퐸퐿) = 푋 (Δ퐿) − 푋 (퐿). 3.2. SYMMETRIES AND CONSTANTS OF THE MOTION 43
Combining these last two equations yields
푣 푐 푐 휉퐿,훽 (푋 퐿) = 푋 (퐿) − 훽(푋 ). In particular, the right-hand side vanishes if and only if the left-hand side does. In other words, the following result holds. Theorem 10 (Noether’s theorem for forced Lagrangian systems). Let 푋 be a vector eld on 푄. Then 푋푐 (퐿) = 훽(푋푐) if and only if 푋 푣 (퐿) is a constant of the motion. A vector eld 푋 on 푄 such that 푋푐 (퐿) = 훽(푋푐) will be called a symmetry of the forced Lagrangian (퐿, 훽). Analogously to the conservative case, a vector eld 푋 on 푄 will be called a Lie 푐 symmetry if [푋 , 휉퐿,훽] = 0. Observe that
휄 푐 휔 푐 휄 휔 휄 푐 휔 [푋 ,휉퐿,훽 ] 퐿 =L푋 ( 휉퐿,훽 퐿) − 휉퐿,훽 (L푋 퐿)
푐 퐸 훽 휄 푐 훼 . =L푋 (d 퐿 + ) + 휉퐿,훽 d(L푋 퐿)
Suppose that d(L푋푐 훼퐿) = 0. Then 푋 is a Lie symmetry if and only if
푐 L푋푐 훽 = −d(푋 (퐸퐿)). (3.4)
Moreover, suppose that L푋푐 훼퐿 is exact. Then there exists a function 푓 on 푇푄 such that ∗ d푓 = L푋푐 훼퐿 = 휄푋푐 (d훼퐿) + d(휄푋푐 훼퐿) = 휄푋푐 (d훼퐿) + d(휄푋푐 푆 d퐿) 푣 = 휄푋푐 (d훼퐿) + d(휄푆푋푐 d퐿) = 휄푋푐 (d훼퐿) + d(푋 퐿), so 푣 푣 휄 휄 푐 훼 휄 푓 푋 퐿 휉 푓 푋 퐿 , 휉퐿,훽 푋 (d 퐿) = 휉퐿,훽 (d( − )) = 퐿,훽 ( − ( )) but 푐 푐 휄 휄 푐 훼 휄 푐 휄 휔 휄 푐 퐸 훽 푋 퐸 훽 푋 . 휉퐿,훽 푋 d 퐿 = 푋 휉퐿,훽 퐿 = 푋 (d 퐿 + ) = ( 퐿) + ( ) 푣 푐 푐 This implies that 푓 − 푋 (퐿) is a constant of the motion if and only if 푋 (퐸퐿) + 훽(푋 ) 푐 푐 vanishes. A vector eld 푋 on 푄 such that L푋푐 훼퐿 is exact and 푋 (퐸퐿) + 훽(푋 ) vanishes will be called a Noether symmetry. It is worth remarking that, unlike in the conservative case, Noether symmetries are not necessarily Lie symmetries. As a matter of fact, if 푋 is a Noether symmetry,
푐 L푋푐 훽 = 휄푋푐 (d훽) + d(휄푋푐 훽) = 휄푋푐 (d훽) + d(훽(푋 )) 푐 = 휄푋푐 (d훽) − d(푋 (퐸퐿)), and obviously L푋푐 훼퐿 is closed. Then, by equation (3.4), 푋 is a Lie symmetry if and only if 휄푋푐 (d훽) vanishes. Observe that every symmetry of the forced Lagrangian is a Noether symmetry. In fact, if 푓 is a constant function, clearly 푋 푣 (퐿) is a conserved quantity. Moreover,
L푋푐 훼퐿 = 0, so 푐 푐 0 = (L푋푐 훼퐿)(휉퐿,훽) = 푋 훼퐿 (휉퐿,훽) − 훼퐿 ([푋 , 휉퐿,훽 ]) 푐 푐 푐 = 푋 (Δ퐿) − 푆 [푋 , 휉퐿,훽 ]퐿 = 푋 (Δ퐿), 44 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES and thus,
푐 푐 푐 푐 푐 푐 푐 0 = 푋 (퐸퐿) + 훽(푋 ) = 푋 (Δ퐿) − 푋 (퐿) + 훽(푋 ) = −푋 (퐿) + 훽(푋 ).
These results can be summarized in the following way. Proposition 11. Let (퐿, 훽) be a forced Lagrangian system with forced Euler-Lagrange vector eld 휉퐿,훽 . Let 푋 be a vector eld on 푄. Then the following results hold. i) 푋 is a symmetry of the forced Lagrangian if and only if 푋 푣 (퐿) is a constant of the motion.
ii) If L푋푐 훼퐿 is closed, then 푋 is a Lie symmetry if and only if
푐 L푋푐 훽 = −d(푋 (퐸퐿)).
iii) If 푋 is a Noether symmetry, it is also a Lie symmetry if and only if
휄푋푐 d훽 = 0.
푣 iv) If L푋푐 훼퐿 = d푓 , then 푋 is a Noether symmetry if and only if 푓 − 푋 (퐿) is a conserved quantity.
v) If 푋 is a Noether symmetry, it is also a symmetry of the forced Lagrangian if and only if L푋푐 훼퐿 = 0. The results just exposed apply for point-like symmetries, that is, innitesimal sym- metries on 푄. Analogous results can be obtained for non-point-like symmetries, that is, innitesimal symmetries on 푇푄. A vector eld 푋˜ on 푇푄 will be called a dynamical symmetry if [푋,˜ 휉퐿,훽 ] = 0. It will be called a Cartan symmetry if 푋˜ (퐸퐿) + 훽(푋˜ ) = 0 and L푋푐 훼퐿 is exact. Proposition 12. Consider a forced Lagrangian system (퐿, 훽)with forced Euler-Lagrange vector eld 휉퐿,훽 . Let 푋 be a vector eld on 푄, and let 푋˜ be a vector eld on 푇푄. Then the following statements hold.
i) 푋 is a Lie symmetry if and only if 푋푐 is a dynamical symmetry.
ii) 푋 is a Noether symmetry if and only if 푋푐 is a Cartan symmetry. 훼 푋˜ iii) If L푋˜ 퐿 is closed, then is a dynamical symmetry if and only if 푋˜ 퐸 훽. d( ( 퐿)) = −L푋˜
iv) A Cartan symmetry is a dynamical symmetry if and only if 휄 훽 . 푋˜ d = 0
훼 푓 푋˜ 푓 푆푋˜ 퐿 v) If L푋˜ 퐿 = d , then is a Cartan symmetry if and only if − ( )( ) is a constant of the motion. 3.2. SYMMETRIES AND CONSTANTS OF THE MOTION 45
The rst two assertions are derived straightforwardly from the denitions. The rest can be proven in a completely analogous manner to the corresponding results for point- like symmetries. As it was discussed in the conservative case (see Section 2.5), Theorem 10 cannot be generalised for symmetries on 푇푄, since [휉퐿,훽, 푋˜ ] is not a vertical vector eld for a general 푋˜ on 푇푄.
