Low-order Hamiltonian operators having momentum (Q1947465)

The author gives a classification of fifth-order Hamiltonian operators in one dependent and one independent variable that possess momentum. The existence of momentum is useful for averaging the corresponding Hamiltonian systems. Further interest lies in the possibility of generating hierarchies of commuting flows. The main result is Theorem 1: Theorem 1. (1) A fifth-order Hamiltonian operator which is not translation-invariant cannot have momentum. (2) A fifth-order translation-invariant Hamiltonian operator that can be transformed using a special contact transformation into an operator with the leading coefficient \(\pm 1\) cannot have momentum. (3) Any fifth-order translation-invariant Hamiltonian operator that can be transformed using a special contact transformation into an operator with the leading coefficient \(\pm1/u_1^4\) has momentum. Theorem 1 is to be understood in conjunction with Proposition 3: Any fifth-order translation-invariant Hamiltonian operator can be reduced by a special contact transformation to an operator with the leading coefficient equal to either \(\pm 1\) or \(\pm1/u_1^4\).