In this section we're going to use the Hamilton-Jacobi
equation in order to derive analytical formulae for
a motion of a material point in the central Newtonian *M*/*r* potential.

The derived formulae can then be used to compare an approximate numerical solution against an analytical, i.e., exact solution to the problem.

The problem is easiest to describe in spherical coordinates,
*r*,
and .
In these coordinates the Hamiltonian
assumes the following form:

(4.41) |

where we have assumed that the gravitational constant

The Hamilton-Jacobi equation that corresponds to that Hamiltonian
is

(4.42) |

The method that is commonly use here is called the

- time,
*t* - angle

(4.43) |

where

Substituting this into the Hamilton-Jacobi equation yields:

(4.44) |

Multiply this equation by 2

(4.45) |

Because expressions on both sides of this equation depend on different variables, the equality can hold only if they are equal to the same constant,

(4.46) |

And this implies that:

= | (4.47) | ||

= | (4.48) |

These, in turn, are first order ordinary differential equations, which can be readily integrated.

But recall that there are additional conditions that
function S must satisfy. In this case

(4.49) |

where

= | Q_{r} |
(4.50) | |

= | (4.51) | ||

= | (4.52) |

where

The last equation (4.59) yields:

(4.56) |

It can be proven that this relation is satisfied by a flat motion, i.e., that the material point moves in a plane with vector

(4.57) |

Equation (4.58) yields the shape of the orbit:

(4.58) |

with the following solution:

(4.59) |

where

(4.60) |

Other parameters pertaining to the orbit are:

(4.61) |

Remember that for a trapped particle the energy is negative.

(4.62) |

(4.63) |

is the distance of the closest approach.

Equation (4.57) yields the time dependence of *r* versus *t*:

(4.64) |

The solution to this equation is quite complicated and can be given in terms of Bessel functions and harmonic motion with the mean circular frequency of

(4.65) |