Example 11 (Harmonic oscillator with a gyroscopic force). Consider an 푛-dimensional harmonic oscillator, with mass and frequency equal to 1 for simplicity’s sake, so that its Lagrangian is 푛 1 ∑︁ h i 퐿 = 푞¤푖 2 − 푞푖 2 . 2 푖=1 Then the Poincaré-Cartan forms are simply
푖 푖 푖 푖 훼퐿 = 푞¤ d푞 , 휔퐿 = d푞 ∧ d푞¤ , and the energy of the system is
푛 1 ∑︁ h i 퐸 = 푞¤푖 2 + 푞푖 2 . 퐿 2 푖=1 Suppose that it is subject to a gyroscopic force [EGG21], that is, an external force 훽 of the form 푗 푖 훽 = 푠푖 푗 (푞)푞 ¤ d푞 , where 푠푖 푗 is skew-symmetric. The forced Euler-Lagrange vector eld of the system (퐿, 훽) is thus 휕 휕 휉 = 푞¤푖 − 푞푖 + 푠 푞¤푗 퐿,훽 휕푞푖 푖 푗 휕푞¤푖 which can be straightforwardly derived from equations (3.2) and (3.3). Consider now the two-dimensional case. Obviously, in that case there is only one independent non- vanishing component of the gyroscopic force: 푠12(푞) C 푔(푞). Then
훽 = 푔(푞) 푞¤2d푞1 −푞 ¤1d푞2 , and 휕 휕 휕 휕 휉 = 푞¤1 +푞 ¤2 − 푞1 + 푔푞¤2 − 푞2 − 푔푞¤1 . 퐿,훽 휕푞1 휕푞2 휕푞¤1 휕푞¤2 Consider the vector eld 푋˜ on 푇푄 given by 휕 휕 휕 휕 푋˜ = 푔푞¤1 + 푞2 + 푔푞¤2 − 푞1 +푞 ¤2 −푞 ¤1 휕푞1 휕푞2 휕푞¤1 휕푞¤2
Clearly, this vector eld veries
푋˜ (퐸퐿) + 훽(푋˜ ) = 0.
In addition, 휄 훼 푞1 푔푞1 푞2 푞2 푔푞2 푞1 , 푋˜ 퐿 = ¤ ¤ + + ¤ ¤ − 46 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES and 1 2 1 2 1 2 2 1 1 2 휄푋 휔퐿 = 푔푞¤ + 푞 d푞¤ + 푔푞¤ − 푞 d푞¤ −푞 ¤ d푞 +푞 ¤ d푞 If 훽 does not depend on 푞, that is, 푔 is a constant, one can write 1 2 1 2 휄 휔 = d 푔 푞¤1 + 푔 푞¤2 − 푞1푞¤2 + 푞2푞¤1 , 푋 퐿 2 2 and hence 훼 휄 훼 휄 휔 푓 , L푋˜ 퐿 = d( 푋˜ 퐿) − 푋˜ 퐿 = d where 1 1 푓 = 푔 푞¤12 + 푔 푞¤22 . 2 2 Therefore, when 훽 is independent of 푞, 푋˜ is a Cartan symmetry of (퐿, 훽), and 푔 h i ℓ = 푓 − (푆푋˜ )(퐿) = 푞1푞¤2 − 푞2푞¤1 − 푞¤12 + 푞¤22 2 is its associated constant of the motion. Observe that when 푔 = 0 the conservation of angular momentum is recovered. In fact, when 푔 = 0 one can write 푋˜ = 푋푐 for 휕 휕 푋 = 푞2 − 푞1 , 휕푞1 휕푞2 which is the generator of rotations on the (푞1, 푞2) plane. ♦ The types of symmetries for a forced Lagrangian system (퐿, 훽) on 푇푄 are summarized in Figure 3.1. Observe that, unlike in the conservative case (cf. Figure 2.1), Noether symmetries are not a subset of Lie symmetries, and Cartan symmetries are not a subset of dynamical symmetries neither. Obviously, when 훽 vanishes, 휉퐿,훽 = 휉퐿 and the conservative results are recovered. Similar results can be obtained in the Hamiltonian framework. Consider a forced Hamiltonian system on (푇 ∗푄, 휔) with Hamiltonian function 퐻 and external force 훾. Denote by 푋퐻,훾 the dynamical vector eld associated to (퐻,훾).A constant of the motion (or a conserved quantity) is then a function 푓 on 푇 ∗푄 which is a rst integral of 푋퐻,훾 (that is, 푋퐻,훾 (푓 ) = 0). Recall that the Poisson bracket of two functions 푓 and 푔 on (푇 ∗푄, 휔) is given by {푓 ,푔} = 휔 (푋푓 , 푋푔), where 푋푓 and 푋푔 are the Hamiltonian vector elds associated with 푓 and 푔, respectively. Clearly, 푋 푓 휄 푓 휄 휄 휔 휄 휄 휔 퐻,훾 ( ) = 푋퐻,훾 d = 푋퐻,훾 ( 푋푓 푄 ) = − 푋푓 ( 푋퐻,훾 푄 )
= −휔푄 (푋퐻, 푋푓 ) − 훾 (푋푓 ) = {푓 , 퐻 } − 훾 (푋푓 ), and hence 푓 is a constant of the motion if and only if
{푓 , 퐻 } = 훾 (푋푓 ).
