PRESERVATION OF QUADRATIC INVARIANTS BY SEMIEXPLICIT SYMPLECTIC INTEGRATORS FOR NONSEPARABLE HAMILTONIAN SYSTEMS

We prove that the recently developed semiexplicit symplectic integrators for nonseparable Hamiltonian systems preserve any linear and quadratic invariants possessed by the Hamiltonian systems. This is in addition to being symmetric and symplectic as shown in our previous work; hence, it shares the crucial structure-preserving properties with some of the well-known symplectic RungeKutta methods such as the Gauss-Legendre methods. The proof follows two steps: First we show how the extended Hamiltonian system proposed by Pihajoki inherits linear and quadratic invariants in the extended phase space from the original Hamiltonian system. Then we show that this inheritance in turn implies that our integrator preserves the original linear and quadratic invariants in the original phase space. We also analyze preservation/nonpreservation of these invariants by Tao's extended Hamiltonian system and the extended phase space integrators of Pihajoki and Tao. The paper concludes with numerical demonstrations of our results using a simple test case and a system of point vortices.

Automated summary

We prove that the recently developed semiexplicit symplectic integrators preserve linear and quadratic invariants possessed by nonseparable Hamiltonian systems. These integrators share the structure-preserving properties with well-known symplectic Runge-Kutta methods and are shown to be symmetric and symplectic. The proof demonstrates how the extended Hamiltonian system inherits invariants in the extended phase space and how this inheritance preserves the original invariants in the original phase space. The paper also includes an analysis of the preservation/nonpreservation of invariants by other extended Hamiltonian systems and phase space integrators.