Do the Euler-Lagrange equations hold for a time-dependent V?

[Coffee_]

The title basically says it, if I want to use a potential that is time dependent (for example someone is amping up the electric field externally) and keep using the form $L=T-V$ with the standard E-L equations. Can one still use them or not? If no, why? I have seen two derivations of the E-L equations both minimizing action and the virtual work principle and I can't find a reason why this would differ?

 

[Orodruin]

Yes, the EL equations still hold. However, you lose time translation invariance, so energy is no longer conserved.

 

[Coffee_]

Yes, the EL equations still hold. However, you lose time translation invariance, so energy is no longer conserved.

Someone tried to convince me that this was not the case. Are they almost certainly wrong then or is it just that the question can have more technical interpretations?

 

[vanhees71]

You can easily check yourself. The equation of motion you want to find in your case read
$$m\ddot{\vec{x}}=-\vec{\nabla} V(t,\vec{x}).$$
Now check, whether you can derive this equation of motion from the Lagrangian
$$L=\frac{m}{2} \dot{\vec{x}}^2 -V(t,\vec{x}),$$
i.e., whether the Euler-Lagrange equations coincide with the equation of motion (it's almost trivial!).

 

[Coffee_]

You can easily check yourself. The equation of motion you want to find in your case read
$$m\ddot{\vec{x}}=-\vec{\nabla} V(t,\vec{x}).$$
Now check, whether you can derive this equation of motion from the Lagrangian
$$L=\frac{m}{2} \dot{\vec{x}}^2 -V(t,\vec{x}),$$
i.e., whether the Euler-Lagrange equations coincide with the equation of motion (it's almost trivial!).

Oh now I understand. Since when defining the Lagrangian we are putting the same info into out formalism as Newton laws that's all one needs to check. Thanks