# Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system?

• Assuming you mean a continuous system? A discrete dyamical system is a trivial example of a non-Hamiltonian system which can have conserved quantities. Mar 8 '13 at 6:59
• Cross-posted from math.stackexchange.com/q/324348/11127 Mar 8 '13 at 18:44
• Dear @user1544418. In general, it is frown upon to cross-post simultaneously, because it may waste potential answerer's time. As a minimum OP should mention the cross-post (on both sites!). The preferred procedure is to not cross-post, and if the post hasn't received an acceptable answer after, say, a couple of days, then OP could flag for migration. Mar 8 '13 at 18:45

Answer: No. Take a system $M$ that has a constant of motion and another system $N$ that doesn't have a Hamiltonian formulation. Then the combined system $M\times N$ (where the two parts don't talk to each other) will have a constant of motion, but the full system will not have a Hamiltonian formulation.