To clarify, the relevance is that deriving the vector field from the Hamiltonian requires taking partial derivatives; see the linked section from Structure and

To clarify, the relevance is that deriving the vector field from the Hamiltonian requires taking partial derivatives; see the linked section from Structure and Interpretation of Classical Mechanics (SICM) by Sussman and Wisdom w/ Mayer. Closures lets one do this by composing functions in a way that mirrors the mathematical structure. Note in the SICM implementation, the partial derivatives are computed exactly, not numerically, but the resulting ODEs are integrated numerically.