Is f(q1, p1) an Integral of Motion in This Hamiltonian System?

[Oreith]

I am stuck on a proof. I need to show that if a Hamiltonian only depends on q₁ and p₁ though a function $f(q_1,p_1)$, that is; $$H(f(q_1, p_1), q_2, p_2, q_3, p_3, ... q_n, p_n)$$ then $$f(q_1, p_1)$$ is an integral of motion.

My attempt at a solution is as rather simplistic but I'm stuck making the final jump. Since the Hamiltonian is not directly a function of q₁ and p₁ it must commute;

$$[q_1, H] = [p_1, H] = 0$$

so q₁ and p₁ are integrals of motion. How do i then say that a function of those two variables is also an integral of motion, it feels intuitive but I cannot figure out how to write it down.

 

[Orodruin]

Have you tried simply computing the Poisson bracket between $f$ and $H$?

 

[Oreith]

I have now, thanks =)