# Find the lagrangian of a particle given the force

A particle is submitted to a time dependent force $$F(x,t)=\dfrac{k}{x^2}e^{-t/\tau}$$

Which is the Lagrangian of the particle?

I think that the force is derived from the potential $V$ and this potential has not explicit dependence of $\dot x$. So i can write
$$\dfrac{d}{dt}\dfrac{\partial \mathcal L}{\partial \dot x} = m \ddot x$$

$$\mathcal L = T-\int \dfrac{\partial \mathcal L}{\partial x} dx$$ Then the lagrangian is $$\mathcal L = \dfrac{m}{2}\dot x^2 + \dfrac{k}{x}e^{-t/\tau}$$

Am i right?

• Does the Euler-Lagrange equation coincide with the desired equation of motion? If so, what else do you need? – Emilio Pisanty May 2 '17 at 1:08
• the statement does not say too much. I think that $F=\nabla V$ but have the time implicit. I just need the lagrangian. – Gabriel Sandoval May 2 '17 at 1:21

• why that is not $F$, if the potential has not $\dot x$ explicitly then $\nabla V = -\dfrac{\partial \mathcal L}{\partial x}$. My problem is the time dependence. – Gabriel Sandoval May 2 '17 at 1:27