Applied analysis of constrained Lagrangian ODEs and PDEs

Twist-to-bend waves in a 3D inextensible/unshearable elastic ring.

Related Articles

O. Gonzalez, J.H. Maddocks & R.L. Pego, ``Multi-Multiplier Ambient Space Formulations of Constrained Dynamical Systems, with an Application to Elastodynamics,'' Archive for Rational Mechanics and Analysis, accepted.


``Multiplier formulations for the numerical treatment of constrained Lagrangian systems in finite and infinite dimensions,'' working manuscript.


``Three-dimensional dynamics of inextensible and unshearable elastic rods: ambient-space formulation and discretization,'' working manuscript.

Overview

There are many ways to formulate the equations of motion for a Lagrangian system subject to configuration (holonomic) constraints. If the system is finite-dimensional, one typically has a choice between:

• ODEs in local coordinates (traditionally used in analytical studies)
• DAEs in ambient coordinates with multipliers (traditionally used in numerical studies)
  • index-3 (obtained directly from Hamilton's Principle)

  • index-2 (various ways to obtain from index-3 form)

  • index-1 <=> ODE with invariant manifold (obtained by "eliminating" multipliers from equations; physically meaningful solutions reside in the invariant manifold)

If the system is infinite-dimensional, then choices are limited. Local coordinates are often unavailable, and so one typically formulates equations of motion in an ambient function space with explicit constraints and multipliers. The resulting equations are analogous to the DAEs of the finite-dimensional case and so are called PDAEs or partial differential-algebraic equations. Moreover, the notion of "index" can also be formally extended to the infinite-dimensional case.

I am particularly interested in the description of constrained systems by ODEs/PDEs with invariant manifolds. I call these descriptions ambient-space formulations. A given constrained system can be described by many such formulations, and these can have very different properties.

• ODEs/PDEs can be Hamiltonian or non-Hamiltonian
• invariant manifolds can be "strongly invariant" (one of a family; level set of an integral) or "weakly invariant" (a single object; not level of an intergal)
• invariant manifolds can be exponentially stable/unstable or neutrally stable

Describing constrained systems by an ambient-space formulation can be useful in both numerical and analytical studies. On the numerical side, such formulations

• can assist in the analysis of discretizations of higher-index equations (e.g. identify discretization of a DAE as a consistent discretization of ambient-space ODE)
• can themselves be discretized to provide a basis for simulation (e.g. simulate ODE with stable invariant set rather than DAE)

• by Moser to study the integrability of geodesic and potential flows on a large class of embedded surfaces. In particular, Moser was able to give remarkable geometric interpretations of various geodesic flows, and in the classic integrable cases, was also able to give explicit, algebraic expressions for the integrals.
• by Dichmann, Maddocks & Pego to study the stability of solitary waves in an inextensible and unshearable elastic rod by applying standard Lyapunov arguments.

My research goals are to develop a unified theory of ambient-space formulations, and to demonstrate that these formulations can provide a useful and sometimes preferable means of studying constrained dynamical systems.