We found that certain choices of coordinates give rise to cyclic coordinates leading to their corresponding conjugate momenta being constants. We will see that it is possible to find a set of coordinate transformations that lead to all coordinates being cyclic.
In the Lagrangian formalism, it is difficult to determine these optimal transformations; this is because both the coordinates and their time derivatives appear in the Lagrangian. In the Hamiltonian formalism, we can treat the coordinates and their conjugate momenta as independent, and we thus have more liberty to seek transformations that make the coordinates cyclic. In general, we call transformations that preserve Hamilton’s canonical equations, “canonical transformations”. The transformed coordinates are denoted by \(Q_i\) and \(P_i\), and the transformed Hamiltonian by \(K\). The conditions satisfied by a canonical transformation are thus: