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Classical Dynamics
Beschreibung:
Since Lagrange laid the foundation of analytical dynamics some two centuries ago, the discipline has continued to evolve and develop, embracing the theories of Hamilton and Jacobi, Einstein's relativity theory and advanced theories of classical mechanics.
This text proposes to give graduate students in science and engineering a strong background in the more abstract and intellectually satisfying areas of dynamical theory. It is assumed that students are familiar with the principles of vectorial mechanics and have some facility in the use of this theory for analysis of systems of particles and for rigid-body rotation in two and three dimensions.
After a concise review of basic concepts in Chapter 1, the author proceeds from Lagrange's and Hamilton's equations to Hamilton-Jacobi theory and canonical transformations. Topics include d'Alembert's principle and the idea of virtual work, the derivation of Langrange's equation of motion, special applications of Lagrange's equations, Hamilton's equations, the Hamilton-Jacobi theory, canonical transformations and an introduction to relativity.
Problems included at the end of each chapter will help the student greatly in solidifying his grasp of the principal concepts of classical dynamics. An annotated bibliography at the end of each chapter, a detailed table of contents and index, and selected end-of-chapter answers complete this highly instructive text.
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