The Hamiltonian counterpart of Proposition 12 is as follows. Proposition 13. Let 푋ˆ be a vector eld on 푇 ∗푄. Then the following assertions hold: 3.2. SYMMETRIES AND CONSTANTS OF THE MOTION 47
휃 푋ˆ 푋 i) If L푋ˆ is closed, then commutes with 퐻,훾 if and only if 푋ˆ 퐻 훾. d( ( )) = −L푋ˆ 휃 푓 푋ˆ 퐻 훾 푋ˆ 푓 휃 푋ˆ ii) If L푋ˆ = d , then ( ) + ( ) = 0 if and only if − ( ) is a constant of the motion.
iii) If the previous statement holds, 푋ˆ commutes with 푋퐻,훾 if and only if 휄 훾 . 푋ˆ d = 0 If the Lagrangian is hyperregular, the symmetries and constants of the motion on 푇푄 can be easily associated with their counterparts on 푇 ∗푄 via the Legendre transform. ∗ Suppose that (퐿, 훽) is a Lagrangian system such that 퐻 ◦ Leg = 퐸퐿 and Leg 훾 = 훽. Let 푋˜ be a vector eld on 푇푄 and 푋ˆ the Leg-related vector eld on 푇 ∗푄. Then:
i) 푋ˆ commutes with 푋퐻,훾 if and only if 푋˜ is a dynamical symmetry of (퐿, 훽). 휃 푓 훼 푔 푔 푓 ii) L푋ˆ = d if and only if L푋˜ 퐿 = d , where = ◦ Leg. 휃 푓 iii) Suppose that L푋ˆ = d . Then the following assertions are equivalent. a) 푋ˆ (퐻) + 훾 (푋ˆ) = 0. b) 푓 − 휃 (푋ˆ) is a conserved quantity.
c) 푋˜ (퐸퐿) + 훽(푋˜ ) = 0.
d) 푓 ◦ Leg −훼퐿 (푋˜ ) is a conserved quantity.
Figure 3.1: Types of symmetries on 푄 and on 푇푄. The complete lifts of Lie symmetries and Noether symmetries correspond to dynamical symmetries and Cartan symmetries, respectively. The symmetries of the forced Lagrangian system are the subset of Noether symmetries for which L푋푐 훼퐿 = 0, so 푓 is a constant that can be taken as 0 without loss of generality. The Noether symmetries that satisfy 휄푋푐 d훽 = 0 are also Lie symmetries, and analogously with the intersection between Cartan and dynamical symmetries. 48 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
3.3 Momentum map and reduction
In this section, the symplectic reduction method explained in Section 2.6 is extended for forced Lagrangian systems. Consider a Lie group 퐺 with action Φ푔 on 푄, and consider the lifted action to 푇푄 by tangent prolongation. In what follows, every group action will be assumed to be free and proper. Denote by 픤 the Lie algebra of 퐺, and by 픤∗ the dual of 픤. As it was previously explained, the symplectic point reduction (see Theorem9) requires the action of 퐺 to be symplectic. In the conservative case, if the Lagrangian 퐿 on 푇푄 is 퐺-invariant, the dynamics given by 휉퐿 on 푇푄 can be reconstructed from the 휉 푇푄 휔 dynamics given by 퐿휇 on the reduced space ( )휇. Since the symplectic structure 퐿 on 푇푄 is induced by the Lagrangian 퐿, in the case of a forced Lagrangian system (퐿, 훽), reduction will require 퐿 and 훽 to be invariant under 퐺 (or a subgroup of 퐺) independently. Hereinafter, suppose that the 퐺-action leaves 퐿 invariant, and hence, 훼퐿 and 휔퐿 are invariant. Recall that then the natural momentum map
퐽 : 푇푄 → 픤∗, 퐽 휉 훼 휉푐 , ( ) = 퐿 푄
∗ 휉 is Ad -equivariant. For each 휉 ∈ 픤 and 푣푞 ∈ 푇푄, 퐽 : 푇푄 → R is the function given by
휉 퐽 (푣푞) = 퐽 (푣푞), 휉 .
In the conservative case, 퐽 휉 is a constant of the motion for each 휉 ∈ 픤. Consider a forced Lagrangian system (퐿, 훽) with forced Euler-Lagrange vector eld 휉퐿,훽 . Clearly,
휉 푐 퐽 = 훼 (휉 ) = 휄 푐 훼 , 퐿 푄 휉푄 퐿 so 휉 d퐽 = d(휄 푐 훼 ) = L 푐 훼 − 휄 푐 d훼 = 휄 푐 휔 , 휉푄 퐿 휉푄 퐿 휉푄 퐿 휉푄 퐿 since 훼퐿 is 픤-invariant. Contracting this equation with 휉퐿,훽 , one gets
휄 퐽 휉 휉 퐽 휉 , 휉퐿,훽 d = 퐿,훽 ( ) on the left-hand side, and
푐 푐 휄 휄 푐 휔 = −휄 푐 휄 휔 = −휄 푐 (d퐸 + 훽) = −휉 (퐸 ) − 훽(휉 ), 휉퐿,훽 휉푄 퐿 휉푄 휉퐿,훽 퐿 휉푄 퐿 푄 퐿 푄
휉 on the right-hand side. Thus 퐽 is a conserved quantity for 휉퐿,훽 if and only if 휉푐 퐸 훽 휉푐 . 푄 ( 퐿) + ( 푄 ) = 0 (3.5)
Now observe that
푐 푐 푐 푐 휉 (퐸 ) = 휉 (Δ(퐿)) − 휉 (퐿) = 휉 (Δ(퐿)) = L 푐 (Δ(퐿)) = L 푐 (휄 d퐿) 푄 퐿 푄 푄 푄 휉푄 휉푄 Δ 푐 = 휄[ 푐 ]d퐿 + 휄 (L 푐 d퐿) = 휄[ 푐 ]d퐿 = [휉 , Δ](퐿), 휉푄,Δ Δ 휉푄 휉푄,Δ 푄 3.3. MOMENTUM MAP AND REDUCTION 49
휉푐 (퐿) 퐺 퐿 [휉푐 , ] since 푄 = 0 by the -invariance of , but 푄 Δ = 0, and thus 휉푐 퐸 푄 ( 퐿) = 0
