# Hi,may potential, time dependent force field be called

1. Sep 23, 2011

### apedcen

Hi,

may potential, time dependent force field be called conservative?

If so, the mechanical energy conservation of an isolated mechanical system does not hold in such field?

Thanks.

2. Sep 23, 2011

### Curl

Re: Conservative?

time dependent means your force field is a function of 4 variables. In math, its equivalent to having a vector field in 4-space. Curl is defined only on R3, but there are some generalizations to the product involving exterior derivatives (which I have not studied).
But when you do a path integral, you have a curve in 4 space which doesn't make physical sense since you can only traverse time in 1 direction, so a closed path is impossible.

3. Sep 25, 2011

### apedcen

Re: Conservative?

Curl,
I don't think that we need to incorporate time in definition of conservative force (its just not there). The force may be conservative in a fixed time moment. I dont have problem in calling time dependent force conservative, however the energy conservation will not work on for such force imho.

4. Sep 25, 2011

### Curl

Re: Conservative?

Ok now I think I see what you're saying. Are you considering a vector field which changes with time, but at any instant in time the force field is conservative, i.e. curl F = 0 at a fixed t?

5. Sep 26, 2011

### G01

Re: Conservative?

I think that's what the OP means.

If that is indeed the case:

A time dependent potential function will not give rise to a conservative force. If we consider the Lagrangian of a system:

$$L=T-V(t)$$

Since, V has an explicit time dependence, L has an explicit time dependence. We can also derive the time dependence of the energy function from Lagrangian mechanics:

$$\frac{dH}{dt}=-\frac{\partial L}{\partial t}$$

Thus, if L has an explicit time dependence, the energy function will vary in time, and is thus not conserved.