Adding a constant to potential energy doesn't change action?

[blaughli]

Hello. I've been watching Susskind's online Stanford lectures on classical mechanics to review the subject, and I believe he said that adding a constant to the potential energy does not change the action of a system. I see how it doesn't change the Euler-Lagrange equations and therefore doesn't affect the equations of motion (and therefore the trajectories), yet the integral of a constant is non-zero so I don't see how adding a constant to U in A = ∫(T-U)dt wouldn't change the action A. Where have I gone wrong? There seems to be an inconsistency in saying that the action changes (implying it's not at a minimum and therefore doesn't describe the true trajectory) while the E-L equations don't change (implying no change in trajectory). Thank you, sorry if I'm missing something basic and have wasted your time by not thinking about this more on my own :)

 

[Orodruin]

It does change the action. However, it does not change the variation of the action. The variation of the action should be taken with respect to the dynamical variables. The potential is not in itself a dynamical variable, but generally a function of them. Adding a constant to the potential does not change the derivatives of the potential.

 

[Dale]

There seems to be an inconsistency in saying that the action changes (implying it's not at a minimum and therefore doesn't describe the true trajectory) while the E-L equations don't change (implying no change in trajectory).

Think about adding a constant to a parabola. The location of the minimum does not change, but the value does. Similarly, the path that is stationary will not change, even though the value of the action does.

 

[anorlunda]

There seems to be an inconsistency in saying that the action changes (implying it's not at a minimum and therefore doesn't describe the true trajectory)

It sounds like you are misinterpreting "minimum" In this case, minimum just means the bottom of a curve, such that any incremental change results in a higher action. For example, see the picture below. The red line in this image could represent the constant. Your comment makes it sound like you think minimum must mean y=0 in the picture.

 

[Orodruin]

It sounds like you are misinterpreting "minimum" In this case

Also, this reminds me of my pet peeve:

implying it's not at a minimum and therefore doesn't describe the true trajectory

The action does not need to be minimal for a path satisfying the equations of motion. The "principle of least action" is a misnomer and it is more accurate to call it the principle of stationary action. Of course, what has been said about minima in this thread also holds for stationary points. A stationary point of an action will continue being a stationary point even if you add a constant to the potential.

 

[anorlunda]

@Orodruin is right. Stationary is a better word than minimum. That allows for the case with multiple local minima, each of which is a stationary point. It also allows for maxima or saddle points.

Huh, should we allow maxima and saddle points? See
https://en.wikipedia.org/wiki/Principle_of_least_action#Jacobi_and_Morse
https://en.wikipedia.org/wiki/Morse_theory
for more discussion of degenerate points.

 

[Jason Bennett]

Hello. I've been watching Susskind's online Stanford lectures on classical mechanics to review the subject, and I believe he said that adding a constant to the potential energy does not change the action of a system. I see how it doesn't change the Euler-Lagrange equations and therefore doesn't affect the equations of motion (and therefore the trajectories), yet the integral of a constant is non-zero so I don't see how adding a constant to U in A = ∫(T-U)dt wouldn't change the action A. Where have I gone wrong? There seems to be an inconsistency in saying that the action changes (implying it's not at a minimum and therefore doesn't describe the true trajectory) while the E-L equations don't change (implying no change in trajectory). Thank you, sorry if I'm missing something basic and have wasted your time by not thinking about this more on my own :)

Hey! Can you recall which # of Susskind's lectures this was from?

 

[etotheipi]

Hey! Can you recall which # of Susskind's lectures this was from?

I'd imagine it be the one titled "Lagrangian, least action, Euler-Lagrange equations"... enjoy!
http://theoreticalminimum.com/courses/classical-mechanics/2011/fall/lecture-3