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Classical Mechanics Classical Mechanics Prof. Dr. Alberto S. Cattaneo and Nima Moshayedi January 7, 2016 2 Preface This script was written for the course called Classical Mechanics for mathematicians at the University of Zurich. The course was given by Professor Alberto S. Cattaneo in the spring semester 2014. I want to thank Professor Cattaneo for giving me his notes from the lecture and also for corrections and remarks on it. I also want to mention that this script should only be notes, which give all the definitions and so on, in a compact way and should not replace the lecture. Not every detail is written in this script, so one should also either use another book on Classical Mechanics and read the script together with the book, or use the script parallel to a lecture on Classical Mechanics. This course also gives an introduction on smooth manifolds and combines the mathematical methods of differentiable manifolds with those of Classical Mechanics. Nima Moshayedi, January 7, 2016 3 4 Contents 1 From Newton's Laws to Lagrange's equations 9 1.1 Introduction . 9 1.2 Elements of Newtonian Mechanics . 9 1.2.1 Newton's Apple . 9 1.2.2 Energy Conservation . 10 1.2.3 Phase Space . 11 1.2.4 Newton's Vector Law . 11 1.2.5 Pendulum . 11 1.2.6 The Virial Theorem . 12 1.2.7 Use of Hamiltonian as a Differential equation . 13 1.2.8 Generic Structure of One-Degree-of-Freedom Systems . 13 1.3 Calculus of Variations . 14 1.3.1 Functionals and Variations . 14 1.3.2 Extremals . 15 1.3.3 Shortest Path . 15 1.3.4 Multiple Functions . 16 1.3.5 Symmetries and Conservation Laws . 17 1.4 The Action principle . 18 1.4.1 Coordinate-Invariance of the Action Principle . 18 1.4.2 Central Force Field Orbits . 19 1.4.3 Systems with Constraints and Lagrange Multipliers . 21 2 Differential forms 23 2.1 Notations . 23 2.2 Definitions . 24 2.2.1 The wedge product . 24 2.2.2 The exterior derivative . 24 2.2.3 The Pullback . 24 2.2.4 The Lie derivative . 24 2.2.5 The contraction . 25 2.3 Properties . 25 3 Hamiltonian systems 27 3.1 Introduction . 27 3.2 Legendre Transform . 27 3.2.1 Derivatives and Convexity . 28 3.2.2 Involution . 28 3.2.3 Total Differential of Legendre Transform . 29 3.2.4 Local Legendre Transformation . 29 3.2.5 Multivariable Case . 30 3.3 Canonical Equations . 30 5 6 CONTENTS 3.3.1 Hamiltonian Function . 30 3.3.2 Canonical Action Principle . 31 3.3.3 Previous Examples in Canonical Form . 32 3.4 The Poisson bracket . 33 3.4.1 Constants of motion . 34 3.4.2 The Poisson bracket in coordinate-free language . 34 4 Symplectic integrators 37 4.1 Introduction . 37 4.2 The Euler method . 37 4.3 Hamiltonian systems . 38 5 The Noether Theorem 41 5.1 Introduction . 41 5.2 Symmetries in Lagrangian mechanics . 41 5.2.1 Symmetries and the Lagrangian function . 42 5.2.2 Examples . 43 5.2.3 Generalized symmetries . 43 5.3 From the Lagrangian to the Hamiltonian formalism . 47 5.4 Symmetries in Hamiltonian mechanics . 48 5.4.1 Symplectic geometry . 49 5.4.2 The Kepler problem . 50 6 The Hamilton{Jacobi equation 51 6.1 Introduction . 51 6.2 The Hamilton{Jacobi equation . 51 6.3 The action as a function of endpoints . 52 6.4 Solving the Cauchy problem for the Hamilton{Jacobi equation . 54 6.5 Generating functions . 57 7 Introduction to Differentiable Manifolds 59 7.1 Introduction . 59 7.2 Manifolds . 59 7.3 Maps . 61 7.3.1 Submanifolds . 62 7.4 Topological manifolds . 62 7.5 Differentiable manifolds . 64 7.5.1 The tangent space . 64 7.5.2 The tangent bundle . 66 7.6 Vector bundles . 67 7.6.1 Constructions on vector bundles . 68 7.6.2 Differential forms . 69 7.7 Applications to mechanics . 69 7.7.1 The Noether 1-form . 69 7.7.2 The Legendre mapping . 70 7.7.3 The Liouville 1-form . 70 7.7.4 Symplectic geometry . 71 Appendices CONTENTS 7 Appendix A Topology and Derivations 75 A.1 Topology . 75 A.2 Derivations . 