Open Access Journal of Mathematical and Theoretical Physics Review Article Open Access To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models Abstract Volume 3 Issue 1 - 2020 The geodesic has a fundamental role in physics and in mathematics: roughly speaking, it represents the curve that minimizes the arc length between two points on a manifold. We Rizzuti BF, Vasconcelos Júnior GF, Resende analyze a basic but misinterpreted difference between the Lagrangian that gives the arc MA length of a curve and the one that describes the motion of a free particle in curved space. Department of Physics, Federal University of Juiz de Fora, Brazil They are taken as equivalent but it is not the case from the beginning. We explore this difference from a geometrical point of view, where we observe that the non-equivalence is Correspondence: Mateus Antanio Resende, Department of nothing more than a matter of symmetry. The equivalence appears, however, after fixing Physics, Federal University of Juiz de Fora, MG, Brazil, a particular condition. As applications, some distinct models are studied. In particular, we Email: explore the standard free relativistic particle, a couple of spinning particle models and also Received: January 14, 2020 | Published: March 04, 2020 the forceless mechanics formulated by Hertz. Introduction mechanics of Hertz4,12 H Hertz12 has provided a particular description of Newtonian classical mechanics without forces. His clever idea was Differential geometry is extensively used as the basic tool for to insert one more nonphysical dimension into the game, together with description of many distinct areas of physics, after all it is widely the introduction of the Newtonian potential inside the metric. In this accepted the space-time possesses the very structure of a differentiable case, second Newton law appears as the geodesic equation of a free 1 manifold. Besides that, it goes from special and general relativity particle in a curved manifold. These applications were chosen because 2 3 theories, till electrodynamics and gauge theories. We would like to they represent very clearly the difference between systems described focus our attention to its application even in the less advanced topic, S S by 1 and 2 . Finally, Section 6 is left for conclusions. Throughout all though not less important, of classical mechanics. One may find an the paper, Einstein notation of sum over repeated indexes is adopted. extensive literature on the subject, see 4,5 and references therein. The central idea of these notes is to draw attention to an underlying fact, L or L ? which is often misinterpreted: given a (semi-)riemannian manifold Let us consider a (semi-)riemannian m-dimensional manifold 1 (, g ) , the action functionals: m m ( , g ) and a chart (ϕ,U ) . U is an open set U ⊂ and S = ∫ dT g (,) v v (1) 1 mm1 ϕϕ:U→⊂ A( open ) P ( P) : =( x( P) ,..., x( P)) . (3) and 1 We write collectively s 2 = ∫ dt g ( v , v ) (2) 2 ϕii(P) = xPi( ); = 1,..., m (4) are not necessarily equivalent. In the literature, this fact is usually ignored. However, it is a subtle result that cannot be taken for granted. and elements of m will be denoted by capital letters. This problem will become clear along the text and it is intimately m Let X( ) be the set of all smooth vector fields on m , that related to what is known as pseudo-classical mechanics,6 that is, is, the set of mappings V which assigns to each point P on m models whose number of configuration variables is less than the a tangent vector V in the tangent space T m . In coordinatesϕ number of physical degrees of freedom. It turns out that these models P P ( ) ,VV=i ∂ . We also fix the notation for the (pseudo-)metric g in the are useful to describe quantum phenomena even before quantization, i coordinatesϕ , see7,8 justifying, thus, our presentation. The paper is divided as follows. i j ij In Section 2, we will show the non-equivalence between S1 and S2 L:,= gVV( ) = gV( ∂i , V ∂= j) gVV ij . (5) and under what conditions they become equivalent. Sections 3 and 4 are dedicated to examples on the lack of equivalence between actions For the case, there is a unique connection mm m S1 and S2 . In particular, we discuss in the former the free relativistic D :,XX()×→( ) X( ) (6) particle, while the latter contains examples of classical models of spinning particles. These consist of an example of a semiclassical known as the Levi-Civita connection, such that inϕ , it reads 9 model whose quantization leads to the Pauli equation and another k i1 il D∂ (∂j) =Ã ij ∂ k ; Ã jk =gg( ∂j kl +∂ k g jl −∂ l g jk ). (7) model for the relativistic rotator, where the particle is considered i 2 to resemble a moving clock with the pointer characterizing its gil denotes the elements of g −1 . A particle shall be denoted by the spin.10,11 Then, we devote the entire Section 5 to explore the forceless corresponding curve it traces2 1Here V is a tangent vector. Submit Manuscript | http://medcraveonline.com Open Acc J Math Theor Phy. 2020;3(1):8‒15. 