휉 for each 휉 ∈ 픤, that is, 퐸퐿 is 퐺-invariant. By equation (3.5), 퐽 is a conserved quantity for 휉퐿,훽 if and only if 훽 휉푐 . ( 푄 ) = 0 (3.6) This result can also be obtained from a variational approach (see reference [MW01, Theorem 3.1.1]). In addition, 푐 L 푐 훽 = d(휄 푐 훽) + 휄 푐 d훽 = d(훽(휉 )) + 휄 푐 d훽. 휉푄 휉푄 휉푄 푄 휉푄
If equation (3.6) holds, then 훽 is 휉-invariant (that is, L 푐 훽 = 0) if and only if 휉푄
휄 푐 d훽 = 0. (3.7) 휉푄
This motivates dening the vector subspace 픤훽 of 픤 given by
n 푐 o 픤 = 휉 ∈ 픤 | 훽(휉 ) = 0, 휄 푐 d훽 = 0 . 훽 푄 휉푄
Furthermore, for each 휉, 휂 ∈ 픤훽 ,
푐 푐 푐 훽 [휉 , 휂 ] = 훽 [휉 , 휂 ] = 휄[ 푐 푐 ] 훽 = L 푐 휄 푐 훽 − 휄 푐 L 푐 훽 푄 푄 푄 푄 휉푄,휂푄 휉푄 휂푄 휂푄 휉푄 푐 푐 푐 푐 = 휉 (훽(휂 )) − 휂 (훽(휉 )) − 휄 푐 (휄 푐 d훽) = 0, 푄 푄 푄 푄 휂푄 휉푄 and similarly, 휄[ ]푐 d훽 = 휄[ 푐 푐 ]d훽 = L 푐 휄 푐 d훽 − 휄 푐 L 푐 d훽 휉푄,휂푄 휉푄,휂푄 휉푄 휂푄 휂푄 휉푄 = L 푐 휄 푐 d훽 − 휄 푐 dL 푐 훽 = 0. 휉푄 휂푄 휂푄 휉푄
Then [휉, 휂] ∈ 픤훽 for each 휉, 휂 ∈ 픤훽 , and hence 픤훽 is a Lie subalgebra of 픤. Since 훼퐿 is invariant, L 푐 훼 = 0 휉푄 퐿 for each 휉 ∈ 픤. In combination with equation (3.5), this implies that 휉푄 is a Noether symmetry of (퐿, 훽). In fact, it is also a symmetry of the forced Lagrangian (see Proposition 11 v) ). By equation (3.7), 휉푄 is also a Lie symmetry. As a matter of fact, 픤훽 is the subspace of 휉 ∈ 픤 such that 휉푄 is both a Lie symmetry and a Noether symmetry of (퐿, 훽). Symplectic point reduction (Theorem9) can be generalized for forced Lagrangian systems as follows.
퐺 ⊂ 퐺 픤 퐽 푇푄 → 픤∗ Theorem 14. Let 훽 be the Lie subgroup generated by 훽 and 훽 : 훽 the 휇 ∈ 픤∗ 퐽 (퐺 ) reduced momentum map. Let 훽 be a regular value of 훽 and 훽 휇 the isotropy group in 휇. Then the following results hold:
퐽 −1(휇) 푇푄 휉 i) 훽 is a submanifold of and 퐿,훽 is tangent to it. 50 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
(푇푄) 퐽 −1(휇)/(퐺 ) ii) The quotient space 휇 B 훽 훽 휇 is endowed with an induced symplectic structure 휔휇, namely 휋∗휔 푖∗휔 , 휇 휇 = 휇 퐿 휋 퐽 −1(휇) → (푇푄) 푖 퐽 −1(휇) ↩→ 푇푄 where 휇 : 훽 휇 and 휇 : 훽 denote the projection and the inclusion, respectively.
iii) 퐿 induces a function 퐿휇 : (푇푄)휇 → R dened by
퐿휇 ◦ 휋휇 = 퐿 ◦ 푖휇 . (3.8) 퐸 푇푄 퐸 퐿 퐿 iv) One can also introduce a function 퐿휇 : ( )휇 → R, given by 퐿휇 = Δ휇 ( 휇) − 휇, which satises 퐸 휋 퐸 푖 . 퐿휇 ◦ 휇 = 퐿 ◦ 휇 (3.9)
v) 훽 induces a reduced semibasic 1-form 훽휇 on (푇푄)휇, given by 휋∗훽 푖∗훽. 휇 휇 = 휇
Proof. The proof of the rst three statements can be found in references [AM08; MMR90; OR04]. If Δ is the Liouville vector eld on 푇푄, let Δ휇 is a 휋휇-related vector eld on (푇푄)휇, namely 휋휇∗Δ휇 = 푖휇∗Δ. Observe that 퐿 푖 푖∗ 퐿 푖∗ 휄 퐿 휄 푖∗ 퐿 푖 푖∗퐿 Δ( ) ◦ 휇 = 휇Δ( ) = 휇 ( Δd ) = 푖휇∗Δ ( 휇d ) = ( 휇∗Δ)( 휇 )
= (푖휇∗Δ)(퐿 ◦ 푖휇) = (푖휇∗Δ)(퐿휇 ◦ 휋휇) = (휋휇∗Δ휇)(퐿휇 ◦ 휋휇) (3.10) 휋∗ 퐿 퐿 휋 . = 휇 (Δ휇 ( 휇)) = Δ휇 ( 휇) ◦ 휇 Combining equations (3.8) and (3.10), it is clear that equation (3.9) holds. On the other hand, 푐 훽(휉 ) = 휄 푐 훽 = 휄 푐 휄 휔 = −휄 휄 푐 휔 , 푄 휉푄 휉푄 휉훽 퐿 휉훽 휉푄 퐿 but 휉 휄 푐 휔 = −L 푐 훼 + d(휄 푐 훼 ) = d(휄 푐 훼 ) = d퐽 , 휉푄 퐿 휉푄 퐿 휉푄 퐿 휉푄 퐿 since 훼퐿 is 픤-invariant, and thus 훽(휉푐 ) −휄 퐽 휉 −휉 (퐽 휉 ). 푄 = 휉훽 d = 훽
The left-hand side vanishes for any 휉 ∈ 픤훽 by denition, and hence the ow of 휉훽 preserves 휉 −1 퐽 . In other words, the ow 퐹푡 of 휉훽 leaves invariant the connected components of 퐽 (휇). 퐹 휇 푇푄 One can then dene a ow 푡 on ( )휇 given by 휋 퐹 푖 퐹 휇 휋 . 휇 ◦ 푡 ◦ 휇 = 푡 ◦ 휇 푌 푇푄 퐹 휇 푌 휋 Let be the vector eld on ( )휇 whose ow is 푡 . By construction, is 휇-related to 휉훽 , so one can introduce a 1-form 훽휇 B 휄푌 휔휇 on (푇푄)휇 that satises 휋∗훽 휋∗ 휄 휔 휄 휋∗휔 휄 푖∗휔 푖∗ 휄 휔 푖∗훽. 휇 휇 = 휇 ( 푌 휇) = 휋휇∗푌 ( 휇 휇) = 푖휇∗휉훽 ( 휇 퐿) = 휇 ( 휉훽 퐿) = 휇 3.4. RAYLEIGH SYSTEMS 51
푇푄 휉 The dynamics on ( )휇 are determined by the vector eld 퐿휇,훽휇 , dened by 휄 휔 = 퐸 + 훽 . 휉퐿휇,훽휇 휇 d 퐿휇 휇
−1 This vector eld is 휋휇-related to 휉퐿,훽. The dynamics on 퐽 (휇) ⊂ 푇푄 can be reconstructed by the same procedure as in the conservative case (see Remark2).
