Does non-conservative force have potential energy?

Question

I was reading Classical Dynamics of Particles and Systems, Marion 5th edition, and faced a conundrum.

According to the book, if potential energy is not an explicit function of time, which is $\frac{\partial U}{\partial t} $ = $0$. Then it is conservative force, and if is not equal to $0$, which is $\frac{\partial U}{\partial t} \neq 0 $. Then it is nonconservative force.

I was wondering if a nonconservative force has potential energy. I tried to google this and most of them say there is no potential energy associated with nonconservative forces.

If the potential energy of nonconservative force can't be defined, then how can we differentiate potential energy with respect to time since we don't have a potential energy function for nonconservative force?

Please provide me with an example if there is a potential energy function for nonconservative force.

$\textbf{My Idea:}$

It doesn't matter if we have a potential energy function of nonconservative force or not, we just have to check if the kinetic energy and the potential energy ( conservative force) are conserved.

Regards,

Answer 1

According to the book, if potential energy is not an explicit function of time, which is $\frac{\partial U}{\partial t} $ = $0$. Then it is conservative force, and if is not equal to $0$, which is $\frac{\partial U}{\partial t} \neq 0 $.

Essentially what they are saying is the work done by a conservative force, which is the negative of the change in potential energy, is independent of the path between the initial and final position over which the work is done. This "independence" includes the rate at which the work is done, i.e., it means it is not a function of time.

I was wondering if a nonconservative force has potential energy.

A force does not "have" potential energy. The work done by a non conservative force may or may not result in a change in potential energy, whereas the work done by a conservative force always results in a change in potential energy.

Please provide me with an example if there is a potential energy function for nonconservative force.

While it is not a "potential energy function" the force exerted by you, which is a non conservative force, doing positive work lifting an object results in an increase in gravitational potential energy. But the reason is gravity simultaneously does negative work taking your energy and storing it as gravitational potential energy. The work done by gravity is a potential energy function.

On the other hand, the kinetic friction force is a non conservative force. When an object sliding on a horizontal surface is brought to rest by the opposing kinetic friction force, there is no change in potential energy, only a loss of kinetic energy in the form of heat.

Hope this helps.

Answer 2

So when addressing the "big principle" questions, I like to start with a "toy" system.

For instance, let's have a 1D frictionless mass $m$ moving on an infinite ($x$) horizontal ($g$ exists) surface. We got:

$$ U(x) = 0 $$ $$ T = \frac 1 2 m \dot x^2 $$

$$ L = T-U $$

Looks trivial, but: $L$ is invariant under translations in $t$ and $x$, et voila: energy and momentum are conserved. An example of Noether's theorem.

Now make $x$-finite, a box of size $2L$:

$$ U(x) = \infty \ \ \ \ |x| \ge L $$

zero otherwise.

That broke space translation symmetry, and momentum is no longer conserved (the mass bounces back and forth).

Energy is still conserved, though.

Now add a quadratic potential for $|x| \lt L$:

$$ U(x) = \frac 1 2 mg\alpha x^2 $$

were $\alpha$ is some constant.

So again, momentum is not conserved, but energy:

$$ E = T + U $$

is. The mass moves according harmonic motion (until it hits an edge). No big deal.

We have a conservative force:

$$ F = -\frac{dU}{dx} = -kx $$

with $k = \alpha mg$.

Now imagine the we have our box on a little rocker that oscillates in the small angle limits (on a pivot at $x=0$):

$$ U(x,t) = \frac 1 \alpha mg x^2 + x\sin(\omega t) $$

Now:

$$ U(x, t+\delta t) \ne U(x, t) $$

Time invariance is broken, and energy is no longer conserved.

But that's Noether's theorem, you asked about conservative forces:

$$ F(x, t) = -kx + \sin(\omega t) $$

has time dependence (but not path dependence--kind of hard to do that in 1D). It's not conservative. idk what else to say about it.

A counter problem that just occurred to me, suppose you're lifting the box at a uniform velocity $v$:

$$ U(x, t) = U_{SHO}(x) + mgvt $$

Here:

$$ \frac{dU}{dt} = mgv \ne 0 $$

but the force is conservative. irl it's a Galilean transformation, but it's in a dimension that doesn't exist in my toy Lagrangian's world.

Anyway, I think applying these physics principles to simple toy systems helps learning, and you can keep building until you have all the bells and whistles, such as canonical momentum with a vector potential in it, friction, and so on.