Kepler problem in parabolic coordinates

Thread starter Kate_12
Start date Jun 18, 2021
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In summary, the Kepler problem in parabolic coordinates is a mathematical problem used to describe the motion of a particle under the influence of a central force. It is unique in that it allows for the consideration of both bound and unbound orbits and simplifies the equations of motion. The key equations used in solving this problem are derived from Newton's laws of motion and have applications in astrodynamics and celestial mechanics. However, there are limitations to this problem, such as assuming a perfect spherical central mass and not accounting for other forces.

Jun 18, 2021

Kate_12

Homework Statement: kepler problem
H=1/2m(px^2+py^2+pz^2)-k/(x^2+y^2+z^2)^1/2
with parabolic coordinates (a,b,c)
x=sqrt(ab)cos c
y=sqrt(ab)sin c
z=(a-b)/2
1) rewrite H as a function of new canonical variables (a,b,c, pa,pb,pc)
2) Hamilton-Jacobi equation in this coordinate system turns out to be completely separable. Using the Ansatz S=Wa(a)+Wb(b)+Wc(c)-Et, write the partial differential equation for each Wa, Wb, Wc with suitable separation constants.

Relevant Equations: Hamilton Jacobi equation

I solve (1).
But to solve (2), What should be the suitable separation constants?
I am so confused...

E=2/(m*(a+b)) * (a*(dWa/da)^2+b*(dWb/db)^2-k)+l^2/(2mab)
where l(constant) is pc since c is cyclic.

What should I do to solve the problem?

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Jun 20, 2021

anuttarasammyak

Gold Member

if you do not mind could you write down H of your solution 1 ?

1. What is the Kepler problem in parabolic coordinates?

The Kepler problem in parabolic coordinates is a mathematical model used to describe the motion of a particle under the influence of a central force, such as gravity. It is based on the laws of motion and gravitation proposed by Johannes Kepler in the 17th century.

2. How are parabolic coordinates used in the Kepler problem?

Parabolic coordinates are a type of coordinate system that is particularly useful for solving the Kepler problem. They consist of two focal points and a third coordinate, which is the distance from the focus line. By using these coordinates, the equations of motion for a particle can be simplified and solved analytically.

3. What is the significance of the Kepler problem in parabolic coordinates?

The Kepler problem in parabolic coordinates is significant because it provides a simplified and elegant solution to the motion of a particle under a central force. It is also a fundamental problem in classical mechanics and has applications in many fields, such as celestial mechanics, astrophysics, and aerospace engineering.

4. What are some real-world examples of the Kepler problem in parabolic coordinates?

One real-world example of the Kepler problem in parabolic coordinates is the motion of planets and satellites around a central body, such as the Earth orbiting around the Sun. Another example is the motion of comets around the Sun, which can be approximated as a parabolic trajectory.

5. Are there any limitations to the Kepler problem in parabolic coordinates?

While the Kepler problem in parabolic coordinates provides a simplified solution to the motion of a particle under a central force, it does have some limitations. For example, it assumes that the central force is constant and does not take into account any external forces or perturbations. Additionally, it only applies to two-body systems and cannot be used for more complex systems with multiple interacting objects.