W. A. Horowitz

A. Rothkopf, W. A. Horowitz

Variational principles play a central role in classical mechanics, providing compact formulations of dynamics and direct access to conserved quantities. While holonomic systems admit well-known action formulations, non-holonomic systems -- subject to non-integrable velocity constraints or position inequality constraints -- have long resisted a general extremized action treatment. In this work, we construct an explicit and general action for non-holonomic motion, motivated by the classical limit of the quantum Schwinger-Keldysh action formalism, rediscovered by Galley. Our formulation recovers the correct dynamics of the Lagrange-d'Alembert equations via extremization of a scalar action. We validate the approach on canonical examples using direct numerical optimization of the novel action, bypassing equations of motion. Our framework extends the reach of variational mechanics and offers new analytical and computational tools for constrained systems.

Alexander Rothkopf, W. A. Horowitz, Jan Nordström

In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value Problems directly on the action level, bypassing governing equations. We show that by including coordinate maps as dynamical degrees of freedom together with propagating fields two key results emerge. Space-time symmetries remain protected even after discretization, leading to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. We stress that as long as SBP operators are used for the discretization, our results are independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order. As proof-of-principle for our approach we present its application to scalar wave-propagation in 1+1 dimensions.