Angular Momenta and Actions of Orbits

[tade]

Consider a tiny planet orbiting a massive star.

If the value of the planet's angular momentum (w.r.t. to star) is fixed, does the action of the planet's orbit depend on the eccentricity of the orbit?

 

[drvrm]

Consider a tiny planet orbiting a massive star.

If the value of the planet's angular momentum (w.r.t. to star) is fixed, does the action of the planet's orbit depend on the eccentricity of the orbit?

pl.explain the term action of ...

 

[tade]

pl.explain the term action of ...

the integral of momentum w.r.t. displacement for one loop of the orbit

 

[drvrm]

If the value of the planet's angular momentum (w.r.t. to star) is fixed, does the action of the planet's orbit depend on the eccentricity of the orbit?

the value of angular momentum defines the trajectory of the body as per Keplar's laws and thus has a defined eccentricity -so the action integral will have a stationary value for actual/ particular path . For different eccentricities the paths will be different and the action should have another stationary value,
for example;

The path in space of a body in a gravitational field can be found using the action principle. for a free falling body, this trajectory is a geodesic.
but different geodesics can exist for different initial conditions.
similarly if a body is projected in space with certain energy and falls in the field of another body it can move on a variety of paths but the actual path will have a stationary action.

 

[tade]

Let's keep it to classical inverse square law.

Because we can have different eccentricities for the same value of angular momentum.