In this paper, we study the dual transformation applied to N particles and N −1 springs that are no longer constrained to the real line. For simplicity, assume the Hamiltonian system formed to be embedded in Euclidean n−space with the standard metric and also assume that N < ∞ and that all N particles are connected to at least one spring.

Our interpretation of one possible extension is to create graphs that can act as dynamical systems. We decide to take a graph (a finite one, for simplicity), position masses at the vertices of the graph, and exchange the edges of the graphs with springs. Again, for simplicity, we assume that the springs behave uniformly, although not necessarily linearly. We then re-express the Hamiltonian with respect to the graphs rather than focusing on the underlying manifold. We focus on graphs which may be neatly embedded into Euclidean n−dimensional space for the purpose of this paper. Hence we now have information regarding how graphs behave when “equipped” with a symplectic 2−form.

with Elinor Velasquez. Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium 169 (2004), 127–139.   MR2122063     70H06 (05C45 37J35 37K60 70H15 82C05)

An Introduction to Symplectic Maps and Generalizations of the Toda Lattice