Kepler problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them. The force may be either attractive or repulsive. The problem is to find the position or speed of the two bodies over time given their masses, positions, and velocities. Using classical mechanics, the solution can be expressed as a Kepler orbit using six orbital elements.
The Kepler problem is named after Johannes Kepler, who proposed Kepler's laws of planetary motion (which are part of classical mechanics and solved the problem for the orbits of the planets) and investigated the types of forces that would result in orbits obeying those laws (called Kepler's inverse problem).[1]
For a discussion of the Kepler problem specific to radial orbits, see Radial trajectory. General relativity provides more accurate solutions to the two-body problem, especially in strong gravitational fields.
Applications
The Kepler problem arises in many contexts, some beyond the physics studied by Kepler himself. The Kepler problem is important in
The Kepler problem and the
Mathematical definition
The
where k is a constant and represents the unit vector along the line between them.[2] The force may be either attractive (k<0) or repulsive (k>0). The corresponding scalar potential is:
Solution of the Kepler problem
The equation of motion for the radius of a particle of mass moving in a central potential is given by Lagrange's equations
and the angular momentum is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force equals the centripetal force requirement , as expected.
If L is not zero the definition of angular momentum allows a change of independent variable from to
giving the new equation of motion that is independent of time
The expansion of the first term is
This equation becomes quasilinear on making the change of variables and multiplying both sides by
After substitution and rearrangement:
For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written
The orbit can be derived from the general equation
whose solution is the constant plus a simple sinusoid
where (the eccentricity) and (the phase offset) are constants of integration.
This is the general formula for a conic section that has one focus at the origin; corresponds to a circle, corresponds to an ellipse, corresponds to a parabola, and corresponds to a hyperbola. The eccentricity is related to the total energy (cf. the Laplace–Runge–Lenz vector)
Comparing these formulae shows that corresponds to an ellipse (all solutions which are closed orbits are ellipses), corresponds to a parabola, and corresponds to a hyperbola. In particular, for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius).
For a repulsive force (k > 0) only e > 1 applies.
See also
- Action-angle coordinates
- Bertrand's theorem
- Binet equation
- Hamilton–Jacobi equation
- Laplace–Runge–Lenz vector
- Kepler orbit
- Kepler problem in general relativity
- Kepler's equation
- Kepler's laws of planetary motion
References
- Addison Wesley.
- ISBN 978-0-387-96890-2.