On a $2N$ dimensional phase space, consider the set of functions which are purely quadratic in the generalised coordinates and momenta.

$($a$)$ Show that the Poisson bracket of any two members of the set is again in the set.

$($b$)$ Let the Hamiltonian be given by$,$

$H=\frac{1}{2}{\displaystyle \underset{i=1}{\overset{N}{}}}(p_i^2+q_i^2)$

which functions in the above set constitute infinitesimal symmetries of the system? How many independent symmetrics are there?