# Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?

Given the Hamiltonian $$H(x,p)$$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $$K(x',p')=K(p')$$?

• Locally, assuming various regularity conditions, one could in principle solve Hamilton-Jacobi (HJ) equation to perform a canonical transformation (CT) and reach a vanishing Kamiltonian $$K\equiv 0$$.