The question I have is, what conditions must satisfy an external force dependent on velocity so it can be a part of the lagrangian and Euler-Lagrange equations are still true.
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Essentially a duplicate of Velocity-Dependent Potential and Helmholtz Identities – Qmechanic May 31 '20 at 17:25
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1Does this answer your question? Velocity-Dependent Potential and Helmholtz Identities – Jon Custer Jun 01 '20 at 02:11
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The (non-relativistic) Lagrangian force is $\frac{\partial L}{\partial q}$, with variables $q$ and $\dot{q}$ considered independent, thus term of the form $\delta L = q F(\dot{q})$ is interpreted as velocity dependent force $ F(\dot{q})$ on coordinate $q$. Notice that as long as $F$ doesn't explicitly depends on coordinate $q$ (but may be fependent on others) - $F$ can be function of velocities of all the coordines. For example $F = F(|\dot{\vec{q}}|) = -\gamma \sqrt{\dot{q_1}^2+\dot{q_2}^2+\dot{q_3}^2}$
Alexander
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