Classical Mechanics 220 Eric D'Hoker Department of Physics and Astronomy, UCLA Los Angeles, CA 90095, USA 9 November 2015 1 Contents 1 Review of Newtonian Mechanics 5 1.1 Some History . .5 1.2 Newton's laws . .6 1.3 Comments on Newton's laws . .7 1.4 Work . .8 1.5 Dissipative forces . .9 1.6 Conservative forces . 10 1.7 Velocity dependent conservative forces . 12 1.8 Charged particle in the presence of electro-magnetic fields . 14 1.9 Physical relevance of conservative forces . 15 1.10 Appendix 1: Solving the zero-curl equation . 15 2 Lagrangian Formulation of Mechanics 17 2.1 The Euler-Lagrange equations in general coordinates . 17 2.2 The action principle . 20 2.3 Variational calculus . 21 2.4 Euler-Lagrange equations from the action principle . 23 2.5 Equivalent Lagrangians . 24 2.6 Symmetry transformations and conservation laws . 24 2.7 General symmetry transformations . 26 2.8 Noether's Theorem . 28 2.9 Examples of symmetries and conserved charges . 29 2.10 Systems with constraints . 31 2.11 Holonomic versus non-holonomic constrains . 33 2.12 Lagrangian formulation for holonomic constraints . 36 2.13 Lagrangian formulation for some non-holonomic constraints . 38 2.14 Examples . 38 3 Quadratic Systems: Small Oscillations 41 3.1 Equilibrium points . 41 3.2 Mechanical stability of equilibrium points . 42 3.3 Small oscillations near a general solution . 44 3.4 Magnetic stabilization . 45 3.5 Lagrange Points . 46 3.6 Stability near the non-colinear Lagrange points . 49 2 4 Hamiltonian Formulation of Mechanics 51 4.1 Canonical position and momentum variables . 51 4.2 Derivation of the Hamilton's equations . 52 4.3 Some Examples of Hamiltonian formulation . 53 4.4 Variational Formulation of Hamilton's equations . 54 4.5 Poisson Brackets and symplectic structure . 55 4.6 Time evolution in terms of Poisson brackets . 56 4.7 Canonical transformations . 57 4.8 Symmetries and Noether's Theorem . 59 4.9 Poisson's Theorem . 59 4.10 Noether charge reproduces the symmetry transformation . 60 5 Lie groups and Lie algebras 62 5.1 Definition of a group . 62 5.2 Matrix multiplication groups . 63 5.3 Orthonormal frames and parametrization of SO(N).............. 65 5.4 Three-dimensional rotations and Euler angles . 67 5.5 Definition of a Lie group . 68 5.6 Definition of a Lie algebra . 68 5.7 Relating Lie groups and Lie algebras . 69 5.8 Symmetries of the degenerate harmonic oscillator . 70 5.9 The relation between the Lie groups SU(2) and SO(3) . 73 6 Motion of Rigid Bodies 74 6.1 Inertial and body-fixed frames . 74 6.2 Kinetic energy of a rigid body . 75 6.3 Angular momentum of a rigid body . 76 6.4 Changing frames . 77 6.5 Euler-Lagrange equations for a freely rotating rigid body . 78 6.6 Relation to a problem of geodesics on SO(N)................. 79 6.7 Solution for the maximally symmetric rigid body . 80 6.8 The three-dimensional rigid body in terms of Euler angles . 80 6.9 Euler equations . 82 6.10 Poinsot's solution to Euler's equations . 82 7 Special Relativity 84 7.1 Basic Postulates . 84 7.2 Lorentz vector and tensor notation . 86 7.3 General Lorentz vectors and tensors . 88 3 7.4 Relativistic invariance of the wave equation . 90 7.5 Relativistic invariance of Maxwell equations . 91 7.6 Relativistic kinematics . 94 7.7 Relativistic dynamics . 96 7.8 Lagrangian for a massive relativistic particle . 97 7.9 Particle collider versus fixed target experiments . 98 7.10 A physical application of time dilation . 99 8 Fluid Mechanics 100 8.1 Conservation of mass . 101 8.2 Conservation of momentum and Newton's third law . 101 8.3 Euler's equations . 103 8.4 Ideal fluids . 103 8.5 Viscous fluids . 110 9 Completely integrable systems 114 9.1 Criteria for integrability, and action-angle variables . 114 9.2 Standard examples of completely integrable systems . 114 9.3 More sophisticated integrable systems . 115 9.4 Elliptic functions . 116 9.5 Solution by Lax pair . 117 9.6 The Korteweg de Vries (or KdV) equation . 119 9.7 The KdV soliton . 119 9.8 Integrability of KdV by Lax pair . 121 10 Global properties of Hamiltonian mechanics 123 10.1 The Poincar´eRecurrence Theorem . 123 10.2 Definition of a manifold . 124 10.3 Examples . 125 10.4 Maps between manifolds . 126 10.5 Vector fields and tangent space . 127 10.6 Poisson brackets and Hamiltonian flows . 129 10.7 Stokes's theorem and grad-curl-div formulas . 130 10.8 Differential forms: informal definition . 131 10.9 Structure relations of the exterior differential algebra . 132 10.10Integration and Stokes's Theorem on forms . 134 10.11Frobenius Theorem for Pfaffian systems . 