76 Appendix B Vector fields as derivations 79 Bibliography 80 8 CONTENTS Chapter 1 From Newton's Laws to Lagrange's equations 1.1 Introduction Classical mechanics is a very peculiar branch of physics. It used to be considered the sum total of our theoretical knowledge of the physical universe (Laplace's daemon, the Newtonian clockwork), but now it is known as an idealization, a toy model if you will. Classical Mechanics still describes the world pretty well in the range of validity, which is for example that of our everyday experience. So it is still an indispensable part of any physicist's or engineer's education. It is so useful because the more accurate theories that we know of (general relativity and quantum mechanics) make corrections to classical mechanics generally only in extreme situations (black holes, neutron stars, atomic structure, superconductivity, and so forth). Given that GR and QM are much more harder theory to use and apply it is no wonder that scientists will revert to classical mechanics whenever possible. So, what is classical mechanics? 1.2 Elements of Newtonian Mechanics In the title classical means that there are no quantum effects. The simplest mechanical system is a mass point, which is a single point moving in space that has a finite mass m attached to it. You can think of the matter field belonging to a mass point as a delta-function in space: an infinitely concentrated, featureless lump of matter. The equation of motion for the mass point comes from physics and is expressed by Newton's law: force = mass × accelaration; F = ma: Our first mass point system comes right at the beginning of mechanics, it is Newton's apple. 1.2.1 Newton's Apple Newton's apple is a mass point with mass m and vertical position z(t) at time t. Here z is a Cartesian coordinate pointing upwards. From physics it is known that the gravitational force on the apple is given by mg and points downwards. Here g is the gravity constant with typical value 9:81ms−1. Thus Newton's law for the apple is mz¨ = −mg () z¨ + g = 0; where the dot denotes a time derivative. We see that the mass of the apple does not affect its motion in the gravitational field. What is the solution to the equation given above? This is 9 10 CHAPTER 1. FROM NEWTON'S LAWS TO LAGRANGE'S EQUATIONS a second-order ODE in time, so the solution to the initial-value problem requires specifying two initial conditions. In our case these are given by the initial position z(0) and the initial velocity z_(0). Given these two numbers the solution to the above ODE is g z(t) = z(0) +z _(0)t − t2; 2 which is the equation for a parabola that you may recognize. It is worth reflecting about what we have done so far. Newton's law tells us how to evolve a mechanical system in time. More specifically, let us define the state of our system as the collection of variables that completely specifies the conditions of our system at a moment in time. This is a key definition in mechanics. In the present case we have that state = fposition, velocityg = fz; z_g because these were the initial conditions needed for our ODE. If our ODE is well posed for t 2 [0;T ] with some T > 0 then there is a unique map such that state(0) −! state(t) for all t 2 [0;T ]. Thus, in principle, the present state contains all the information needed to compute any future state; in other words the classical mechanical universe is deterministic and the future can in principle be predicted by solving a well-posed differential equation. 1.2.2 Energy Conservation To derive the energy conservation law from the solution of our ODE, we use a standard proce- dure: multiply the equation of motion by the velocityz _ and manipulate. This yields d z_2 z_z¨ + gz_ = 0 , + gz = 0; dt 2 which is a conservation law for the energy function z_2 H(z; z_) = + gz = const = E: 2 This function is defined up to an integration constant. The meaning of it is that.
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