Corinaldesi E. Classical Mechanics for Physics Graduate Students

World Scientific Publishing, Singapore, 1998, 286 pages, ISBN10: 9810236255This book is intended for first year physics graduate students who wish to learn about analytical mechanics. Lagrangians and Hamiltonians are extensively treated following chapters where particle motion, oscillations, coordinate systems, and rigid bodies are dealt with in far greater detail than in most undergraduate textbooks. Perturbation theory, relativistic mechanics, and two case studies of continuous systems are presented. Each subject is approached at progressively higher levels of abstraction. Lagrangians and Hamiltonians are first presented in an inductive way, leading up to general proofs. Hamiltonian mechanics is expressed in Cartan's notation not too early; there is a self-contained account of the traditional formulationNumerous problems with detailed solutions are provided. Graduate students studying for the qualifying examination will find them very useful.Motion in phase space
Motion of a particle in one dimension
Flow in phase space
The action integral
The Maupertuis principle
Thetime
Fermat’s principle
Chapter 1 problemsExamples of particle motion
Central forces
Circular and quasi-circular orbits
Isotropic harmonic oscillator
The Kepler problem
The L-R-L vector
Open Kepler-Rutherford orbits
Integrability
Chapter 2 problemsFixed points. Oscillations. Chaos
Fixed points
Small oscillations
Parametric resonance
Periodically jerked oscillator
Discrete maps, bifurcation, chaos
Chapter 3 problemsCoordinate systems
Translations and rotations
Some kinematics
Fictitious forces
Chapter 4 problemsRigid bodies
Angular momentum
Euler’s equations
Euler angles
Spinning top
Regular precession of top
Sleeping top
Irregular precessions of top
Gyrocompass
Tilted disk rolling in a circle
Chapter 5 problemsLagrangians
Heuristic introduction
Velocity-dependent forces
Equivalent Lagrangians
Invariance of Lagrange equations
Proofs of the Lagrange equations
Constraints
Invariance of L and constants of motion
Invariance of L up to total time derivative
Noether’s theorem
Chapter 6 problemsHamiltonians
First look . 1
First look . 2
H as Legendre transform of L
Liouville’s theorem
Cartan’s vectors and forms
Lie derivatives
Time-independent canonical transformations
Generating functions
Lagrange and Poisson brackets
Hamilton-Jacobi equation revisited
Time-deDendent canonical transformations
Time-dependent Hamilton-Jacobi equation
Stokes’ theorem and some proofs
Hamiltonian flow as a Lie-Cartan group
Poincare-Cartan integral invariant
Poincare’s invariants
Chapter 7 problemsAction-angle variables
One dimension
Multiply periodic systems
Integrability, non-integrability, chaos
Adiabatic invariants
Outline of rigorous theory
Chapter 8 problemsPerturbation theory
The operator ll
Perturbation expansions
Perturbed periodic systems
Chapter 9 problemsRelativistic dynamics
Lorentz transformations
Dynamics of a particle
Formulary of Lorentz transformations
The spinor connection
The spin
Thomas precession
Charged particle in static em field
Magnetic moment in static em field
Lagrangian and Hamiltonian
Chapter 10 problemsContinuous systems
Uniform string
Ideal fluids
Chapter 11 problems