Keplerian Planetary Orbits in Multidimensional Euclidian Spaces

Newton's laws of motion are three physical laws that, together, laid the foundation for classical three dimensional mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. They have been expressed in several different ways, over nearly three centuries. Kepler's laws of planetary motion are also three scientific laws describing the motion of planets around the Sun. Kepler's work published between 1609 and 1619 improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles. Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a non uniform field of force by small steps. It is generally not known that Kepler demonstrated the elliptic orbit to be composed of intelligible differential pieces, in modern language, to result from a differential equation. Kepler was first to attribute planetary orbits to a force from the sun, rather than giving them a predetermined geometric shape. Even though neither the force was known nor its relation to motion, he could determine the differential equations of motion from observation. This is one of the most important achievements in the history of physics. When the only force acting on a particle is always directed towards a fixed point, the motion is called central force motion. The laws which govern this motion were first postulated by Kepler and deduced from observation. In this paper, we will see that these laws are a consequence of Newton’s second law even in multidimensional Euclidian spaces.

Keywords:

Central vector field , Differential equation