References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 2.5; Exercise 2.5.4.
The Hamiltonian is given by
where the velocities are expressed in terms of the positions and momenta . In this case, we start with the Lagrangian in terms of the centre of mass position and the relative position of mass 2 to mass 1.
where is the total mass and is the reduced mass.
There are potentially 6 velocity components and 6 coordinate components in the Lagrangian, but the 3 components of do not appear, which simplifies things a bit. To convert to a Hamiltonian, we need the momenta
The component of momentum of the centre of mass is
The other two components of the centre of mass velocity, and of the relative velocity, have a similar form, and in general we can write
In vector notation, this becomes
The Lagrangian thus becomes
The Hamiltonian is
Once we’ve got the Hamiltonian, we can apply Hamilton’s canonical equations to get the equations of motion.
Since does not appear in the Hamiltonian, we have
so the momentum of the centre of mass does not change, as expected.
For , we have
The first equation tells us nothing new, while the second is just Newton’s law for a central force: .