The action for a system with generalized coordinates, $q_{i}(t)$, for a time interval $\left[t_{1}, t_{2}\right]$ is given by

$$S=\int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot{q}_{i}\right) d t$$

where $L$ is the Lagrangian, and where the end point values $q_{i}\left(t_{1}\right)$ and $q_{i}\left(t_{2}\right)$ are fixed at specified values. Derive Lagrange's equations from the principle of least action by considering the variation of $S$ for all possible paths.

What is meant by the statement that a particular coordinate $q_{j}$ is ignorable? Show that there is an associated constant of the motion, to be specified in terms of $L$.

A particle of mass $m$ is constrained to move on the surface of a sphere of radius $a$ under a potential, $V(\theta)$, for which the Lagrangian is given by

$$L=\frac{m}{2} a^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)-V(\theta)$$

Identify an ignorable coordinate and find the associated constant of the motion, expressing it as a function of the generalized coordinates. Evaluate the quantity

$$H=\dot{q}_{i} \frac{\partial L}{\partial \dot{q}_{i}}-L$$

in terms of the same generalized coordinates, for this case. Is $H$ also a constant of the motion? If so, why?