in this talk, we explore the relationship between symplecticity and the preservation of quadratic invariants. symplectic runge-kutta methods, the bread and butter of numerical geometric integration, are exactly the ones that preserve all quadratic first integrals of a system. but when we expand our focus to larger classes of methods, we will find a more nuanced connection. it is also known that for symplectic rk methods, the action of discretization commutes with forming variational equations, and we will discuss the expansion of this result to a larger class of methods.