Question

Write hamiltonian for free particle in one dimensional.

Answer 1

In general case,

$H = p \dot{q} - L$

where p is generalized momentum and $\dot{q}$ is generalized velocity and $L$ is Lagrange function or Lagrangian. In classical mechanics, Lagrange function $L = T - U$ , where T is kinetic energy and U is potential energy.

In case of free particle, there is no potential field, so $\displaystyle L= T = \frac{mv^2}{2} = \frac{m \dot{x}^2}{2}$

$p = mv = m \dot{x}$

$\dot{q} = \dot{x}$

$\displaystyle H = m \dot{x}^2 - \frac{m \dot{x}^2}{2} = \frac{m \dot{x}^2}{2}$

As we see here, Hamiltonian for free particle is coincident with its kinetic energy.