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Classical Mechanics Notes on Classical Mechanics Thierry Laurens Last updated: August 8, 2021 Contents I Newtonian Mechanics 4 1 Newton's equations 5 1.1 Empirical assumptions . .5 1.2 Energy . .7 1.3 Conservative systems . 11 1.4 Nonconservative systems . 13 1.5 Reversible systems . 17 1.6 Linear momentum . 19 1.7 Angular momentum . 21 1.8 Exercises . 23 2 Examples 30 2.1 One degree of freedom . 30 2.2 Central fields . 33 2.3 Periodic orbits . 37 2.4 Kepler's problem . 39 2.5 Virial theorem . 42 2.6 Exercises . 43 II Lagrangian Mechanics 49 3 Euler{Lagrange equations 50 3.1 Principle of least action . 50 3.2 Conservative systems . 55 3.3 Nonconservative systems . 58 3.4 Equivalence to Newton's equations . 60 3.5 Momentum and conservation . 61 3.6 Noether's theorem . 63 3.7 Exercises . 65 4 Constraints 72 4.1 D'Alembert{Lagrange principle . 72 4.2 Gauss' principle of least constraint . 74 1 CONTENTS 2 4.3 Integrable constraints . 77 4.4 Non-holonomic constraints . 77 4.5 Exercises . 79 5 Hamilton{Jacobi equation 83 5.1 Hamilton{Jacobi equation . 83 5.2 Separation of variables . 86 5.3 Conditionally periodic motion . 87 5.4 Geometric optics analogy . 89 5.5 Exercises . 91 III Hamiltonian Mechanics 95 6 Hamilton's equations 96 6.1 Hamilton's equations . 96 6.2 Legendre transformation . 98 6.3 Liouville's theorem . 101 6.4 Poisson bracket . 103 6.5 Canonical transformations . 106 6.6 Infinitesimal canonical transformations . 109 6.7 Canonical variables . 111 6.8 Exercises . 112 7 Symplectic geometry 115 7.1 Symplectic structure . 115 7.2 Hamiltonian vector fields . 116 7.3 Integral invariants . 118 7.4 Poisson bracket . 120 7.5 Time-dependent systems . 121 7.6 Locally Hamiltonian vector fields . 123 7.7 Exercises . 124 8 Contact geometry 127 8.1 Contact structure . 127 8.2 Hamiltonian vector fields . 129 8.3 Dynamics . 130 8.4 Contact transformations . 131 8.5 Time-dependent systems . 134 8.6 Exercises . 136 Bibliography 139 Preface Although many mathematical tools originated in the study of classical mechan- ics, their connection to the foundational physical example is often neglected. The objective of these expository notes is to present the theory of classical me- chanics, develop the concomitant mathematical concepts, and to see the physical systems as examples of the math. These notes include a self-contained presentation of the central physics the- ory of mechanics designed for mathematicians. However, we exclude many topics that commonly appear in a first-year graduate course on mechanics, e.g. rigid bodies. For a thorough study of the theory, techniques, and examples of classical mechanics, see the standard physics texts [Arn89, Gol51, LL76]. Alongside the physics, we will develop the relevant concepts and tools from the mathematical study of differential equations. Note that this does not in- clude mathematical methods used in mechanics, which can be found in the text [AKN06]. We will assume the reader possesses a beginner graduate or advanced un- dergraduate mathematical understanding|specifically, multivariate calculus in Part I, manifolds and tangent bundles in Part II, and cotangent bundles in Part III. Unless otherwise stated, we will always assume all given functions are smooth (i.e. infinitely differentiable). Although many of the results hold under weaker assumptions, we choose to focus on the proofs in the smooth case. 3 4 Part I Newtonian Mechanics The Newtonian perspective is the most fundamental interpretation of the motion of a mechanical system. Newton's principle of determinacy is an ex- perimental observation, the mathematical phrasing of which yields Newton's equations. Together with Galileo's principle of relativity, these form the axioms of classical mechanics. We will also see how additional empirical assumptions (conservative sys- tem, closed system, and the law of action and reaction) yield both physically observable effects (conservation laws and symmetries) and mathematical con- sequences (special properties for Newton's equations and its solutions). This study will require only multivariable calculus and will reveal some of the basic mathematical tools of ODE theory (conserved quantities, Lyapunov functions, and time-reversibility). Although Newtonian mechanics are naturally phrased on Euclidean space, we will introduce and study the fundamental objects (configuration and phase space) which we will later generalize to include a broader class of systems. Chapter 1 Newton's equations We build the physical theory starting from essential empirical observations. We then define and prove fundamental mathematical properties of conservative, Lyapunov, and gradient systems. The physics material is based on [Arn89, Ch. 1], [AKN06, Ch. 1], and [Gol51, Ch. 1], and the math material is based on [Str15, Ch. 5{7] and [JS07, Ch. 10]. 