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Consider a particle moving in the field of a conservative potential $V(\bf{r}$$)$ which is constrained to move on a surface $\sigma(\bf{r},$$t)=0$. Here $\bf{r}$ denotes the radius vector of the particle. Solving the Euler-Lagrange equation by Lagrange multipliers, it is found that the total mechanical energy changes at a rate, $$ \left( \bf{\dot p} -F\right)\cdot {\bf{\dot r}} = \dot W= \lambda \frac {\partial \sigma}{\partial t} $$ Where does the energy go? My belief is that the surface $\sigma$ "moves in time" through continuous elastic transformations given by the explicit time variation, and the change in the elastic P.E. is compensated by the mechanical power. However, I am not sure about this and need some authentic resources telling about this phenomenon. Thanks!

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I haven't checked your formula, but it is trivial that a time dependent constraint can do work. Just think of the floor of an elevator as an example for $\sigma(x,t)$. Do you think the elevator raises the energy of you moving in the cabin? I would say yes. Is the energy state of the elevator/environment part of the described system? No, it is not, the surface is just a constraint of the action minimization and as such not part of the Lagrangian. Hence the energy that you gain cannot be reflected by an equal but opposite energy change of your surroundings within the Lagrange multiplier approach.

If you included the reaction forces of the moving surface into the Lagrangian such that the motion of the system is exactly as it was previously constrained, energy conservation would return. But then it would be very difficult to solve because you don't know the reaction forces.

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  • $\begingroup$ then how do we conserve energy? $\endgroup$ Mar 14 at 13:27
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    $\begingroup$ I don't know what you mean. Energy is of course conserved "in the background", but by applying optimization+constraints you are explicitly aiming at a solution that does not conserve energy, but has certain properties otherwise difficult to enforce. $\endgroup$
    – oliver
    Mar 14 at 13:32
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    $\begingroup$ @AweKumarJha Think of the elevator example: an external force moves the elevator up. We describe the system in such a way that this external force looks the same as a constraint. If we want energy to be conserved we need to describe the entire system including the engine which pulls up the elevator. Another example is friction. Friction doesn't destroy energy but moves it to the microscopic degrees of freedom. If you ignore the microscopic degrees of freedom it looks like energy is lost. $\endgroup$ Mar 14 at 13:34
  • $\begingroup$ Constraints usually either represent very big environments (e.g. the earth), or controlled systems (e.g. elevator, where controller powers the motor such that the trajectory is as desired). In both cases the "missing energy" is of no practical interest. $\endgroup$
    – oliver
    Mar 14 at 13:35
  • $\begingroup$ The question is: do you want a method that solves practical problems or do you want a method that looks nice (from the POV of energy) and could solve your specific problem if you knew everything you need to put into the method. In the first case choose action principle + constraints, in the second case choose just the action principle without constraints. $\endgroup$
    – oliver
    Mar 14 at 13:38

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