Strong degenerate constraining in Lagrangian dynamics
Juan Manuel Burgos
TL;DR
This work addresses the limit behavior of strongly constraining, degenerate potentials in Lagrangian mechanics with codimension-one constraints. It models stiff constraints via $L_\varepsilon(x,v)=K_x(v)-\varepsilon^{-2}U(x)$ with $U=g\circ f$ and analyzes the high-frequency limit $\varepsilon\to0$, deriving an effective intrinsic dynamics on the constraint manifold $M={f=0}$. The main result is that the limit curve $x$ on $M$ satisfies $\nabla_{\dot x}\dot x+\theta\|\mathsf{grad}_\rho f(x)\|^{2/(2\alpha+1)}\,\kappa(x)=0$, where $\kappa$ is the equipotential distortion and $\theta$ is an adiabatic invariant given by $\theta=\frac{1}{2\alpha+1}\|v_{\perp}\|^2\|\mathsf{grad}_\rho f(p)\|^{-{2/(2\alpha+1)}}$; the paper also develops the radial energy analysis, yielding a weak virial relation and an effective potential $U_{eff}$ that governs transverse dynamics. It further provides criteria for when an ideal constraint is real (vanishing equipotential distortion) and extends previous nondegenerate results to degenerate potentials in codimension one. Overall, the findings illuminate how high-frequency effects shape constrained Lagrangian dynamics in degenerate settings and yield a precise, geometrically interpretable limit model.
Abstract
We study the strong constraining problem in Lagrangian dynamics in the degenerate codimension one case. This is the first time that degenerate potentials at the constraint are considered for this problem. These new results cover several real analytic potentials that the previous do not. Some counterintuitive effects of the degenerate constraining problem are discussed.