How Do You Find A Central Force Orbit?

In summary: Then again, in most situations, you'll just want to solve the Euler-Lagrange equation.In summary, if you are given a central force field and an initial velocity of a particle in this field, you would go about finding the orbit of the particle in polar coordinates by calculating the Lagrangian and solving the Euler-Lagrange equation.
  • #1
Chronothread
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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.
 
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  • #2
Well the easiest way would probably be to calculate the Lagrangian, and then solve the Euler-Lagrange equation in polar coordinates.
 
  • #3
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.
 
  • #4
there are only a few central force laws that allow orbits, most do not
 
  • #5
By orbits you mean "closed orbits" only?
 
  • #6
LogicalTime said:
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?
 
  • #7
yes, sry I meant stable closed orbits

how is that lagrangian coming along chrono?
 
  • #8
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.
 
  • #9
Redbelly98 said:
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.
 
  • #10
LogicalTime said:
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.
 

1. What is a central force orbit?

A central force orbit is a type of orbit in which the force acting on a body is directed towards a fixed point, known as the center of force. Examples of central forces include gravity, electrostatic forces, and magnetic forces.

2. How is a central force orbit different from other types of orbits?

A central force orbit is different from other types of orbits because the force acting on the body is always directed towards the center of force, resulting in a curved path. In contrast, other types of orbits may have varying forces and result in different types of paths, such as ellipses, parabolas, or hyperbolas.

3. How do you calculate the trajectory of a central force orbit?

The trajectory of a central force orbit can be calculated using the principles of classical mechanics, specifically Newton's laws of motion. The specific equations used will depend on the type of central force present and may involve concepts such as angular momentum and energy conservation.

4. Can central force orbits exist in non-circular shapes?

Yes, central force orbits can exist in non-circular shapes. In fact, most central force orbits are not perfectly circular but rather elliptical in shape. The shape of the orbit will depend on the strength of the central force and the initial conditions of the body, such as its velocity and position.

5. How do you find the stable orbits in a central force field?

The stable orbits in a central force field can be found by analyzing the potential energy function of the central force. The stable orbits will correspond to the local minima of the potential energy function, where the force acting on the body is balanced and the orbit remains stable over time.

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