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# What happens when appell-chetaev'sAppell-Chetaev's rule for constainedconstrained mechanical systems is not applicable?

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.

2 added 51 characters in body

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 mildly mild assumption, a prescription to find $X_C$ is the Appel-Chetaev Appell-Chetaev rule, which extends the Lagrange-D'Alembert principle in the context of constraints that are 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 following Appell-Chetaev's rulebased on Appell-Chetaev?
And in such a case, what rules are the alternative rules alternatives in prescribing the constraint forces? and what are the domains of applicability of such other rules?

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# What happens when appell-chetaev's rule for constained mechanical systems is not applicable?

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 mildly assumption a prescription to find $X_C$ is the Appel-Chetaev rule, which extends the Lagrange-D'Alembert principle in the context of constraints that are 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 following Appell-Chetaev's rule? And in such a case what are the alternative rules in prescribing the constraint forces?