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Hamiltonian Mechanics Hamiltonian Mechanics Marcello Seri Bernoulli Institute University of Groningen m.seri (at) rug.nl Version 1.0 May 12, 2020 This work is licensed under a Creative Commons “Attribution-NonCommercial-ShareAlike 4.0 Inter- national” license. Contents 1 Classical mechanics, from Newton to Lagrange and back 3 1.1 Newtonian mechanics .................... 3 1.1.1 Motion in one degree of freedom ........... 5 1.2 Hamilton’s variational principle ................ 7 1.2.1 Dynamics of point particles: from Lagrange back to Newton 13 1.3 Euler-Lagrange equations on smooth manifolds ........ 18 1.4 First steps with conserved quantities ............. 24 1.4.1 Back to one degree of freedom ............ 24 1.4.2 The conservation of energy .............. 25 1.4.3 Back to one degree of freedom, again ......... 27 2 Conservation Laws 34 2.1 Noether theorem ....................... 34 2.1.1 Homogeneity of space: total momentum ....... 38 2.1.2 Isotropy of space: angular momentum ......... 40 2.1.3 Homogeneity of time: the energy ........... 44 2.1.4 Scale invariance: Kepler’s third law .......... 44 2.2 The spherical pendulum ................... 45 2.3 Intermezzo: small oscillations ................. 47 2.3.1 Decomposition into characteristic oscillations ..... 50 2.3.2 The double pendulum ................. 51 2.3.3 The linear triatomic molecule ............. 52 2.3.4 Zero modes ...................... 53 2.4 Motion in a central potential ................. 54 2.4.1 Kepler’s problem ................... 57 2.5 D’Alembert principle and systems with constraints ...... 62 3 Hamiltonian mechanics 67 3.1 The Legendre Transform in the euclidean plane ........ 68 3.2 The Legendre Transform ................... 69 3.3 Poisson brackets and first integrals of motion ......... 77 3.3.1 The symplectic matrix ................ 80 i Contents 3.3.2 A brief detour on time-dependent hamiltonians .... 80 3.4 Variational principles of hamiltonian mechanics ........ 81 3.5 Canonical transformations .................. 86 3.5.1 Hamiltonian vector fields ............... 89 3.5.2 The hamiltonian Noether theorem ........... 93 3.6 The symplectic structure on the cotangent bundle ...... 95 3.6.1 Symplectomorphisms and generating functions .... 100 3.6.2 Darboux Theorem .................. 104 3.6.3 Liouville theorem ................... 106 4 Integrable systems 108 4.1 Lagrangian submanifolds ................... 108 4.2 Canonical transformations revisited .............. 111 4.2.1 The generating function S1 .............. 112 4.2.2 The generating functions S2 and S3 .......... 115 4.2.3 Infinitesimal canonical transformations ........ 117 4.2.4 Time-dependent hamiltonian systems ......... 119 4.3 Hamilton-Jacobi equations .................. 122 4.3.1 Separable hamiltonians ................ 125 4.3.2 A few exercises .................... 129 4.4 The Liouville-Arnold theorem ................. 130 4.4.1 Action-angle variables ................. 136 5 Hamiltonian perturbation theory 143 5.1 Small oscillations revisited .................. 143 5.2 Birkhoff normal forms .................... 147 5.3 A brief look at KAM theory ................. 150 5.4 Nekhoroshev theorem .................... 155 6 Conclusion 158 ii Preface The main reference for this course is [1], although we will deviate from the structure of the book or use alternative approaches for some of the proofs. The literature on classical (or analytical, as it was called by Lagrange) me- chanics is full of good material. Different sources present the topic from different perspectives and with different points of view, and may suite different people dif- ferently. In addition to [1], these lecture notes have also been heavily influenced by [10, 14, 17, 18, 22, 23]. The book [14], in addition to being a good introduction to many topics in classical mechanics that we will not have time to discuss during the lectures, can be freely accessed from within the university using the SpringerLink service. Most of the topics covered in this course can be found on [14] in chapters 1, 6, 8, 10, 11, 13, 15. In particular, it presents a more accessible approach for the proof of the KAM theorem. These notes are not (yet) exhaustive. They will be updated throughout the course, and it will likely take a few years of course iterations before they stabilize. Some topics, examples and exercises discussed in class will not be in these notes, but may appear as examples, exercises or problems in the literature presented above. Throughout the course, we will discuss the main ideas in Newtonian, La- grangian and Hamiltonian mechanics and the relations between them. We