3.4 Rayleigh systems
This section presents a special type of external forces: the so-called Rayleigh dissipation (see references [Cli21; Gan70; Gol87; Lur02; Min15; Ray71]). As far as the author is aware, these forces have not been previously studied in a geometric formalism. The results regarding symmetries and reduction from sections 3.2 and 3.3, respectively, are now particularized for Rayleigh dissipative forces. Rayleigh’s hypothesis is a non-conservative force which is linear in the velocities. Let 푅 be a bilinear form on 푇푄 given by
푅 : 푇푄 × 푇푄 → R, 푅 푞, 푞 , 푞 푅 푞 푞푖 푞 푗 . ( ¤1 ¤2) = 푖 푗 ( ) ¤1 ¤2 In other words, 푅 is a symmetric (0, 2)-tensor eld on 푄. Consider the quadratic form R on 푇푄 given by1 1 1 R(푞, 푞¤) = 푅(푞, 푞,¤ 푞¤) = 푅 (푞)푞 ¤푖푞¤푗, 2 2 푖 푗 which is called Rayleigh dissipation function. This dissipation function generates an external force 푅¯, whose component along the 푖-th direction is given by 휕R 푅¯ = = 푅 (푞)푞 ¤푗, 푖 휕푞¤푖 푖 푗 that is, 푗 푖 푅¯ = 푅푖 푗 (푞)푞 ¤ d푞 . Since 푅 is a (0, 2)-tensor on 푄, it denes a linear mapping
푅˜ : 푇푄 → 푇 ∗푄 by 푅˜ 푣 휄 푅, ( 푞) = 푣푞 that is, 푅˜(푣푞)(푤푞) = 푅(푣푞,푤푞). Therefore 푖 푖 푖 푖 푗 푅˜(푞 , 푞¤ ) = (푞 , 푅푖 푗 (푞)푞 ¤ 푞¤ ). Recall that, given a morphism of bre bundles 푇푄 → 푇 ∗푄, there is always an associated semibasic 1-form on 푇푄 and conversely. Hence 푅¯ can also be dened as the semibasic 1-form on 푇푄 associated with the bre morphism 푅˜. 1Here the 1/2 factor is introduced for convenience. 52 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
퐿, 푅¯ 푇푄 휉 푇푄 Consider a forced Lagrangian system ( ) on . If 푅¯ is the vector eld on given by 휄 휔 = 푅,¯ 휉푅¯ 퐿 then locally 휕 휉 = −푅 푞¤푘푊 푖 푗 . 푅¯ 푖푘 휕푞¤푗 퐿, 푅¯ 휉 The dynamics of the system ( ) is given by the vector eld 퐿,푅¯, given by 휉 휉 휉 , 퐿,푅¯ = 퐿 + 푅¯ where 휉퐿 is the Hamiltonian vector eld of 퐿 on (푇푄, 휔퐿). The equations of motion of the system are given by d 휕퐿 휕퐿 − = −푅 (푞)푞 ¤푗 . d푡 휕푞¤푖 휕푞푖 푖 푗 Notice that 휉 (퐸 ) = 휄 퐸 = 휄 휄 휔 = −휄 휄 휔 = −휄 ( 퐸 + 푅¯), 퐿,푅¯ 퐿 휉퐿,푅¯ d 퐿 휉퐿,푅¯ 휉퐿 퐿 휉퐿 휉퐿,푅¯ 퐿 휉퐿 d 퐿 but 휄 퐸 휄 휄 휔 , 휉퐿 d 퐿 = 휉퐿 휉퐿 퐿 = 0 so 휉 퐸 휄 푅¯ 푅 푞 푞푖푞 푗 . 퐿,푅¯ ( 퐿) = − 휉퐿 = − 푖 푗 ( ) ¤ ¤ = −2R 휎 휉 Suppose that is a trajectory of the system (that is, an integral curve of 퐿,푅¯). Then d 퐸 (휎 (푡)) = 휉 ¯ (퐸 )(휎 (푡)) = −2R(휎 (푡)). d푡 퐿 퐿,푅 퐿 Therefore R expresses the rate at which energy is dissipated away along the evolution of the system. Of course, Rayleigh dissipation can also be described in the Hamiltonian formalism. Indeed, since 퐿 is assumed to be hyperregular, the external force 푅ˆ on 푇 ∗푄 can be dened by 푅¯ = Leg∗ 푅,ˆ and the Hamiltonian function by 퐻 ◦ Leg = 퐸퐿. Locally 푅ˆ can be written as
푗 푅ˆ = 푅푖 푗 (푞)푝푖d푞 . 푍 If 푅ˆ is the vector eld dened by 휄 휔 = 푅,ˆ 푍푅ˆ 푄 then the equations of motion of the system are the integral curves of the vector eld 푋 퐻,푅ˆ, given by 푋 푋 푍 , 퐻,푅ˆ = 퐻 + 푅ˆ ∗ where 푋퐻 is the Hamiltonian vector eld of 퐻 on (푇 푄, 휔). In canonical coordinates, 휕 푍 푅 푞 푝 . 푅ˆ = − 푖 푗 ( ) 푖 휕푝 푗 3.4. RAYLEIGH SYSTEMS 53
The equations of motion are d푞푖 휕퐻 = , d푡 휕푝푖 d푝 휕퐻 푖 = − + 푅 (푞)푝 . d푡 휕푞푖 푖 푗 푗
If a forced Lagrangian system (퐿, 푅¯) is subject to an external force which is linear in the velocities, a Rayleigh dissipation function R from which 푅¯ is derived can be dened. In that case, the system can be characterized by the pair of functions 퐿 and R on 푇푄. As a matter of fact, an external force does not have to be linear in the velocities for this to be the case. Consider a function R on 푇푄 (not necessarily a quadratic form) and the external force 푅¯ given by 휕R 푅¯ = d푞푖 . (3.11) 휕푞¤푖 Geometrically, this can be expressed as
푅¯ = 푆∗(dR).
