2018-78.md

course	course_year	question_number	tags	title	year
Variational Principles	IB	78	IB 2018 Variational Principles	Paper 3, Section I, B	2018

For a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is

$$L=\frac{1}{2} \dot{\theta}^{2}+\frac{1}{2} \sin ^{2} \theta \dot{\phi}^{2}$$

Determine the equations of motion. Show that $l=\sin ^{2} \theta \dot{\phi}$ is a conserved quantity, and use this result to simplify the equation of motion for $\theta$. Deduce that

$$h=\dot{\theta}^{2}+\frac{l^{2}}{\sin ^{2} \theta}$$

is a conserved quantity. What is the interpretation of $h$ ?