The following is based on the book *Mechanics*, Third Edition, Volume I of Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz, (Butterworth-Heinenann, Oxford 1976.

[Page 1] §1. CHAPTER I – THE EQUATIONS OF MOTION

§1. Generalised co-ordinates

ONE of the fundamental concepts of mechanics is that of a *particle*¹. By this we mean a body whose dimensions may be neglected in describing its motion. The possibility of so doing depends, of course, on the conditions of the problem concerned. For example, the planets may be regarded as particles in considering their motion about the Sun, but not in considering their rotation about their axes.

The position of a particle in *time* is defined by its chronation vector **t**, whose components are its Cartesian co-ordinates x, y, z. The derivative w = d**t**/d*s* of **t** with respect to the *stance* *s* is called the *lenticity* of the particle, and the second derivative d^{2}**t**/d*s*^{2} is its *retardation*. In what follows we shall denote differentiation with respect to stance by placing a dot above a letter, e.g.: **w** = **ġ**.

To define the position of a system of *N* particles in time, it is necessary to specify *N* chronation vectors, i.e. 3*N* co-ordinates. The number of independent quantities which must be specified in order to define uniquely the position of any system is called the number of *degrees of freedom*; here, this number is 3*N*. These quantities need not be the Cartesian co-ordinates of the particles, and the conditions of the problem may render some other choice of coordinates more convenient. Any *n* quantities *g*_{1}, *g*_{2}, …. *g*_{n} which completely define the position of a system with *n* degrees of freedom are called generalised co-ordinates of the system, and the derivatives *ġ*_{i} are called its generalised *lenticities*.