Planetary Systems

  • I demonstrate by means of philosophy that the earth is round, and is inhabited on all sides; that it is insignificantly small, and is borne through the stars. (Johannes Kepler)

The equations of motion for a planet (viewed as a pointlike particle) of unit mass orbiting a fixed Sun of unit mass centered at the origin, take the form

  • dx=vdt,\quad v=Fdt,


  • F(x)=-\frac{x}{\vert x\vert^3}         (Newton’s Gravitation Law)

is the gravitational force. This is a force acting at distance, because the Sun at the origin acts at the point x at the distance \vert x\vert from the origin.

Note that Newton’s Gravitation Law is the famous inverse square law of gravitation stating that the magnitude of the gravitational force F between two bodies with mass M_1 and M_2 at distance r, is given by

  • F=G\frac{M_1M_2}{r^2},

where G is the gravitational constant.

We shall prove below that Newton’s Gravitation Law is a consequence of the fact that the gravitational potential satisfies a certain differential equation named Laplace’s equation, and we shall uncover the assumptions leading to Laplace’s equation. We can this way motivate that the exponent in Newton’s Law is 2 and nothing else.


Crab nebulosa: A macroscopic particle system.}

                          Galileo presenting mathematical arguments to disbelieving Catholic priests.

Galileo's telescope


To Think About

To Read

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