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What happens when appellAppell-chetaev'sChetaev'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.

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 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?

What happens when Appell-Chetaev's rule for constrained 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.

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agt
agt
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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 mildlymild assumption, a prescription to find $X_C$ is the AppelAppell-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 followingbased on Appell-Chetaev's ruleChetaev? And
And in such a case, what rules are the alternative rulesalternatives in prescribing the constraint forces? and what are the domains of applicability of such other rules?

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?

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?

Source Link
agt
agt
  • 4.3k
  • 2
  • 36
  • 51

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?