Solution
= Solution
Time and azimuthal invariance give energy
$$
E=\frac{7m}{10}(\dot r^2\csc^2\alpha+r^2\dot\phi^2)+mgr\cot\alpha
$$
and <angular momentum> $J=(7m/5)r^2\dot\phi$. They represent mechanical energy and vertical <angular momentum>.
= Solution
Time and azimuthal invariance give energy
$$
E=\frac{7m}{10}(\dot r^2\csc^2\alpha+r^2\dot\phi^2)+mgr\cot\alpha
$$
and <angular momentum> $J=(7m/5)r^2\dot\phi$. They represent mechanical energy and vertical <angular momentum>.