Constructing the Hamiltonian from the behaviour of a dynamical system by proper symplectic decomposition

TL;DR

This paper introduces a symplectic formalism-based method to reconstruct Hamiltonians from dynamical system data, enabling model reduction and analysis even when the Hamiltonian form is unknown or complex.

Contribution

It develops a general symplectic decomposition approach to approximate Hamiltonians from structural evolution data, with proven convergence and applicability to high-dimensional models.

Findings

Method converges with refined time discretization.

Allows Hamiltonian approximation from limited data.

Enables model order reduction for complex systems.

Abstract

The modal analysis is revisited through the symplectic formalism, what leads to two intertwined eigenproblems. Studying the properties of the solutions, we prove that they form a canonical basis. The method is general and works even if the Hamiltonian is not the sum of the potential and kinetic energies. On this ground, we want to address the following problem: data being given in the form of one or more structural evolutions, we want to construct an approximation of the Hamiltonian from a covariant snapshot matrix and to perform a symplectic decomposition. We prove the convergence properties of the method when the time discretization is refined. If the data cloud is not enough rich, we extract the principal component of the Hamiltonian corresponding to the leading modes, allowing to perform a model order reduction for very high dimension models. The method is illustrated by a numerical example.