# Gradient of a time-dependent potential energy

1. Jul 6, 2011

### kuntalroy

Say, we have potential energy of the form $U = cos (\theta(t)) H(t)$.

H denotes a magnetic field that is time-dependent and it's an input variable to the system. Now when you take gradient of potential energy, would you write

$\nabla U = \left[ - sin (\theta(t)) H(t) + cos (\theta(t)) \frac{\partial H(t)}{\partial \theta(t)}\right] \hat{e}_{\theta} = \left[ - sin (\theta(t)) H(t) + cos (\theta(t)) \frac{dH(t)/dt}{d\theta(t)/dt}\right] \hat{e}_{\theta}$ ?

Should the 2nd term be present?

Last edited: Jul 6, 2011
2. Jul 7, 2011

### lanedance

this doesn't totally make sense to me - is the potential only a function of theta or does H vary with position?

also if you want to take the gradient in cylindrical or spherical coords, you have to account for the basis vectors changing, and so use the form of the gradinet operator in those coordinates

3. Jul 7, 2011

### kuntalroy

Well, theta is a state variable while H is a input variable. Usually, the potential energy U is a function of only the state variables - (r,theta,phi) - in spherical coordinate system - for the simplified case it's only a function of theta. It's valid for only a 'closed' system.

If some external agent modifies the potential landscape and does an energy exchange in time, it gets coupled with the dynamics of the system - here it's theta. This can be produce an additional non-conservative field and subsequently a torque. This may have a self-consistent nature with the dynamics of the system.

This is not a straigtforward problem - first we have catch the problem - and providing a solution would not be straigtforward at all.

4. Jul 7, 2011