Q1.
(a) Let \mathbfx ∈ \mathbbRn × \mathbbRn. Define the Lagrangian \mathcalL(\mathbfx, \dot{\mathbfx}, t) and the Hamiltonian \mathcalH(\mathbfx, \mathbfp, t). Show that (d\mathbfr)/(dt) = (1)/(2π)(d\mathbfr)/(dt))2 is not a valid equation of motion.
(b) Consider the equation (d\mathbfr)/(dt) = (1)/(2π)(d\mathbfr)/(dt))2. Discuss the physical implications of this equation and its relation to conservation laws.
(c) Explain the concept of phase space and its significance in classical mechanics. Discuss the properties of trajectories in phase space.
(d) Derive the Hamilton's equations of motion from the Lagrangian formulation.
(e) Discuss the canonical transformations and their properties. Provide an example of a canonical transformation.
