Symplectic Representation of Legendre Dynamics

Abstract

Modern learning systems act on internal representations of data, yet how these representations encode underlying physical or statistical structure is often left implicit. In physics, symplecticity keeps Hamiltonian systems faithful to their phase-space geometry. Recent learning methods impose such geometric structure either in the dynamics or through training losses. Here we ask a different question: what would it mean for the representation itself to obey a symplectic conservation law? We pose this representation-level constraint through Legendre duality: the relation $p = dψ(q)$ between primal and dual coordinates, which in exponential family models is the information-geometric pairing of natural and expectation parameters. We formalize Legendre dynamics as stochastic processes whose trajectories remain on Legendre graphs, where the evolving primal-dual parameters stay Legendre dual. We show that this class includes linear time-invariant Gaussian process regression and Ornstein-Uhlenbeck dynamics. Geometrically, we characterize the symplectomorphisms of cotangent bundles that preserve all Legendre graphs. We show that these maps are exactly cotangent lifts of base diffeomorphisms followed by exact fibre translations. This gives an explicit normal form for Legendre-preserving representation updates. Dynamically, we prove that the normal form is realized by Hamiltonians that are at most linear in the momentum. This realization principle is used to construct linear and nonlinear Hamiltonian Symplectic Reservoirs (SR) whose recurrent updates preserve Legendre graphs by construction. This is the only normal form that preserves Legendre duality, so the architecture follows from the invariant. Numerical experiments confirm the normal-form identities and distinguish Legendre preserving Hamiltonian SRs from generic symplectic and standard reservoir baselines.