Classical Mechanics (Kibble & Berkshire) · Problem 14.8

The general theory of small oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · relativistic-hamiltonian

Problem

Relativistic Hamiltonian The relativistic Lagrangian for a particle of rest mass m0m_0 moving along the xx-axis under the potential field V(x)V(x) is given by L=m0c2(1−(1−x˙2c2)1/2)−V(x).L = m_0 c^2\left(1 - \left(1 - \frac{\dot{x}^2}{c^2}\right)^{1/2}\right) - V(x). Show that the corresponding Hamiltonian is given by H=m0c2(1+(pxm0c)2)1/2−m0c2+V(x),H = m_0 c^2\left(1 + \left(\frac{p_x}{m_0 c}\right)^2\right)^{1/2} - m_0 c^2 + V(x), where pxp_x is the generalised momentum conjugate to xx.

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