Energy- and Quadratic Invariants--Preserving Integrators Based upon Gauss Collocation Formulae

We introduce a new family of symplectic integrators for canonical Hamiltonian systems. Each method in the family depends on a real parameter $\alpha$. When $\alpha=0$ we obtain the classical Gauss collocation formula of order $2s$, where $s$ denotes the number of the internal stages. For any given non-null $\alpha$, the corresponding method remains symplectic and has order $2s-2$; hence it may be interpreted as an $O(h^{2s-2})$ (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the parameter $\alpha$ may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution, as well as to maintain the original order $2s$ as the generating Gauss formula.