Classical Mechanics (Kibble & Berkshire) · Problem 12.16

The calculus of variations and Hamilton's principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · lagrangian-mechanics-relativistic-lagrangian

Problem

Relativistic Lagrangian The relativistic Lagrangian for a particle of rest mass

m_0

moving along the

x

-axis under the simple harmonic potential field

V = \tfrac{1}{2}m_0\Omega^2 x^2

is given by

L = m_0 c^2 \left(1 - \left(1 - \frac{\dot{x}^2}{c^2}\right)^{1/2}\right) - \tfrac{1}{2}m_0\Omega^2 x^2.

Obtain the energy integral for this system and show that the period of oscillations of amplitude

a

is given by

\tau = \frac{4}{\Omega}\int_0^{\pi/2} \frac{1 + \tfrac{1}{2}\epsilon^2\cos^2\theta}{\left(1 + \tfrac{1}{4}\epsilon^2\cos^2\theta\right)^{1/2}}\, d\theta,

where the dimensionless parameter

\epsilon = \Omega a/c

. Deduce that

\tau = \frac{2\pi}{\Omega}\left[1 + \tfrac{3}{16}\epsilon^2 + O\left(\epsilon^4\right)\right],

when

\epsilon

is small.