Classical Dynamics

Chapter Two:

Lagrangian Formulation of Mechanics

Constraints and configuration manifolds

1. constraints: constraint equations, constraints and work
2. generalized coordinates
3. examples of configuration manifolds: the finite line, the circle, the plane, the two-sphere S², the double pendulum, discussion

Lagrange's equations

1. derivation of Lagrange's equations
2. transformations of Lagrangians: equivalent Lagrangians, coordinate independence, Hessian condition
3. conservation of energy
4. charged particle in an electromagnetic field: the Lagrangian, a time-dependent coordinate transformation

Central force motion

1. the general central force problem: statement of the problem; reduced mass, reduction to two freedoms, the equivalent one-dimensional problem
2. the Kepler problem
3. Bertrand's theorem

The tangent bundle TQ

1. dynamics on TQ: velocities do not lie in Q, tangent spaces and the tangent bundle, Lagrange's equations and trajectories on TQ
2. TQ as differentiable manifold: differentiable manifolds, tangent spaces and tangent bundles, application to Lagrange's equations

Problems