Time independent potential and mechanical energy conservatio

[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 anyone 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) -∫_r0^rF(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 ∇∫_r0^rF(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?

 

[mfb]

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

Can you see what goes wrong with the argument leading to Equation (4.19), that is, conservation of energy?

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

 

[C. Lee]

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?

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?

 

[mfb]

Alright. So in the case where r0 ≠ r, that means the object should move finite distance without change in time. This is impossible

Right.

so therefore this definition of time-dependent potential energy cannot lead to conservation of mechanical energy.

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