Mechanics

Author: L.D.Landau et al.
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Publish Date: 2003-01-01
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Fragment: THE EQUATIONS OF MOTION? Generalised Co-ordinates
ONE of the fundamental concepts of mechanics is that of a particle. 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 coordinates x, y, z. The derivative v = dr/dt of r with respect to time is called the velocity of the particle, and the second derivative d2r/dt2 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 = r. To define the position of a system of N particles in space, it is necessary to specify N radius vectors, i.e., 3N coordinates. 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 coordinates of the particles, and the conditions of the problem may render some other choice of coordinates more convenient. Any quantities q?, q?, ..., q?, which completely define the position of a system with s degrees of freedom, are called the generalised coordinates of the system, and the derivatives q? are called its generalised velocities. When the values of the generalised coordinates 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 coordinates, the system can have any velocities, and these affect the position of the system after an infinitesimal time interval dt. If all the coordinates 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 coordinates q and velocities q are given at some instant, the accelerations q at that instant are uniquely defined.
The Equations of Motion
The relations between the accelerations, velocities, and coordinates 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.
The principle of least action
The most general formulation of the law governing the motion of mechanical systems is the principle of least action or Hamilton's principle, according to which every mechanical system is characterised by a definite function L(q?, q?, ..., q?, q??, q??, ..., q??, t), or briefly L(q, q?, t), and the motion of the system is such that a certain condition is satisfied. Let the system occupy, at the instants t? and t?, positions defined by two sets of values of the coordinates, q(1) and q(2). Then the condition is that the system moves between these positions in such a way that the integral takes the least possible value. The function L is called the Lagrangian of the system concerned, and the integral (2.1) is called the action. The fact that the Lagrangian contains only q and q?, but not the higher derivatives q?, q?, etc., expresses the result already mentioned, that the mechanical state of the system is completely defined when the coordinates and velocities are given. Let us now derive the differential equations which solve the problem of minimising the integral (2.1). For simplicity, we shall at first assume that the system has only one degree of freedom, so that only one function q(t) has to be determined. Let q = q(t) be the function for which S is a minimum. This means that S is increased when q(t) is replaced by any function of the form where q(t) is a function which is smaller everywhere in the interval of time from t? to t?; q(t) is called a variation of the function q(t). Since, for t = t? and for t = t?, all the functions (2.2) must take the values q(1) and q(2) respectively, it follows that