Gradient of a time-dependent potential energy

[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?

 

[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

 

[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.

 

[lanedance]

i'm a little confused, can you describe the system clearly... does $\theta(t)$ represent the path of some particle?

also did you consider the comments on using a polar form off the gradient operator?
http://en.wikipedia.org/wiki/Gradient#Expression_in_3-dimensional_spherical_polar_coordinates

 

[CellCoree]

Yes, the second term should be present in the gradient of the time-dependent potential energy. The gradient of a function with respect to a variable is defined as the vector of partial derivatives of the function with respect to each of the variables. In this case, the potential energy has two variables - time (t) and the magnetic field (H). Therefore, the gradient would have two components, one with respect to time and one with respect to the magnetic field.

The second term in the gradient represents the rate of change of the magnetic field with respect to the angle (theta), which is an important factor in determining the overall potential energy. This term takes into account the fact that the magnetic field is changing with time and how that change affects the potential energy. Without this term, the gradient would not accurately represent the potential energy function.

In summary, the second term is necessary in the gradient of the time-dependent potential energy as it takes into account the rate of change of the magnetic field with respect to the angle, which is an important factor in determining the overall potential energy.