I'm here to ask a rather vague question and I know that may upset some people. I'm reading about symplectic geometry and the last sentence of a theorem is that the geodesic flow preserves the form $\omega$ (a symplectic form).
I want to verify this myself, but was let to this broader question of what types of computations are we generally looking for when we want to verify that some 1-parameter family of flowlines preserves a form?
I imagine that if we are looking at a 1-form/vector field, this means that we differentiate the flows and get back the vector field. But how about more complex forms?
Suppose $(M,\omega)$ is a symplectic manifold. A flow (say generated by a vector field $X$) preserves $\omega$ iff $\mathcal{L}_X\omega = 0$. Here $\mathcal{L}_X\omega$ is the Lie derivative of $\omega$ with respect to $X$. By Cartan's homotopy formula, $\mathcal{L}_X\omega = d\iota_X\omega + \iota_Xd\omega= d(\iota_X\omega)$. $d\omega =0$ because $\omega$ is closed. So, the condition is equivalent to $d(\iota_X\omega) = 0$ which can in principle be checked in local coordinates.
By the way, this condition on $X$ is sometimes referred to as $X$ being a symplectic vector field. A symplectic vector field $X$ is called Hamiltonian if $\iota_X \omega$ is exact, i.e. if there exists $H\in C^\infty(M)$ such that $\iota_X\omega = dH$.