Consider a particle of constant mass $m$ with Lagrangian
$$L (x, y, z, \dot{x}, \dot{y}, \dot{z}) = \frac{1}{2}m((\dot{x}-wy)^2+(\dot{y}+wx)+\dot{z}^2)$$
where $(x(t), y(t), z(t))$ is the location of the particle relative to an observer rotating with constant angular velocity ω around the z-axis, and dots denote derivatives with respect to time $t$.
(a) State whether $x$, $y$ or $z$ are cyclic coordinates and determine the corresponding constant(s) of motion.
(b) Using time translation symmetry of the Lagrangian, find another constant of motion for the particle.
(c) Use Hamilton’s principle of least action to determine the equations of motion for $x(t)$, $y(t)$ and $z(t)$.
(d) Use your answer to part (c) to show that
$$Q = x\dot{y} − y\dot{x} + ω(x^2+y^2)$$
is a constant of motion. Explain this result using a symmetry of the Lagrangian.
for (a) since only $z$ is a cyclic coordinate $\frac{dL}{dz}$=0
therefore its corresponding conservation law is
$$\frac{d}{dt}\frac{dL}{d\dot{z}}=0$$
$$=\frac{d}{dt}(m\dot{z})=0$$
I don't know what to do with $\frac{d}{dt}$ and would appreciate help with what to do from here as well as corrections on any mistakes I may have made so far, please add tags if you think they are relevant.