will discuss symmetries and phase space reduction, normal modes and small oscil- lations. We will then move to study action–angle coordinates and integrability, and finally we will discuss perturbation theory. On the latter, we will only briefly discuss resonances and mostly focus on KAM theory and Nekhoroshev theorem. The final lectures of the course will focus on the role of hamiltonian mechanics in current research. They are open to discussion and will be chosen by focusing on the main interests of the class. Possible topics include quantization, semi- classical analysis, contact and/or sub-riemannian geometry, numerical methods, mechanics of non-conservative systems, rigid bodies, mathematical billiards, ... Before concluding, let me emphasize two non-standard notational conventions that I decided to adopt from [1]: in the rest of this manuscript we will denote (·; ·) the inner products, also in Euclidean spaces when they correspond to the 1 Contents 3 standard scalar product, and [·; ·] the exterior products, also in R when I could have instead used the cross product. Please don’t be afraid to send me comments to improve the course or the text and to fix the many typos that will surely be in this first draft. They will be very appreciated. 2 1 Classical mechanics, from Newton to Lagrange and back In this chapter we will briefly review some basic concepts of classical mechanics, in particular we will only briefly discuss variational calculus and Newtonian me- chanics. And we will present some simple examples as spoilers for some of the material that will follow. For a deeper and more detailed account, we refer the readers to [1, 14]. 1.1 Newtonian mechanics Our main interest will be in describing equations of motion of a idealized point particle. This is a point–like object obtained by ignoring the dimensions of the physical object. Note that this can be done in many cases, for example when describing planetary motion around the sun, we can consider the planet as point particles. Of course this is not a universal simplification, for example we cannot do it if we want to describe the motion of a planet around its axes. For us, a point particle usually carries a mass m. Its position in space is 3 described by the position vector x = (x; y; z). Keep in mind that x : R ! R is a function of time, denoted by t, as the instantaneous state of the system is supposed to evolve, but we will omit the dependence on t in most cases. The velocity of the point particle is given by the rate of change of the position vector, i.e. its derivative with respect to time d x v = =: x_ = (x; _ y;_ z_): d t We call acceleration, the rate of change of the velocity, i.e. the second derivative d 2x a = =: x¨ = (¨x; y;¨ z¨): d t2 3 1 Classical mechanics, from Newton to Lagrange and back The mechanics of the particle is encoded in Newton’s second law of motion, which states that there exist frames of reference (i.e systems of coordinates) in which the motion of the particle is described by a differential equation involving the forces F acting on the point particle, its mass m and its acceleration as follows F = mx¨: (1.1) Let’s leave it here for now, we will come back to it in the next part of the lecture. In general we will consider systems of N point particles. These will be described by a set of N position vectors xk = (xk ; yk ; zk ) with masses mk , 3N k = 1;:::;N. For convenience we will denote x = (x1;:::; xN ) 2 R , and N m = (m1; : : : ; mN ) 2 R . We call x(t) the configuration of the system at time 3N t in the configuration space R . We say that a system of N particles has n degrees of freedom if we need n independent parameters to uniquely specify the system configuration. Note that, in general, the n degrees of freedom do not have to be the cartesian co- ordinates of the point particles. We call generalized coordinates any set of n parameters q = (q1; : : : ; qn) that uniquely determine the configuration of a sys- tem with n degrees of freedom, and generalized velocities their time derivatives q_ = (q _1;:::; q_n). The state of the system is then characterized by the set of (generalized) coordinates and velocities (q; q_) = `q1; : : : ; qn; q_1;:::; q_n´. If you recall differential geometry, you may correctly guess that the generalized coor- dinates will be points on some manifold q 2 M and the state will be a point in the tangent bundle (q; q_) 2 TM, i.e. q_ 2 TqM (see section 1.3). We have now all the elements to translate the Newtonian principle of deter- minacy in mathematical terms. In 1814, Laplace [16] wrote We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
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