I was reading up on Hamiltonian Mechanics and came across the following:
If a generalized coordinate $q_j$ doesn't explicitly occur in the Hamiltonian, then $p_j$ is a constant of motion (meaning, a constant, independent of time for a true dynamical motion). $q_j$ then becomes a linear function of time. Such a coordinate $q_j$ is called a cyclic coordinate.
The above quote is taken from p. 4 in Ref. 1.
What I don't understand is why $q_j$ is a linear function of time if $p_j$ is constant in time. In other words, why does $p_j$ constant in time imply partial $\frac{\partial H}{\partial p_j}$ is a constant? (In particular, $\frac{\partial H}{\partial p_j}$ could depend on any of the other coordinates or momenta.)
Reference:
- Patrick Van Esch, Hamilton-Jacobi Theory in Classical Mechanics, Lecture notes. The pdf file is available from the author's homepage here.