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Is there any theorem about for which velocity-dependent forces a velocity-dependent generalized potential of the form $$F_k=\frac d {dt} \left(\frac {\partial U}{\partial \dot q_k}\right)-\frac {\partial U}{\partial q_k}$$

can be found, e.g. can it be said that for energetically conservative velocity-dependent and therefore mathematically non-conservative, path dependent forces have a velocity-dependent potential? Can someone please give an example of a velocity-dependent potential other than the one for the Lorentz-force?

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Let us assume that that the Lagrangian is polynomial in the velocities $\dot{q}_i$. A quadratic term in the velocities $\dot{q}_i$ would properly be thought of as part of the kinetic energy rather than the potential energy. And terms that are higher order in the velocities are equivalent (via integration by parts) to terms that involve higher-order derivatives of $q_i$, which will generically lead to Ostrogradski instabilities, so it is also reasonable to eliminate those from consideration.

Let us also assume that the Lagrangian is independent of $t$. This ensures the existence of a conserved quantity that can be identified as the energy of the system.

But this means that the only option left is for $U$ to be quasi-linear in $\dot{q}_i$, i.e., of the form $$ U(q_i, \dot{q}_i) = \sum_k A_k(q_i) \dot{q}_k + \tilde{U}(q_i) $$ where the functions $A_k$ and $\tilde{U}$ only depend on the coordinates. This is exactly the same form as the Lorentz force (if the $q_i$ coordinates are Cartesian coordinates, this is just $\vec{A} \cdot \vec{v}$.) And the resulting force is of the form $$ F_k = \sum_j \left[\left( \frac{\partial A_k}{\partial q_j} - \frac{\partial A_j}{\partial q_k} \right) \dot{q}_j\right] + \frac{\partial \tilde{U}}{\partial q_k} $$ which again looks an awful lot like the Lorentz force.

So under the above assumptions, it does not seem like there are a lot of options other than "Lorentz-like" forces to arise from velocity-dependent potentials.

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  • $\begingroup$ here, you made no assumptions other than the Lagrangian not being explicitly dependent on time, and the velocity dependence of the force on velocity to be quasi-linear. under these circumstances, doesn't your comment imply the existence of a velocity dependent potential for the friction force which was falsified [here][1] [1]: physics.stackexchange.com/questions/20929/… $\endgroup$
    – gluon
    Commented Aug 30, 2023 at 15:28
  • $\begingroup$ @gluon: No, I showed that if the Lagrangian is time-independent and polynomial in the velocities, then the only allowable velocity-dependent forces are "Lorentz-like"; in particular, you cannot get "friction-like" forces from a Lagrangian of this form. The answer you linked shows that there does not exist a Lagrangian (of any form) that leads to "friction-like" forces. The two statements aren't inconsistent. $\endgroup$ Commented Aug 30, 2023 at 20:48

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