What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable?

2011-10-11T10:46:45Z

Background: Let be given a mechanical system whose configuration space is a manifold $Q$, and the kinetic energy is a metric $K$ on $Q$, in presence of a potential function $V$.
Let us identify the tangent bundle $TQ$ with the phase space $T^\ast Q$ through the Legendre map $\mathcal{L}_K$. Let $X_L$ be the second order differential equation on $M$ corresponding to the lagrangian $L=K+\pi^\ast V$.

If the system is subject to a kinematical constraint represented by a submanifold $M$ of $TQ$, then we need to find the vector field $X$ on $M$ which determines the dynamics of such a constrained system, the difference $X_C=X-X_L$ being the contribution of the constraint force.

Under mild assumption, a prescription to find $X_C$ is the Appell-Chetaev rule, which extends the Lagrange-D'Alembert principle in the context of constraints not necessarily linear on the velocities. This rule imposes that $X_C$ has to lie on $(TM)^{0}$, the annihilator of $TM$.

Question: Are there constrained mechanical systems whose dynamics is not in agree with the previsions based on Appell-Chetaev?
And in such a case, what rules are the alternatives in prescribing the constraint forces? and what are the domains of applicability of such other rules?

As usual any feedback is welcome, thank you.

What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable? - MathOverflow

What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable?