# Effective potential concept

Hi,
I'm looking for some information about effective potential, but I haven't found any (Wikipedia, Googled...). I was just willing to get a rough understanding of the concept, and understand what it is.
Thank you very much.

pervect
Staff Emeritus
It's at least mentioned in the context of GR at http://www.fourmilab.ch/gravitation/orbits/. This is pretty much taken from the textbook "Gravitation" by Misner, Thorne, Wheeler which goes into more detail along the same lines.

The fundamental idea here is that a body orbiting a central mass obeys certain differential equations, known as the geodesic equations. Furthermore, due to the presence of symmetries of the problem, certain quantities of the orbital motion analogous to Newtonian energy and angular momentum are conserved.

The term is also sometimes used in the context of analyzing Newtonian orbits. There are some references in Goldstein, "Classical mechanics", I think, but I haven't found any references for the Newtonian usage online. (The idea is definitely disucssed in Goldstein - I think the name is used as well, but I'm not positive).

In the Newtonain version of "effective potential", one observes that the differential equation for the radial part of the motion of a body orbiting a central mass can be separated into two different differential equations (i.e. the equations are separable) - one for the radial part of the motion, and the other for the angular part. The differential equation for only the radial part of the motion is a 1-d problem that can be physically re-interpreted as a mass and a (fictitious) non-linear spring. The nonlinear spring can be modelled by an "effective potential".

The GR usage is very similar - the "effective potential" is similar to the above fictitious 1-d potential in the Newtonian case. The "energy at infinity" in GR is more closely analogous to the usual Newtonian potential (but one must be careful not to assume the analogy goes too far).

I hope this helps some - I've tried to be clear, but it's early in the morning here :-(. I've got to run now, but I'll check back later.

However there's something that's not clear in my mind: in fourmilab.ch, the author talks about "the position of the test mass on the gravitational energy curve". Does this mean that effective potential is gravitationnal potential energy?

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pervect
Staff Emeritus
However there's something that's not clear in my mind: in fourmilab.ch, the author talks about "the position of the test mass on the gravitational energy curve". Does this mean that effective potential is gravitational potential energy?
Not really. I'd suggest interpreting what they wrote as "the position of the test mass on the effective potential curve" instead. The authors of this webpage also call it the "gravitational effective potential" curve elsewhere, so they haven't "polished" their webpage, calling the same concept by several slightly different names.

The idea of separating kinetic and potential energy makes some sense in Newtonian theory, which has an "absolute space". In GR, one is better off avoiding this sort of separation, and dealing only with total energy, rather than attempting to separate it into "kinetic" and "potential" parts, because GR has no concept of "absolute space".

Thus the best approach in the spirit of GR for this particular case would be to saythat GR's concept of "energy at infinity" is approximately the same as the Newtonian concept of "kinetic plus potential energy".

On the webpage, the "energy at infinity" is represented by the symbol $$\~{E}$$ which is a constant number for any particle following a geodesic.

Note that "the position of the test mass on the gravitational energy curve" that the authors were talking about isn't the same as the "energy at infinity" - the former is a fictitious number that results when one reduces the problem to one dimension.

Also note that in the interest of trying to keep things simple, I've been talking about only a static, Schwarzschild geometry. The concept of energy in GR has other nuances which I've deliberately avoided.

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OK, so as far as I've understood, effective potential is a field where those geodesic equations apply and any object is subject to energy at infinity. Is that right?
Thanks.

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pervect
Staff Emeritus