Hi,may potential, time dependent force field be called

[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.

 

[Curl]

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.

 

[apedcen]

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 don't have problem in calling time dependent force conservative, however the energy conservation will not work on for such force imho.

 

[Curl]

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?

 

[G01]

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?

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.