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It's clear to me that in acquiring the equations of motion using the Euler-Lagrange equations and the Lagrangian, defined as$^\dagger$ $$L \equiv T - V,$$ the potential energy may have an explicit time dependence if the generalized coordinates depend on the regular coordinates and time. It's not clear to me whether or not the potential energy is allowed to depend explicitly on time even if we're using regular coordinates, i.e., it's not clear whether or not a potential energy of the form $$V = V\left(\mathbf{r}_1, \dots, \mathbf{r}_N, t\right)$$ can be used to arrive at the equations of motion the "normal way".$^\ddagger$ Can such a potential energy be used? More generally, what kind of potential energies are allowed?

$^\dagger$ For the purposes of my question, this is the definition, though I am aware that there are other Lagrangians.

$^\ddagger$ By the "normal way" I mean via application of the Euler-Lagrange equations, $$\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} = \frac{\partial L}{\partial q_i}.$$

1 Answers1

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  1. Within Lagrangian point mechanics, a local Lagrangian $L(q,\dot{q}, \ddot{q}, \ldots;t)$ could in principle be higher-order, but often it is assumed to only depend on generalized positions $q$, generalized velocities $\dot{q}$, and time $t$, cf. e.g. this & this Phys.SE posts.

  2. The Lagrangian $L$ does not need to be of the form of kinetic energy minus potential energy $T-U$, cf. e.g. this Phys.SE post.

  3. A generalized potential $U(q,\dot{q},t)$ could in principle depend on all variables. Think e.g. of a charged point particle in a time-dependent E&M background, cf. my Phys.SE answer here and above comment by user kkm.

  4. A system with an external time-dependent background source is a prime example of a potential $U$ with explicit time-dependence, cf. e.g. this Phys.SE post.

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