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title: 1. Generalized co-ordinates
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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 space is defined by its radius vector $r$, whose components are its Cartesian co-ordinates $x$, $y$, $z$. The derivative $\v{v} = \dd{\v{r}}/\dd{t}$. of $r$ with respect to the time $t$ is called the velocity of the particle, and the second derivative $\dd[2]{\v{r}}/\dd[2]{t}$ is its acceleration. In what follows we shall, as is customary, denote differentiation with respect to time by placing a dot above a letter: $\v{v} = \dot{\v{r}}$.

To define the position of a system of N particles in space, it is necessary to specify $N$ radius vectors, i.e. $3N$ 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 $3N$. These quantities need not be the Cartesian co-ordinates of the particles, and the conditions of the problem may render some other choice of co-ordinates more convenient. Any $S$ quantities $q_1$, $q_2$, ..., $q_s$ which completely define the position of a system with $S$ degrees of freedom are called generalised co-ordinates of the system, and the derivatives $q_i$ are called its generalised velocities.

When the values of the generalised co-ordinates are specified, however, the "mechanical state" of the system at the instant considered is not yet determined in such a way that the position of the system at subsequent instants can be predicted. For given values of the co-ordinates, the system can have any velocities, and these affect the position of the system after an infinitesimal time interval $\dd{t}$.

If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically, this means that, if all the co-ordinates $q$ and velocities $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined[^2].

The relations between the accelerations, velocities and co-ordinates are called the equations of motion. They are second-order differential equations for the functions $q(t)$, and their integration makes possible, in principle, the determination of these functions and so of the path of the system.

[^1]: Sometimes called in Russian a material point
[^2]: For brevity, we shall often conventionally denote by q the set of all the co-ordinates $q_1,q_2,...,q_s$ and similarly by $\dot{q}$ the set of all velocities.