Here R takes the role of a “potential” from which the external force is derived. In what follows, this type of forces will be referred to as Rayleigh forces, and R will be called its Rayleigh dissipation function (or Rayleigh potential). A Rayleigh system (퐿, R) will denote a forced Lagrangian system (퐿, 푅¯) with Rayleigh force 푅¯ derived from R. Consider a Rayleigh system (퐿, R). If 휉퐿 is the Hamiltonian vector eld of 퐿, then 휄 푅¯ 휄 푆∗ 휄 휄 . 휉퐿 = 휉퐿 ( dR) = 푆휉퐿 dR = ΔdR = Δ(R) The result above generalizes straightforwardly for non-linear Rayleigh forces as follows. Proposition 15 (Dissipation of energy). Consider a Rayleigh system (퐿, R) on 푇푄. If 휎 : 푇푄 → R is a trajectory of the system, then d 퐸 ◦ 휎 (푡) = −Δ(R) ◦ 휎 (푡), d푡 퐿 where 퐸퐿 is the energy function associated with 퐿 and Δ is the Liouville vector eld on 푇푄. There is a certain “gauge freedom” in the choice of R. Remark 3. For any function 푓 on 푄, R and R + 푓 dene the same Rayleigh force 푅¯. In other words, (퐿, R) (퐿, R + 푓 ).
Constants of the motion for Rayleigh systems Consider a Rayleigh system (퐿, R) on 푇푄. Let 푋˜ be a vector eld on 푇푄 and 푋 a vector eld on 푄. Observe that 푅¯ 푋˜ 휄 푅¯ 휄 푆∗ 휄 푆푋˜ . ( ) = 푋˜ = 푋˜ ( dR) = 푆푋˜ dR = ( )(R) In particular, 푆푋푐 = 푋 푣 , so 푅¯(푋푐) = 푋 푣 (R). 54 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
By equation (3.11), locally
휕2R 휕2R d푅¯ = d푞푖 ∧ d푞 푗 + d푞¤푗 ∧ d푞푖, 휕푞푖푞¤푗 휕푞¤푖푞¤푗 so 휕2 휕2 휕푋 푗 휕2 휕2 푗 R 푗 R 푘 R 푖 푖 R 푗 휄 푐 d푅¯ = 푋 − 푋 +푞 ¤ d푞 − 푋 d푞¤ . 푋 휕푞 푗푞¤푖 휕푞¤푗푞푖 휕푞푘 휕푞¤푖푞¤푗 휕푞¤푖푞¤푗 In addition, 휕R 푋 푣 (R) = 푋 푖 , 휕푞¤푖 so 휕2R 휕푋 푗 휕R 휕2R d(푋 푣 (R)) = 푋 푗 + d푞푖 + 푋 푗 d푞¤푖 . 휕푞푖푞¤푗 휕푞푖 휕푞¤푗 휕푞¤푖푞¤푗 Therefore 푅¯ 휄 푅¯ 휄푐 푅¯ 휄 푅¯ 푅¯ 푋푐 휄 푅¯ 푋 푣 L푋푐 = 푋푐 d + d( 푋 ) = 푋푐 d + d( ( )) = 푋푐 d + d( (R)) 휕2R 휕푋 푗 휕2R 휕푋 푗 휕R = 푋 푗 +푞 ¤푘 + d푞푖 . 휕푞 푗푞¤푖 휕푞푘 휕푞¤푖푞¤푗 휕푞푖 휕푞¤푗
On the other hand, 휕R 휕푋 푗 휕R 푋푐 (R) = 푋 푗 +푞 ¤푘 , 휕푞 푗 휕푞푘 휕푞¤푗 and then 휕(푋푐 (R)) 휕2R 휕푋 푗 휕2R 휕푋 푗 휕R 푆∗(d(푋푐 (R))) = d푞푖 = 푋 푗 +푞 ¤푘 + d푞푖 . 휕푞¤푖 휕푞 푗푞¤푖 휕푞푘 휕푞¤푖푞¤푗 휕푞푖 휕푞¤푗
This shows that ∗ 푐 L푋푐 푅¯ = 푆 (d(푋 (R))). The results from Section 3.2 can now be expressed in the following way.
Proposition 16. The following assertions are equivalent:
i) 푋 is a symmetry of the forced Lagrangian (퐿, 푅¯),
ii) 푋푐 (퐿) = 푋 푣 (R),
iii) 푋 푣 (퐿) is a conserved quantity.
