How Do You Find A Central Force Orbit?

[Chronothread]

If you are given a central force field and an initial velocity of a particle in this field, how would you go about finding the orbit of the particle in polar coordinates?

Thanks for you help and time.

 

[nicksauce]

Well the easiest way would probably be to calculate the Lagrangian, and then solve the Euler-Lagrange equation in polar coordinates.

 

[D H]

What exactly do you mean by "finding the orbit of the particle in polar coordinates"? It sounds like you are talking about Keplerian orbital elements.

 

[LogicalTime]

there are only a few central force laws that allow orbits, most do not

 

[nasu]

By orbits you mean "closed orbits" only?

 

[Redbelly98]

there are only a few central force laws that allow orbits, most do not

Any central force that depends on distance only, and not angle, would allow for a circular closed orbit. Or do you mean orbits that are stable against slight perturbations?

 

[LogicalTime]

yes, sry I meant stable closed orbits

how is that lagrangian coming along chrono?

 

[Chronothread]

Oops, sorry for never replying. I got caught up with work on other things and haven't visited physics forums for a bit.

We were doing some things involving orbits in central force (not necessarily closed) in my mechanics class and I had posted this originally to see if I could get a way to understand the concept. We weren't at Lagrangian stuff yet so I wasn't to use that yet. I eventually ended up using the energy equation of an orbit and a bunch of other energy equations. I still haven't used Lagrangian stuff too much yet, but it definitely seems like it'll be more useful in a situation like this.

And sorry for not replying and thank you for all your responses.

 

[flatmaster]

Any central force that depends on distance only, and not angle, would allow for a circular closed orbit. Or do you mean orbits that are stable against slight perturbations?

This is interesting, the idea of a central force with an angular dependence. The definition of a central force means the force vector will always point toward the center, but there is some angular dependence. Physically, when would something experience some kind of force? My first instinct was an angular dependence due to the gravity of some additional mass (such as jupiter), however, this vector component is not necessarily near to being in the radial direction.

 

[Vanadium 50]

yes, sry I meant stable closed orbits

You only get that with two force laws: k/x^2 and kx.

There's a treatment of this in the first book of Landau and Lifschitz. Like most everything they do, the material is excellent, but quite terse. They also don't hold you hand by giving you a lot of intermediate steps in their derivations.