Classical Dynamics

Preface 1. Introductory concepts 1.1 The Mechanical System. Equations of motion. Units 1.2 Generalized Coordinates. Degrees of freedom. Generalized Coordinates. Configuration space. Example. 1.3 Constraints. Holonomic constraints. Nonholonomic constraints. Unilateral constraints. Example. 1.4 Virtual Work. Virtual displacement. Virtual work. Principle of virtual work. D'Alembert's principle. Generalized force. Examples. 1.5 Energy and Momentum. Potential energy. Work and kinetic energy. Conservation of energy. Equilibrium and stability. Kinetic energy of a system. Angular momentum. Generalized momentum. Example. 2. Lagrange's Equations 2.1 Derivation of Lagrange's Equations. Kinetic energy. Lagrange's equations. Form of the equations of motion. Nonholonomic systems. 2.2 Examples. Spherical pendulum. Double pendulum. Lagrange multipliers and constraint forces. Particle in whirling tube. Particle with moving support. Rheonomic constrained system. 2.3 Integrals of the Motion. Ignorable coordinates. Example--the Kepler problem. Routhian function. Conservative systems. Natural systems. Liouville's system. Examples. 2.4 Small Oscillations. Equations of motion. Natural modes. Principal coordinates. Orthogonality. Repeated roots. Initial conditions. Example. 3. Special applications of Lagrange's Equations 3.1 Rayleigh's Dissipation function 3.2 Impulsive Motion. Impulse and momentum. Lagrangian method. Ordinary constraints. Impulsive constraints. Energy considerations. Quasi-coordinates. Examples. 3.3 Gyroscopic systems. Gyroscopic forces. Small motions. Gyroscopic stability. Examples. 3.4 Velocity-Dependent Potentials. Electromagnetic forces. Gyroscopic forces. Example. 4. Hamilton's Equations 4.1 Hamilton's Principle. Stationary values of a function. Constrained stationary values. Stationary value of a definite integral. Example--the brachistochrone problem Example--geodesic path. Case of n dependent variables. Hamilton's principle. Nonholonomic systems. Multiplier rule. 4.2 Hamilton's Equations. Derivation of Hamilton's equations. The form of the Hamiltonian function. Legendre transformation. Examples. 4.3 Other Variational Principles. Modified Hamilton's principle. Principle of least action. Example. 4.4 Phase Space. Trajectories. Extended phase space. Liouville's theorem. 5. Hamilton-Jacobi Theory 5.1 Hamilton's Principal Function. The canonical integral. Pfaffian differential forms. 5.2 The Hamilton-Jacobi Equation. Jacobi's theorem. Conservative systems and ignorable coordinates. Examples. 5.3 Separability. Liouville's system. Stäckel's theorem. Example. 6. Canonical Transformations 6.1 Differential Forms and Generating Functions. Canonical transformations. Principal forms of generating functions. Further comments on the Hamilton-Jacobi method. Examples. 6.2 Special Transformations. Some simple transformations. Homogeneous canonical transformations. Point transformations. Momentum transformations. Examples. 6.3 Lagrange and Poisson Brackets. Lagrange brackets. Poisson brackets. The bilinear covariant. Example. 6.4 More General Transformations. Necessary conditions. Time transformations. Examples. 6.5 Matrix Foundations. Hamilton's equations. Symplectic matrices. Example. 6.6 Further Topics. Infinitesimal canonical transformations. Liouville's theorem. Integral invariants. 7. Introduction to Relativity 7.1 Introduction. Galilean transformations. Maxwell's equations. The Ether theory. The principle of relativity. 7.2 Relativistic Kinematics. The Lorentz transformation equations. Events and simultaneity. Example--Einstein's train. Time dilation. Longitudinal contraction. The invariant interval. Proper time and proper distance. The world line. Example--the twin paradox. Addition of velocities. The relativistic Doppler effect. Examples. 7.3 Relativistic dynamics. Momentum. Energy. The momentum-energy four-vector. Force. Conservation of energy. Mass and energy. Example--inelastic collision. The principle of equivalence. Lagrangian and Hamiltonian formulations. 7.4 Accelerated Systems. Rocket with constant acceleration. Example. Rocket with constant thrust. Answers to Selected Problems. Index