Introduction

Abstract

In 1842 – 1843, when Jacobi was a professor at Königsberg University, he delivered a special series of lectures on dynamics. The lectures were devoted to the dynamics of a system of n mass points whose motion depends only on the distances between them and is independent of velocities. In this connection, by deriving the law of conservation of energy from the equations of motion of mass points for a conservative system, where the force function is a homogeneous function of space co-ordinates, Jacobi gave this law an unusual form and a new content. In transforming the equations of motion, he introduced an expression for the system’s centre of mass. Then, following Lagrange, he separated the motion of the centre of mass from the relative motion of the mass points. Making the centre of mass coincide with the origin of the co-ordinate system, he obtained the following equation (Jacobi, 1884):

$$ \frac{{{{d}^{2}}}}{{d{{t}^{2}}}}\left( {\sum {{{m}_{i}}} r_{i}^{2}} \right) = - (2k + 4)U + 4E$$

((1.1))

where m_i is their mass point i; $r_{i}=\sqrt{x_{i}^{2}+y_{i}^{2}+z_{i}^{2}}$ the distance between the points and the centre of mass; k the degree of homogeneity of the force function; U the system’s potential energy; and E its total energy.