Vol 1 - L D Landau, E.M. Lifshitz - Mechanics · Problem 40.3
The Canonical Equations
Physics — Landau & Lifshitz · hamiltons-equations
Problem
Find the Hamiltonian for a system comprising one particle of mass and particles each of mass , excluding the motion of the centre of mass (see §13, Problem).Intuitive explanation
The key insight is that the total Hamiltonian includes kinetic energy of the center-of-mass motion, which we must separate out and remove. Transform to the center-of-mass frame using relative coordinates: express momenta in terms of the center-of-mass momentum (which we exclude) and internal momenta. The resulting Hamiltonian will depend only on the relative positions and conjugate momenta, giving you an effective system that describes only the internal dynamics.
Intuitive explanation
The key insight is that the total Hamiltonian includes kinetic energy of the center-of-mass motion, which we must separate out and remove. Transform to the center-of-mass frame using relative coordinates: express momenta in terms of the center-of-mass momentum (which we exclude) and internal momenta. The resulting Hamiltonian will depend only on the relative positions and conjugate momenta, giving you an effective system that describes only the internal dynamics.