What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:41:40Z http://mathoverflow.net/feeds/question/77801 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77801/what-happens-when-appell-chetaevs-rule-for-constrained-mechanical-systems-is-not What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable? Giuseppe 2011-10-11T10:46:45Z 2011-10-11T13:53:55Z <p><strong>Background:</strong> 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 $V$.<br> 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 $L=K+\pi^\ast V$. </p> <p>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.</p> <p>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 $TM$.</p> <p><strong>Question:</strong> Are there constrained mechanical systems whose dynamics is not in agree with the previsions based on Appell-Chetaev?<br> 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?</p> <p>As usual any feedback is welcome, thank you.</p>