Dirac and Lagrange algebraic constraints in nonlinear port-Hamiltonian systems

TL;DR

This paper extends nonlinear port-Hamiltonian systems by introducing Lagrange algebraic constraints, generalizing Dirac constraints, and demonstrating conversions between them using additional variables and Morse families.

Contribution

It introduces Lagrange algebraic constraints in port-Hamiltonian systems, expanding the theoretical framework beyond Dirac constraints and providing methods for converting between the two.

Findings

Lagrange constraints generalize Dirac algebraic constraints.

Conversion between Dirac and Lagrange constraints is possible via extra variables and Morse families.

Extended class of port-Hamiltonian systems broadens modeling capabilities.

Abstract

After recalling standard nonlinear port-Hamiltonian systems and their algebraic constraint equations, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the Hamiltonian function by a general Lagrangian submanifold of the cotangent bundle of the state space manifold, motivated by developments in Barbero-Linan et al., and extending the linear theory as developed in Van der Schaft et al., Beattie et al.. The resulting new type of algebraic constraints equations are called Lagrange constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, conversely, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families.