Classical Dynamics(C)
CLASSICAL DYNAMICS (C)
Part IB Variational Principles is essential.
Review of Newtonian mechanics
- Newton’s second law. Motion of N particles under mutual interactions. Euclidean and Galilean symmetry. Conservation laws of momentum, angular momentum and energy. [2]
Lagrange’s equations
- Configuration space, generalized coordinates and velocities. Holonomic constraints. Lagrangian, Hamilton’s principle and Euler-Lagrange equations of motion. Examples: N particles with potential forces, planar and spherical pendulum, charged particle in a background electromagnetic field, purely kinetic Lagrangians and geodesics. Ignorable coordinates. Symmetries and Noether’s theorem. [5] Quadratic Lagrangians, oscillations, normal modes. [1]
Motion of a rigid body
- Kinematics of a rigid body. Angular momentum, kinetic energy, diagonalization of inertia tensor. Euler top, conserved angular momentum, Euler equations and their solution in terms of elliptic integrals. Lagrange top, steady precession, nutation. [6]
Hamilton’s equations
- Phase space. Hamiltonian and Hamilton’s equations. Simple examples. Poisson brackets, conserved quantities. Principle of least action. Liouville theorem. Action and angle variables for closed orbits in 2-D phase space. Adiabatic invariants (proof not required). Mention of completely integrable systems, and their action-angle variables. [7] Hamiltonian systems in nonlinear phase spaces, e.g. classical spin in a magnetic field. 2-D motion of ideal point vortices. *Connections between Lagrangian/Hamiltonian dynamics and quantum mechanics.* [3]
Appropriate books
- † L.D. Landau and E.M. Lifshitz Mechanics. Butterworth-Heinemann 2002
- F. Scheck Mechanics: from Newton’s laws to deterministic chaos. Springer 1999
- L.N. Hand and J.D. Finch Analytical Mechanics. CUP 1999
- H. Goldstein, C. Poole and J. Safko Classical Mechanics. Pearson 2002
- V.I. Arnold Mathematical methods of classical mechanics. Springer 1978
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