## The Lagrangian and the Classical Hamiltonian of a two-particle system

**Vanessa V. Destura**, MSU-IIT

##### Figure 1. A two-particle system

We consider two particles in a system, that is, and . The *relative coordinate* , is given by

We replace the positions and by the *center of gravity,*

Now, using equations and to solve for the expressions of and , we have

So,

In Classical Mechanics, the two-particle system is described by Lagrangian,

From equations and , we have

Hence, the Lagrangian in terms of and is given by

where (total mass) and (reduced mass).

In order to solve for the classical Hamiltonian, , we alter the expression of the reduced mass from the Lagrangian (equation ) into . So,

Now, we differentiate in terms of and to obtain the conjugate momenta of the variables and , that is,

We further expand by substituting the reduced mass, and . So,

where and are the conjugate momenta of the two particles.

Since, and , we have

So, substituting into we find

or

In this case, is the *total momentumĀ *of the system and is theĀ *relative momentum* of the two particles.

Thus, from the Lagrangian, we arrived to