# Time independent potential and mechanical energy conservatio

1. Sep 3, 2015

### C. Lee

Hi,

I was woking on a problem from Taylor Mechanics.(4.27) It reads:

Suppose that the force F(r , t) depends on the time t but stillsatisfies ∇ × F = 0. It is a mathematical fact that the work integral ∫12F(r , t) ⋅ dr (evaluated at any one time t) is independent of the path taken between the points 1 and 2. Use this to show that the time-dependent PE defined by (4.48) -∫r0rF(r' , t) ⋅ dr', for any fixed time t, has the claimed property that F(r , t) = -∇U(r , t). Can you see what goes wrong with the argument leading to Equation (4.19), that is, conservation of energy?

Equation (4.19) is nothing but Δ(T + U) = 0.

I tried to write down the gradient of (4.48) directly, but I failed because I do not know how to handle ∇∫r0rF(r' , t) ⋅ dr'. And, I think, thus I cannot see what goes wrong with the argument leading to (4.19). (I am aware of the fact that mechanical energy is no longer conserved since potential energy U depends on time)
Can somebody help me with that gradient part, or is there any other way to do this?

2. Sep 4, 2015

### Staff: Mentor

For the potential energy you can just consider t as unknown but fixed parameter.

The work integral is evaluated at a specific time. Does this represent realistic motion?

3. Sep 5, 2015

### C. Lee

Alright. So in the case where r0 ≠ r, that means the object should move finite distance without change in time. This is impossible, so therefore this definition of time-dependent potential energy cannot lead to conservation of mechanical energy.

Is this right?

4. Sep 5, 2015

### Staff: Mentor

Right.
At least you cannot guarantee conservation of energy for a realistic trajectory that needs some time.