The Geometric Approach to Sets of Ordinary Differential Equations and Hamiltonian Dynamics

Abstract

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field ${\bf V}$ in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along ${\bf V}$: scalar fields (first integrals), forms (such as symplectic tensors), vectors (CT generators, Lie’s invariance generators), integrals over subspaces (which are not limited to Poincaré’s absolute and relative invariants). Most classical results are easily understood as simple relations between these. It is shown first that to any field ${\bf V}$ can be associated a Hamiltonian structure of forms (if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added). Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton’s variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms (here a Hamiltonian structure) must include the characteristics of the set (here the trajectories of ${\bf V}$). Auxiliary vector fields are also carefully discussed: the hierarchy of invariance generators of (i) the vector field ${\bf V}$, of (ii) its trajectories, of (iii) any associated Hamiltonian structure of forms. An Appendix summarizes the manipulation of vectors and forms, as a tool for applied mathematics.