134 4 1 Review of Newtonian Mechanics A basic assumption of classical mechanics is that the system under consideration can be understood in terms of a fixed number Np of point-like objects. Each such object is labeled by an integer n = 1; ;N , has a mass m > 0, and may be characterized by a position ··· p n xn(t). The positions constitute the dynamical degrees of freedom of the system. On these objects and/or between them, certain forces may act, such as those due to gravity and electro-magnetism. The goal of classical mechanics is to determine the time-evolution of the position xn(t) due to the forces acting on body n, given a suitable set of initial conditions. A few comments are in order. The point-like nature of the objects described above is often the result of an approximation. For example, a planet may be described as a point-like object when studying its revolution around the sun. But its full volume and shape must be taken into account if we plan to send a satellite to its surface, and the planet can then no longer be approximated by a point-like object. In particular, the planet will rotate as an extended body does. This extended body may be understood in terms of smaller bodies which, in turn, may be treated as point-like. A point-like object is often referred to as a material point or a particle, even though its size may be that of a planet or a star. In contrast with quantum mechanics, classical mechanics allows simultaneous specifica- tion of both the position and the velocity (or momentum) for each of its particles. In contrast with quantum field theory, classical mechanics assumes that the number of particles is fixed, with fixed masses. In contrast with statistical mechanics, classical mechanics assumes that the positions and velocities of all particles can (in principle) be known to arbitrary accuracy. 1.1 Some History Historically, one of the greatest difficulties that needed to be overcome was to observe and describe the motion of bodies in the absence of any forces. Friction on the ground and in the air could not easily be reduced with the tools available prior to the Renaissance. It is the motion of the planets which would produce the first reliable laws of mechanics. Based on the accurate astronomical observations which Tycho Brahe (1546-1601) made with the naked eye on the positions of various planets (especially Mars), Johannes Kepler (1571- 1630) proposed his quantitative and precise mathematical laws of planetary motion. Galileo Galilei (1564-1642) investigated the motion of bodies on Earth, how they fall, how they roll on inclined planes, and how they swing in a pendulum. He demonstrated with the help of such experiments that bodies with different masses fall to earth at the same rate (ignoring air friction), and deduced the correct (quadratic) mathematical relation between height and elapsed time during such falls. He may not have been the first one to derive such laws, but 5 Galileo formulated the results in clear quantitative mathematical laws. Galileo proposed that a body in uniform motion will stay so unless acted upon by a force, and he was probably the first to do so. Of course, some care is needed in stating this law precisely as the appearance of uniform motion may change when our reference frame in which we make the observation is changed. In a so-called inertial frame, which we shall denote by , the motion of a body on which no forces act is along a straight line at constant velocity R and constant direction. A frame 0 which moves with constant velocity with respect to R R is then also inertial. But a frame 00 which accelerates with respect to is not inertial, as R R a particle in uniform motion now sweeps out a parabolic figure. Galileo stated, for the first time, that the laws of mechanics should be the same in different inertial frames, a property that is referred to as the principle of Galilean Relativity, and which we shall discuss later. Isaac Newton (1642-1727) developed the mathematics of differential and integral calculus which is ultimately needed for the complete formulation of the laws of mechanics. These laws form the foundation of mechanics, and were laid out in his Philosophae Naturalis Principia Mathematica, or simply the Principia, published in 1687. Remarkably, the mathematics used in the Principia is grounded in classical Greek geometry, supplemented with methods borrowed from infinitesimal calculus. Apparently, Newton believed that a formulation in terms of Greek geometry would enjoy more solid logical foundations than a formulation in terms of Descartes analytic geometry and his own calculus..