Equation of motion in harmonic oscillator hamiltonian

[oristo42]

See attached photo please.

So, I don't get how equations of motion derived. Why is it that x dot is partial derivative of H in term of p but p dot is negative partial derivative of H in term of x.

 

[Orodruin]

So, I don't get how equations of motion derived. Why is it that x dot is partial derivative of H in term of p but p dot is negative partial derivative of H in term of x.

This follows from the definition of the Poisson bracket and is the basic definition of equations of motion in Hamiltonian mechanics. Any textbook covering Hamiltonian mechanics should tell you this at most 10 pages after starting the discussion on the subject.

The phase space evolution of a system in canonical coordinates is given by
$$
\dot x^i =\{x^i,H\},\quad \dot p_j = \{p_j, H\}.
$$
Indeed, for any time-independent function $f$ on phase space
$$
\dot f = \{f,H\}.
$$