1.1 Empirical assumptions Classical mechanics is the study of the motion of particles in a Euclidean space Rd.A particle (or point mass) consists of two pieces of information: the body's mass m > 0 and its position x 2 Rd. This model for physical objects neglects the object's spatial dimensions. Given a system of N particles with d positions xi 2 R , the collection of all positions x = (x1;:::; xN ) constitute the configuration space Rd × · · · × Rd = RNd, whose dimension Nd is the system's degrees of freedom. The evolution of a system is described by the particles' trajectories, N d maps xi : I ! R for I ⊂ R an interval, which together constitute the motion of the system. In order to determine the system's evolution we will use the velocity x_ i , acceleration x¨i, and momentum pi = mix_ i (1.1) _ df of the ith particle. (We use the notation f = dt for time derivatives.) The positions and momenta together span the phase space RNd × RNd, whose dimension 2Nd is always twice the degrees of freedom. We will refer to the tra- jectories of the system plotted in phase space as the system's phase portrait. Newton's principle of determinacy is the experimental observation that the initial state of a Newtonian system|all the positions x(t0) and velocities x_ (t0) at some moment t0 2 R in time|uniquely determines the system's mo- tion. We formulate this mathematically as follows. 5 CHAPTER 1. NEWTON'S EQUATIONS 6 Definition 1.1 (Newton's principle of determinacy). A Newtonian system Nd Nd Nd is given by N particles and a function F :(R r ∆) × R × R ! R , called the force, such that p_ = F(x; x_ ; t): (1.2) S Here, ∆ = i<jfxi = xjg is the union of diagonals, so that we do not consider two particles occupying the same position. Equation (1.2) is called Newton's equation (or Newton's second law). Unless otherwise noted, we will always assume the particle masses mi are all constant. Equation (1.2) then takes the form mix¨i = Fi(x; x_ ; t); i = 1; : : : ; N: (1.3) Historically, one would experimentally observe that a particle's acceleration is inversely proportional to its mass in order for momentum to merit it's own defini- tion (1.1). Treating the velocity independently of position is a common practice in math to reduce a system of ODEs to first-order, so that instead of a system of N second-order differential equations on configuration space we consider a system of 2N first-order differential equations on phase space, equations (1.1) and (1.2). Unless otherwise stated, we will always assume that the force F is smooth Nd Nd (infinitely differentiable) on (R r∆)×R ×R. The theorem of existence and uniqueness for ODEs then tells us there always exists a unique solution to the system of differential equations (1.2) for any initial state and that the solution is also smooth. Even with this assumption, it may be that the solution exists for a finite amount of time. However, from experimental observation we expect that for any naturally occurring system that the the interval can be extended to all of R, and so we will often work in a regime where the solutions to our mathematical model eq. (1.3) exist for all time. We will formulate one more mathematical assumption: that the physical laws governing the system's motion is independent of the choice of coordinates d d d and origin we impose on R . A transformation R ×Rt ! R ×Rt is Galilean provided that it is an affine transformation (a linear transformation and a trans- lation) that preserves time intervals and for any fixed t 2 R is an isometry of Rd; that is, if we write g(x; t) = (x0; t0) then for each t 2 R the function x0 is an orthogonal transformation and a translation. It is straightforward to show that the set of Galilean transformations forms a group under function composition. Example 1.2. The following are all Galilean transformations: d (1) Translations: g1(x; t) = (x + x0; t + t0) for some fixed x0 2 R , t0 2 R (2) Rotations and reflections: g2(x:t) = (Ax; t) for some fixed orthogonal transformation A 2 O(d) (3) Uniform motion with constant velocity: g3(x; t) = (x + vt; t) for some fixed velocity v 2 Rd. CHAPTER 1. NEWTON'S EQUATIONS 7 In fact, these examples generate the entire Galilean group (cf. Exercise 1.3). Galileo's principle of relativity is the experimental observation that for an isolated system there exists a reference frame|a choice of origin and coordinate axes for Rd|that is invariant under any Galilean transformation. Such a reference frame is called inertial, and the principle also asserts that all coordinate systems in uniform motion with constant velocity with respect to an inertial frame must also be inertial.
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