If we consider the Hamiltonian for the simple harmonic oscillator given by,
$$H(p,x) = \frac{p^2}{2m}+\frac{kx^2}{2}$$
where $m$ is the mass, $k$ is the stiffness and $p$ is the momentum, then the equations of motion for the oscillator can be written as,
$$\frac{dp}{dt}= -\frac{\partial H}{\partial x},\,\,\,\,\,\,\,\,\,\frac{dx}{dt}=\frac{\partial H}{\partial p}$$
How can one express these equations of motion in the Lagrange subsidiary form;
$$\frac{dp}{P(p,x,t)} =\frac{dx}{Q(p,x,t)} = \frac{dt}{R(p,x,t)} $$
I'd like to know the benefits of expressing the equations of motion that way, but in any case we can play with the original ones a little.
$$\frac{dp}{dt}= -\frac{\partial H}{\partial x},\,\,\,\,\,\,\,\,\,\frac{dx}{dt}=\frac{\partial H}{\partial p}$$
Leading to
$$\frac{dp}{dt}= -kx,\,\,\,\,\,\,\,\,\,\frac{dx}{dt}=\frac{p}{m}$$
$$\frac{dp}{-kx}= dt,\,\,\,\,\,\,\,\,\,\frac{dx}{p/m}=dt$$
And
$$\frac{dp}{-kx} =\frac{dx}{p/m} = \frac{dt}{1}$$