Reduction of Order of Hamiltonian Systems

This is a paper on reduction of order of Hamiltonian systems using first integrals, where by order we mean the number of independent variables in the system. In the Hamiltonian case, Noether’s theorem (to be proved below) says that first integrals are equivalent to symmetries of the system and, hence, we could use Sophus Lie’s standard order reduction techniques to accomplish this. The story is, however, more interesting than that: Hamiltonian systems live on manifolds endowed with an additional structure of a Poisson bracket, and exploiting this extra structure will allow us to reduce the order of the system by twice the dimension of our symmetry group while still preserving the Hamiltonian nature of the problem. This is analogous to the case of a system of EulerLagrange equations (where the extra structure exploited is of variational nature). The brother part of the paper exhibits the (easier) case of a system with one known first integral and how we can use it to reduce order by two. The reduction technique in this case uses the same key idea underlying the proof of the important Darboux’ theorem, which we give. The last section is on multi-dimensional symmetry groups, and knowledge of more sophisticated mathematics is needed to fully understand the theory (in particular, Stephen Smale’s Momentum map plays an important role). We shall routinely mention manifolds in this section to make the exposition slicker, but the reader can always think of them as open subsets of Euclidean space. Fortunately, al we need to implement the ideas is basic calculus. Nevertheless, more is required of the reader of the final section while only basic knowledge of Hamiltonian systems and Lie symmetries is assumed for sections leading up to it.