Question #139497
Write hamiltonian for free particle in one dimensional.
1
Expert's answer
2020-10-25T18:31:41-0400

In general case,

H=pq˙LH = p \dot{q} - L

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

In case of free particle, there is no potential field, so L=T=mv22=mx˙22\displaystyle L= T = \frac{mv^2}{2} = \frac{m \dot{x}^2}{2}

p=mv=mx˙p = mv = m \dot{x}

q˙=x˙\dot{q} = \dot{x}

H=mx˙2mx˙22=mx˙22\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.


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