Example 12 (Fluid resistance). Consider a body of mass 푚 moving through a uid that fully encloses it. For the sake of simplicity, suppose that the motion takes place along one dimension. Then the drag force [Bat00; Fal11] is given by 1 푅¯ = 휌퐶퐴푞¤2d푞, 2 where 퐶 is a dimensionless constant depending on the body shape, 휌 is the mass density of the uid and 퐴 is the area of the projection of the object on a plane perpendicular to 3.4. RAYLEIGH SYSTEMS 55 the direction of motion. For further simplication, suppose that the density is uniform, and then 푘 = 퐶퐴휌/2 is constant. The dissipation function is thus 푘 R = 푞¤3. 3
If the body is not subject to forces besides the drag, its Lagrangian is 퐿 = 푚푞¤2/2. Consider the vector eld 푋 = 푒푘푞/푚휕/휕푞. It veries that 푋푐 (퐿) = 푋 푣 (R), so 푋 푣 (퐿) = 푚푒푘푞/푚푞¤ is a constant of the motion. In particular, when 푘 → 0, 푋 is the generator of translations and the conservation of momentum is recovered. ♦
Proposition 17 (Point-like symmetries in Rayleigh systems). Let 푋 be a vector eld on 푄. Then the following results hold:
i) If L푋푐 훼퐿 is closed, then 푋 is a Lie symmetry of (퐿, 푅¯) if and only if
푐 ∗ 푐 d(푋 (퐸퐿)) = −푆 (d(푋 R)).
ii) If L푋푐 훼퐿 = d푓 for some function 푓 on푇푄, then the following statements are equivalent:
a) 푋 is a Noether symmetry, 푐 푣 b) 푋 (퐸퐿) + 푋 (R) = 0, c) 푓 − 푋 푣 (퐿) is a constant of the motion.
iii) If 푋 is Noether symmetry, it is also a Lie symmetry if and only if 휄푋푐 d푅¯ = 0.
iv) If 푋 is Noether symmetry, it is also a symmetry of the forced Lagrangian if and only if L푋푐 훼퐿 = 0.
Proposition 18 (Not-point-like symmetries in Rayleigh systems). Let 푋˜ be a vector eld on 푇푄. Then the following results hold:
훼 푋˜ i) If L푋˜ 퐿 is closed, then is a dynamical symmetry if and only if 푋˜ 퐸 푆푋˜ 휄 푅.¯ d( ( 퐿) + ( )(R)) = − 푋˜ d
훼 푓 ii) If L푋˜ 퐿 = d , then the following statements are equivalent:
a) 푋˜ is a Cartan symmetry,
b) 푋˜ (퐸퐿) + (푆푋˜ )(R) = 0, c) 푓 − (푆푋˜ )(퐿) is a conserved quantity.
푋˜ 휄 푅¯ iii) If is Cartan symmetry, it is also a dynamical symmetry if and only if 푋˜ d = 0.
The following examples of Rayleigh systems are presented in reference [Min15]. Here their symmetries and constants of the motion are obtained. 56 PART 3. MECHANICAL SYSTEMS SUBJECTED TO EXTERNAL FORCES
Example 13 (A rotating disk). Consider a disk of mass 푚 and radius 푟 placed on a horizontal surface. Let 휑 be the angle of rotation of the disk with respect to a reference axis. The Lagrangian of the disk is 퐿 = 푇 = 푚푟 2휑¤2/4 and its Rayleigh dissipation function 2 is R = 휇푚푔푟휑¤/2. The Poincaré-Cartan 1-form is 훼퐿 = 푚푟 휑¤/2 d휑. The Rayleigh force is 푅¯ = 휇푚푔푟/2 d휑. Consider the vector eld 푋˜ = −푟휑휕¤ /휕휑 +휇푔휕/휕휑¤. Clearly, 푋˜ (퐸퐿) = 푋˜ (퐿) = −(푆푋˜ )(R). In addition, 휇푚푔푟 2 푚푟 3 L 훼 = d휑 − 휑¤d휑¤ = d푓 , 푋˜ 퐿 2 2 where 휇푚푔푟 2 푚푟 3 푓 = 휑 − 휑¤2 2 4 modulo a constant, and (푆푋˜ )(퐿) = −푚푟 3휑¤2/2, so 휇푚푔푟 2 푚푟 3 푓 − (푆푋˜ )(퐿) = 휑 + 휑¤2 2 4 푅¯ 휄 푅¯ 푋˜ is a constant of the motion. Since is closed, 푋˜ d = 0 is trivially satised, so is a dynamical symmetry as well as a Cartan symmetry. However, since 푅¯ is closed, it is not strictly an external force. In fact, the Lagrangian 휇푚푔푟 퐿˜ = 퐿 + 휑 2 leads to the same equations of motion as 퐿 with the external force 푅¯. ♦ Example 14 (The rotating stone polisher). Consider a system formed by two concentric rings of the same mass 푚 and radius 푟, which are placed over a rough surface, and rotate in opposite directions. Let (푥,푦) be the position of the centre and 휃 the orientation of the machine. Let 휔 be the angular velocity of the rings. The Rayleigh dissipation function is given by 휇푚푔 R = 2휇푚푔푟휔 + (푥 ¤2 +푦 ¤2), 2푟휔 and the Lagrangian is 퐿 = 푇 = 푚(푥 ¤2 +푦 ¤2 + 푟 2휃¤2 + 푟 2휔2). The Poincaré-Cartan 1-form is 2 ¤ 훼퐿 = 2푚(푥 ¤d푥 +푦 ¤d푦 + 푟 휃d휃) , and the Rayleigh force is 휇푚푔 푅¯ = (푥 ¤ 푥 +푦 ¤ 푦). 푟휔 d d Let 푋˜ (1) = −2푟휔휕/휕푥 + 휇푔휕/휕푥¤ and 푋˜ (2) = −2푟휔휕/휕푦 + 휇푔휕/휕푦¤. One can check that 푋˜ (푖) 퐸 푋˜ (푖) 퐿 푆푋˜ (푖) 푖 , 훼 푓 푓 ( 퐿) = ( ) = −( )(R) (for = 1 2). Moreover, L푋˜ (푖) 퐿 = d 푖 for 1 = (1) (2) 2휇푚푔푥 and 푓2 = 2휇푚푔푦, along with (푆푋˜ )(퐿) = −4푚푟푥¤ and (푆푋˜ )(퐿) = −4푚푟푦¤, so 2푚푟휔푥¤ + 휇푚푔푥 and 2푚푟휔푦¤ + 휇푚푔푦 are constants of the motion. ♦
Reduction of Rayleigh systems with symmetries Consider a Rayleigh system (퐿, R) on 푇푄, and a Lie group 퐺 that acts freely and properly on 푇푄 and leaves 퐿 invariant. Recalling the results from Section 3.3, one can dene a Lie subalgebra 픤R ≡ 픤푅¯ of 픤 by n o 휉 휉푣 , 푆∗ 휉푐 , 픤R = ∈ 픤 | 푄 (R) = 0 d 푄 (R) = 0 3.4. RAYLEIGH SYSTEMS 57
푆∗ 휉푐 ( ) 휉푐 ( ) 푞¤푖 where the condition d 푄 R = 0 means that 푄 R is basic: it does not depend on . For each 휉 ∈ 픤R: i) 퐽 휉 is a constant of the motion,
ii) 휉 leaves 푅¯ invariant. 휉푐 ( ) 휉 ∈ If additionally R is 픤R-invariant, that is, 푄 R = 0 for each 픤R, then it induces a dissipation function R휇 : (푇푄)휇 → R given by
R휇 ◦ 휋휇 = R ◦ 푖휇 .
