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

1. Mar 22, 2015

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?

2. Mar 22, 2015

Orodruin

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

3. Mar 25, 2015

Coffee_

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?

4. Mar 27, 2015

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!).

5. Mar 27, 2015

Coffee_

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