Variational approach to nonholonomic and inequality-constrained mechanics

TL;DR

This paper addresses the lack of a general action principle for nonholonomic and inequality-constrained mechanics by introducing a classical Schwinger-Keldysh-Galley (SKG) action with doubled degrees of freedom and Lagrange multiplier constraints. The generalized action $ ilde{S}_{SKG}$ reproduces the Lagrange-d'Alembert equations for velocity-dependent constraints and extends naturally to inequality constraints modeling normal forces and sliding friction, including a mechanism to recover constraint forces via Chetaev-type terms. It validates the approach on model systems—rolling-spinning disks, tumblers with hard walls, and sliding on inclines with friction—using SBP discretization to locate action critical points, achieving agreement with standard dynamics and automatic treatment of impacts. Overall, the framework broadens variational mechanics to constrained dynamics and offers robust analytical and computational tools for robotics, contact mechanics, and nanoscale systems where nonholonomic and inequality constraints are prevalent.

Abstract

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.

Figures (4)

• Figure 1: Sketch of the rolling-spinning disk on an incline.$x,y$ encode the center of mass ${\bf r}_{\rm c.m.}=x\hat{i} + y\hat{j}+z\hat{k}$ and the angles $\theta,\phi$ encode rolling and spinning motion. Reprinted from flannery:2005, with the permission of AIP Publishing.
• Figure 2: Time evolution of a rolling-spinning disc. Motion obtained via the semi-holonomic (solid gray) and non-holonomic (dashed color) Lagrange-d'Alembert equations of motion as well as via the critical point of our novel classical SKG action \ref{['eq:discrSvc']} (color symbols). Physical degrees of freedom are shown in the top-, the Lagrange multiplier to $g^{\rm D1}$ in the second from top panel. We evaluate the constraint functions $g^{\rm D1}$ and $g^{\rm D2}$ on the solution from our action in the second to bottom panel and the energy in the bottom panel. Due to the presence of the Lagrange multipliers for the connecting condition at the last time slice, one observes benign (diminishing under grid refinement) deviations at the last and second to last time slice. For $\lambda^{\rm D1}_{\rm SKG}$ and the energy close to $t_f$ the last two points are affected, while for the constraints only the last time step is affected.
• Figure 3: Point mass falling in a tumbler. (Top) Motion obtained via Newton's 2nd law (solid colors) and from the critical point of our SBP424 discretized action (gray circles) with $N=256$ and $\sigma=1/110$. (Center) Convergence of the critical point towards the correct solution as the width of the Gaussian $\sigma=1/10\ldots1/110$m is decreased (light to dark gray). (Bottom) Total energy $E$ is shown as blue solid curve. Its kinetic and gravitational contribution are given in red. The gray lines show the change in kinetic and gravitational contributions as $\sigma$ is reduced.
• Figure 4: Sliding motion along an incline of angle $\alpha=\pi/6$ in constant gravity. Trajectories from Newton's second law in gray in the absence (dashed) and presence (solid) of kinetic Coulomb friction $\mu=4/10$. Trajectories from the critical point of action \ref{['eq:discrSF']} are given as open colored symbols, where the normal force is regularized with $\sigma=1/70$m and initial positon is offset by $\delta=1/30$m.