Example 15. Consider the Rayleigh system (퐿, R) on 푇푄. Suppose that 푄 = R푛 \{0}, and that the Lagrangian 퐿 on 푇푄 is spherically symmetric, say 퐿(푞, 푞¤) = 퐿(k푞k, k푞 ¤k). Consider the Lie group 퐺 = SO(푛) acting by rotations on 푄 (see Example 10). Recall that
퐽 (푞, 푞¤) = 푞 ×푞. ¤
휉푣 ( ) Now look for Rayleigh potentials R such that 픤R = 픤. The condition that 푄 R = 0 implies that R must be spherically symmetric on the velocities. Then, the condition that 휉푐 ( ) 푄 R is basic means that the terms which are not spherically symmetric on the positions cannot involve the velocities, that is,
R = 퐴(푞) + 퐵(k푞k, k푞 ¤k).
Without loss of generality (see Remark3), the Rayleigh potential can be taken as
R = 퐵(k푞k, k푞 ¤k). (3.12)
In particular, if R is a quadratic form, say 1 R = 푅 (푞)푞 ¤푖푞¤푗, 2 푖 푗 then requirement (3.12) leads to
푅푖 푗 = 푟푖 (k푞k) 훿푖 푗 .
♦
Part 4
Conclusions and future work
In this master’s thesis, the eld of geometric mechanics has been explored, in particu- lar considering autonomous Lagrangian and Hamiltonian systems. The rst part has presented the geometric framework of mechanical systems. Several results regarding symmetries, constants of the motion and reduction have been reviewed. The second part has generalized these results for systems subjected to external forces. The main original contributions are the following: i) Noether’s theorem has been extended for forced Lagrangian systems. Di erent types of symmetries, and their associated constants of the motion, have also been generalized for forced Lagrangian systems. Their Hamiltonian counterparts have been obtained as well. ii) A theory for the reduction of forced Lagrangian systems invariant under the action of a group of symmetries has been presented. iii) Rayleigh forces have been expressed in a geometric language, which has allowed to obtain particular results regarding the symmetries and reduction of Rayleigh systems. iv) Several examples have been presented. These results have been published in an article [LLL21b]. There are several lines of research to be pursued during the PhD thesis consequent to this master’s thesis. Some of them are the following: i) Hamilton-Jacobi theory. As it was explained in Section 2.3, given a Hamiltonian 퐻 on 푇 ∗푄, by performing a canonical transformation one can obtained a new Hamiltonian 퐾 which satises Hamilton’s equations as well. In particular, choosing the transformation so that 퐾 vanishes identically makes Hamilton’s equations tautological. The relation between 퐻 and 퐾 leads to the so-called Hamilton-Jacobi equation (see references [AM08; Gol87]) 휕푊 퐻 푞푖, = 퐸. 휕푞푖
The Hamilton-Jacobi problem consists in nding a solution푊 , known as the charac- teristic function. This is the formulation of classical mechanics which resembles the
59 60 PART 4. CONCLUSIONS AND FUTURE WORK
most to quantum mechanics. As a matter of fact, Hamilton-Jacobi-type equations are obtained when considering the classical limit of Schrödinger equation [AM08; Car+06; EMS04]. Geometrically, the Hamilton-Jacobi equation can be written as [Car+06; ILD08] (d푊 )∗퐻 = 퐸, where d푊 is a section of 푇 ∗푄. The main interest of the Hamilton-Jacobi theory lies in nding complete solutions, that is, solutions which depend on 푛 = dim푄 parameters. Each value of the parameters provides a trajectory of the system [Car+06]. Moreover, a complete solution provides 푛 constants of the motion and conversely. The Hamilton-Jacobi problem can be extended for forced Hamiltonian systems [ILD08] in a natural manner. However, it has not been studied in detail in the previous literature, for instance complete solutions have not been characterized, and it will be covered in a future paper [LLL21a].
ii) Discrete mechanics. The numerical analysis of a mechanical system requires to discretize it. The naive way to do this is by rst obtaining the continuous equations of motion of the system, and then discretizing them, usually by means of a Runge- Kutta method (see references [New13; Sch17]). However, this approach does not guarantee the constants of the motion of the continuous system to be preserved on its discrete counterpart. Instead, one can approximate the action integral (for instance, by the trapezoidal method [New13]) and, by taking variations on the discretized action, obtain the discrete Euler-Lagrange equations. A similar approach can be carried out for forced Lagrangian systems (see references [Lew+04; MW01]). In the previous literature [LS18; OBL11], a Hamilton-Jacobi theory for discrete systems has been developed. The extension of this theory for forced mechanical systems will be covered in a future paper [LLL21a].
iii) Higher-order systems. Throughout this thesis, the Lagrangians considered only depended on the positions and the velocities. Of course, one can also consider Lagrangians depending on the accelerations and subsequent derivatives, which are called higher-order Lagrangians. Geometrically, a Lagrangian 퐿(푞, 푞,¤ 푞¥) is a function on 푇 2푄 = 푇푇푄, and analogously for higher orders. Higher-order Hamiltonians can be described in a similar manner [PR11; RK96]. Symmetries and constants of the motion for higher-order Lagrangian systems have been studied in references [LM94; LM95; Sco21]. Systems subject to higher-order external forces (such as the Abraham-Lorentz force) will be considered in future papers.
iv) Non-autonomous systems. Lagrangians and Hamiltonians can also depend ex- plicitly on time, in which case they are called non-autonomous [LS17; PR11; Sar13]. Some results regarding non-autonomous forced Lagrangian systems were already obtained in reference [Can82]. A complete classication of the symmetries or the reduction of these systems has not been studied so far. Bibliography
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