Energy Conservation and Time-Dependent Potentials

[paradoxymoron]

In my intro to Quantum Mechanics course, my professor gave a little aside while exploring the analogy between the Schrödinger Equation and Newton's second law: in classical physics, energy is conserved when the potential energy is not a function of time.

I wanted to try to answer this my self, and I thought I had arrived at an answer but then I got stuck again. Can someone explain why this is true? Here is my thought process (for the sake of the question, I leave the velocity as only a function of time and explore its position dependence when its relevant).

If energy is to be conserved, then its time-derivative must be zero, i.e

$\begin{align} E(x,t)&=\frac{1}{2}mv^2+V(x,t)\\ \frac{\partial E}{\partial t}&=mv\frac{dv}{dt}+\frac{\partial V}{\partial x}\frac{dx}{dt}+\frac{\partial V}{\partial t}\\ &=mv\frac{dv}{dt}+\frac{\partial V}{\partial x}v+\frac{\partial V}{\partial t}\\ &=v(m\frac{dv}{dt}+\frac{\partial V}{\partial x})+\frac{\partial V}{\partial t}\\ \end{align}$If all external forces are conservative, then so is the net force, and the potential would be time-independent, so the energy rate would be zero since $m\frac{dv}{dt}=-\frac{\partial V}{\partial x}$.
But, in the general case, if non-conservative forces exist, then would the above statement be true, or would the net force have to be split up as $m\frac{dv}{dt}=F_{cons}+F_{non}$ and then cancel?

$\begin{align} \frac{\partial E}{\partial t}&=v(F_{cons}+F_{non}+\frac{\partial V}{\partial x})+\frac{\partial V}{\partial t}\\ &=vF_{non}+\frac{\partial V}{\partial t} \end{align}$

It looks like the non-conservative force is a source (sink, I guess) of power, and it's somewhat obvious that the potential energy changing would affect the total energy. But what is the nature of the potential? Like, how is it changing? Can you give me a simple physical example?

Also, a little bonus question: What's the difference between power, and the time-derivative of the total energy?

Thanks in advance for your answer! Sorry if it was lengthy.

 

[DustinLiu]

Actually,all your calculation and derivation are right.The wrong is your understanding.Your derivation analyzes the force and the energy of one thing.But conservation of energy is for the system which has no energy exchange with others.

So why you are wrong is the non-conservative forces.When they change the energy you considered,they also change others.For example,considering a falling body,there are two forces--gravity and air friction.When the non-conservative force--air friction--makes the body slower,it also generates heat.

Thats all.



Tips:so sorry that my English is not good enough.If can not understand what I said,you can connect me.

 

[Henryk]

Your math is correct. Potential is only defined for conservative forces but there are other forces that are non-conservative. One example mentioned is friction. The force is given by
F = - f* v
If you plug this into your equation (6) you will see it lead to dissipation of energy (into heat).
Another example of non-conservative force is a force on a charge in the presence of an induced EMF. This is realized in all the transformers in the world. The induced EMF sources energy to the secondary winding of the transformer.

But your question is an example of a time-dependent potential. Here is one: Imagine a charge between two capacitor plates with voltage applied between the plates. At a given instance, there is an electrostatic field between the plates with a well defined potential. Now, instead of a constant voltage, apply a time-varying voltage to the capacitor and you have a time-dependent potential energy. Again, there are devices which use the concept of a charge in a time-dependent potential, they are called quadruple mass spectrometers.

Power is the time derivative of the total energy applied to a system. Consider a plane with engines providing certain power. This power is converted into a sum of kinetic energy of the plane (mV²/2) plus potential energy (mgh) as the plane climbs plus friction losses.

 

[paradoxymoron]

Another example of non-conservative force is a force on a charge in the presence of an induced EMF. This is realized in all the transformers in the world. The induced EMF sources energy to the secondary winding of the transformer.

But your question is an example of a time-dependent potential. Here is one: Imagine a charge between two capacitor plates with voltage applied between the plates. At a given instance, there is an electrostatic field between the plates with a well defined potential. Now, instead of a constant voltage, apply a time-varying voltage to the capacitor and you have a time-dependent potential energy.

When you say the induced EMF sources energy, do you mean it's adding energy to the system? I find that pretty neat, since I've only encountered energy-eating non-conservative forces so far (I haven't taken E&M yet).

In your example of the time-varying voltage, is there a force responsible for this? I would guess it's the electric field, but wouldn't that implicitly be in $m\frac{dv}{dt}$ and cancel with $-\frac{\partial V}{\partial t}$? Or am I missing something else entirely?

 

[Henryk]

Potential is defined only for conservative fields. The definition of the conservative field is ∫F⋅dl = 0 for any integral along a closed loop. The induced EMF is non-conservative. In integral form, the equation is ∫E⋅dl = ^d/_dt∫∫B⋅ds where magnetic field is integrated over the area enclosed by the loop, and it is non-zero for the EMF to develop.
In the transformer, the RHS is the change of the magnetic field of the core. The electric field integral is the induced EMF per turn.