The central force F that varies in strength as the inverse square of the distance r between them: where k is a constant and represents the unit vector along the line between them. The force may be either attractive or repulsive. The corresponding scalar potential is:
Solution of the Kepler problem
The equation of motion for the radius of a particle of massmoving in a central potential is given by Lagrange's equations 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 and 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 Comparing these formulae shows that corresponds to an ellipse, corresponds to a parabola, and corresponds to a hyperbola. In particular, for perfectly circular orbits. For a repulsive force only e > 1 applies.
If we restrict ourselves to the orbiting plane, there is an easy way how to obtain a rough shape of the orbit in pedal coordinates. Remember that a given point on a curve in pedal coordinates is given by two numbers, where is the distance from the origin and is the distance of the origin to the tangent line at . The Kepler problem in a plane asks for a solution of the system of differential equations: where is the product of the gravitational body's mass and gravitational constant. Making the scalar product of the equation with we obtain Integrating we get the first conserved quantity : which corresponds to the energy of the orbiting object. Similarly, making the scalar product with we get with the integral corresponding to the object's angular momentum. Since substituting the above conserved quantities we immediately obtain: which is the equation of the conic section in pedal coordinates. Notice that only 2 conserved quantities are needed to obtain the shape of the orbit. This is possible since the pedal coordinates do not describe a curve in full detail. They are generally indifferent to parametrization and also to a rotation of the curve about the origin—which is an advantage if you care only about the general shape of the curve and do not want to be distracted by details. This approach can be applied to a wide range of central and Lorentz-like force problems as discovered by P. Blaschke in 2017.