Consider an $n$-dimensional dynamical system with generalized coordinates and momenta $\left(q_{i}, p_{i}\right), i=1,2, \ldots, n$.

(a) Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right)$ and $g\left(q_{i}, p_{i}, t\right)$.

(b) Assuming Hamilton's equations of motion, prove that if a function $G\left(q_{i}, p_{i}\right)$ Poisson commutes with the Hamiltonian, that is $\{G, H\}=0$, then $G$ is a constant of the motion.

(c) Assume that $q_{j}$ is an ignorable coordinate, that is the Hamiltonian does not depend on it explicitly. Using the formalism of Poisson brackets prove that the conjugate momentum $p_{j}$ is conserved.