8 ©2020 Rizzuti et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic Copyright: 9 mechanical models ©2020 Rizzuti et al. m we consider the case of the quantity xgx as a function ofτ . Then, we α:.I ⊂→ τ ατ( ) (8) treat it as a constant with respect toτ . I is an open interval on. In this sense, xi (τ ) corresponds to the Let us consider xgx as a function of the parameterτ . Thus, the representation of α in the coordinatesϕ , time derivative in the first term of Eq. (15) implies the equation ii (ϕ ατ)( ) :,= x ( τ) (9) ¨ j Π+ i j kl = (16) d j xÃkl xx 0, as well as xi (τ ) are the components of the tangent vector in dτ the point ατ( ) ik ik iixgjk x i xgjk x where Π=jjδ − , or, by denoting Λ=j , d xgx xgx i |:ατ( ) =x (τ ) ∂i |ατ( ) . (10) dτ iii Πjjj =δ −Λ (17) i dx j We have denoted xi ≡ . Since the state of motion is intuitively It follows that the vector x , whose components are given by x dτ , is in the null space of the operator associated with the matrix Ð . described by the corresponding position together with its velocity, we Thus, the null space is not trivial and Ð is not invertible, implying are naturally led to the notion of a tangent bundle, that det Π=0. Besides that, the Hessian matrix is proportional to mm TM ( ) = U T ( M ) (11) the matrix Ð . pM∈ p ∂2L ii ∼ i whose elements are parametrized by xx, . ijÐ.j (18) ( ) ∂∂xx There are two common ways to find the equations of motion of Therefore, we are dealing with a singular Lagrangian theory. It the particle on the manifold. Although both of them are based on can be shown that this is intimately related with the reparametrization the principle of least action, they are fundamentally different. The invariance of this functional, which we will cover below. Before we equations of motion the distinct actions only become equivalent after consider the case where xgx does not depend ofτ , we can also a particular condition. The main objective of this work is to clarify understand the ambiguity of the singular system in terms of the this point. Let us consider the two actions, already presented in the algebraic properties of the operators Ð and Ë .4 We point out that Introduction, which are linear functionals defined on the space they are projectors because of the following: mm 22 F(T()) ::= { f T ( ) → ; f is smooth }. (12) Π=ΠΠ=Π, ΠΛ = 0 We then take two different smooth functions to write (19) Π+Λ= ττ22 =ττ ≡ ij S1 ∫∫ d L d gij ( x) xx (13) ττ11 So, given an arbitrary vector v , it is uniquely decomposed as v=Π v +Λ vv: = ⊥+ v (20) and ττ i 2211 If we fix the vector x that specifies the projector Ð , then =ττ ≡ ij S 2 ∫∫ d L d g ij ( x ) xx (14) ττ22 iig( xv, ) 11 v= x; g( xv,⊥ ) = 0. (21) g( xx, ) In a sense, S1 has a geometrical meaning, as it describes the arc We observe that v is a projection of v on the direction of xi length of the curve α connecting the points ατ( 1 ) andατ( 2 ) . On the other hand, S2 is analogous to the usual action in classical mechanics whilst v⊥ belongs to the corresponding orthogonal complement. of a free particle, for we are integrating a kinetic energy term. Let us Considering the vector v defined by discuss separately both actions, explaining why one or the other shall v:= xi +Γ ixx jk be elected for further investigations. jk (22) S1 : Geometric point of view our variational problem has led us to the equation of motion (16), which is just the projection Ð0vv=⊥ = . This restricts v⊥ , but v Adopting the standard notion that S1 is related to the arc length remains arbitrary. This is the ambiguity exposed before. of α connecting ατ( 1 ) andατ( 2 ) ,the extremization of S1 provides the following differential equation for the curve which has minimum Now, we work with xgx as if it were a constant with respect to length τ .