Symplectomorphisms and generating functions

A diffeomorphism Φ between symplectic manifolds is symplectic when Φ*ω_target=ω_source. In canonical coordinates, a linear transformation S must satisfy SᵀJS=J, with J the matrix representing ω on tangent vectors. Preserving the top wedge gives det S=1 for real symplectic matrices, but the converse fails in phase dimension four and above. For example, scaling both q₁ and p₁ by two and both q₂ and p₂ by one half has determinant one while multiplying the two canonical area blocks by four and one quarter.

In a single canonical plane the condition reduces to determinant one, so every orientation-preserving area-preserving linear map is symplectic. Hamiltonian time maps give many nonlinear examples, and composing symplectic maps preserves the property. A coordinate change can preserve ω while changing the expression of H; transformed trajectories are generated by the correspondingly transformed Hamiltonian. It does not follow that the numerical values of every observable stay unchanged. The form identifies an invariant structure, not a prohibition on changing coordinates or redistributing Euclidean widths.

Let F₂(q,P) depend on old positions and new momenta, and define p=∂F₂/∂q and Q=∂F₂/∂P. To solve for P as a function of (q,p), require the mixed Hessian F_{qP} to be invertible at the point. The implicit-function theorem then gives a local transformation. Taking the differential of F₂ yields p·dq+Q·dP. Rearranging shows P·dQ−p·dq is the exact differential of P·Q−F₂. Taking another exterior derivative proves the symplectic pullback identity.

The invertibility condition is not a decorative assumption. Without it, the equations may define a relation with several branches or a lower-dimensional constraint instead of a local map. Other types of generating functions use different choices of old and new independent coordinates. For a type-one function F(q,Q), the compatible signs are p=F_q and P=−F_Q. Those minus signs arise from the same tautological-form comparison, not from arbitrary convention. A map whose graph projects badly onto the chosen independent variables can require another generating type.

A quarter-turn Q=p,P=−q is symplectic, but old q and new P satisfy P=−q and cannot serve as independent variables for a type-two representation on an open set. A different generating chart handles it. Even where a local representation exists, global topology can obstruct a single-valued scalar potential for the tautological-form difference. The closed nonexact momentum translation on a circle from the previous lesson is a concrete example. A global exact symplectomorphism is therefore a stronger specification than a merely symplectic one.

Generating functions also organize numerical integrators. Solving derivative relations step by step can produce maps preserving the symplectic form exactly even when the original energy is not exactly preserved. This is a geometric design principle, not an unconditional accuracy or stability theorem. A coarse time step can still yield poor trajectories, implicit equations can have unsuitable branches, and a preserved form does not impose a desired phase error. Verification should include the intended limiting Hamiltonian, solvability and error behavior alongside the canonical-map identity.