Supplementary Material: Appendices A Geometric Numerical Integration Geometric numerical integration is a study on the numerical integrators of ODEs that preserve the

Due to this property, there should exist a corresponding Hamiltonian function, i.e., energy function, As such, the discrete gradient method has achieved great success. As shown above, a discrete gradient is defined in Definition 1. The target equations for this study are the differential equations with a certain geometric structure. A (null u)null v (16) with a matrix A (null u); hence, Eq. (15) is shown to be equivalent to null w This is our target equation in Eq. (1). This section provides the proofs of the Theorems in the main text.