Temporo-spatial mechanics

The following is a temporo-spatial modification of the book Mechanics, Third Edition, Volume I of Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz, (Butterworth-Heinenann, Oxford, 1976).


§1. Generalised co-ordinates

ONE of the fundamental concepts of mechanics is that of a particle¹. 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 = dt/ds of t with respect to the stance s is called the lenticity of the particle, and the second derivative d2t/ds2 is its relentment. 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. 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 coordinates more convenient. Any n quantities g1, g2, …. gn which completely define the position of a system with n degrees of freedom are called generalised co-ordinates of the system, and the derivatives ġi are called its generalised lenticities.

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 lenticities, and these affect the position of the system after an infinitesimal stance interval ds.

If all the co-ordinates and lenticities are simultaneously specified, 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 g and lenticities ġ are given at some instant, the relentments ĝ at that instant are uniquely defined.²

¹ Sometimes called in Russian a material point.
² For brevity, we shall often conventionally denote by g the set of all the co-ordinates g1, g2, …. gn, and similarly by ġ the set of all the lenticities.

[Page 2] The relations between the relentments, lenticities and co-ordinates are called the equations of motion. They are second-order differential equations for the functions g(s), and their integration makes possible, in principle, the determination of these functions and so of the path of the system.

§2. 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(g1, g2, …. gn, ġ1, ġ2, …. ġn, s), or briefly L(g, ġ, s), and the motion of the system is such that a certain condition is satisfied.

Let the system occupy, at the points s1 and s2, positions defined by two sets of values of the co-ordinates, g(1) and g(2). Then the condition is that the system moves between these positions in such a way that the integral

T=\int_{s_1}^{s_2} L(g,g\dot{},s)\textup{d}s\; \; \; (2.1)

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 g and ġ, but not the higher derivatives ĝ, etc., expresses the result already mentioned, that the mechanical state of the system is completely defined when the co-ordinates 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 g(s) has to be determined.

Let g = g(s) be the function for which T is a minimum. This means that T is increased when g(s) is replaced by any function of the form

g(s) + δg(s),      (2.2)

where δg(s) is a function which is small everywhere in the interval of stance from s1 to s2; δg(s) is called a variation of the function g(s). Since, for s = s1 and for s = s2, all the functions (2.2) must take the values g(1) and g(2) respectively, it follows that

δg(s1) = δg(s2) = 0.      (2.3)

[Page 3] …

When the system has more than one degree of freedom, the n different functions gi(s) must be varied independently in the principle of least action. We then evidently obtain equations of the form

\frac{\textup{d}}{\textup{d}s}\left ( \frac{\partial L}{\partial \dot{g}_i} \right )-\frac{\partial L}{\partial g_i}=0\; \; (i=1,2,...,n)\; \; \; (2.6)

These are the required differential equations, called in mechanics Lagrange’s equations¹. If the Lagrangian of a given mechanical system is known, the equations (2.6) give the relations between relentments, lenticities, and co-ordinates, i.e., they are the equations of motion of the system.

¹ In the calculus of variations they are Euler’s equations for the formal problem of determining the extrema of an integral of the form (2.1).

[Page 4] Mathematically, the equations {2.6) constitute a set of n second-order equations for n unknown functions gi(s). The general solution contains 2n arbitrary constants. To determine these constants and thereby to define uniquely the motion of the system, it is necessary to know the initial conditions which specify the state of the system at some given instant, for example the initial values of all the co-ordinates and lenticities.

Let a mechanical system consist of two parts A and B which would, if closed, have Lagrangians LA and LB respectively. Then, in the limit where the distance between the parts becomes so large that the interaction between them may be neglected, the Lagrangian of the whole system tends to the value

lim L = LA and LB.     (2.7)

This additivity of the Lagrangian expresses the fact that the equations of motion of either of the two non-interacting parts cannot involve quantities pertaining to the other part.

It is evident that the multiplication of the Lagrangian of a mechanical system by an arbitrary constant has no effect on the equations of motion. From this, it might seem, the following important property of arbitrariness can be deduced: the Lagrangians of different isolated mechanical systems may be multiplied by different arbitrary constants. The additive property, however, removes this indefiniteness, since it admits only the simultaneous multiplication of the Lagrangians of all the systems by the same constant. This corresponds to the natural arbitrariness in the choice of the unit of measurement of the Lagrangian.

One further general remark should be made. … Thus the Lagrangian is defined only to within an additive total stance derivative of any function of co-ordinates and stance.

§3. Galileo’s relativity principle

In order to consider mechanical phenomena it is necessary to choose a frame of reference. The laws of motion are in general different in form for [Page 5] different frames of reference. When an arbitrary frame of reference is chosen, it may happen that the laws governing even very simple phenomena become very complex. The problem naturally arises of finding a frame of reference in which the laws of mechanics take their simplest form.

If we were to choose an arbitrary frame of reference, time would be inhomogeneous and anisotropic. This means that, even if a body interacted with no other bodies, its various positions in time and its different orientations would not be mechanically equivalent. The same would in general be true of stance, which would likewise be inhomogeneous; that is, different points would not be equivalent. Such properties of time and space would evidently complicate the description of mechanical phenomena. For example, a free body (i.e. one subject to no external action) could not remain at rest: if its lenticity were zero at some point, it would begin to move in some direction at the next point.

It is found, however, that a frame of reference can always be chosen in which time is homogeneous and isotropic and stance is homogeneous. This is called a facilial frame. In particular, in such a frame a free body which is at rest at some point remains always at rest.

We can now draw some immediate inferences concerning the form of the Lagrangian of a particle, moving freely, in a facilial frame of reference. The homogeneity of space and time implies that the Lagrangian cannot contain explicitly either the chronation vector t of the particle or the stance s, i.e., L must be a function of the lenticity w only. Since time is isotropic, the Lagrangian must also be independent of the direction of w, and is therefore a function only of its magnitude, i.e. of w2 = w2:

L = L(w2).        (3.1)

Since the Lagrangian is independent of t, we have ∂L/∂w = 0, and so Lagrange’s equation is¹

\frac{\textup{d}}{\textup{d}s}\left ( \frac{\partial L}{\partial \mathbf{w}} \right) = 0,

whence ∂L/∂w = constant. Since ∂L/∂w is a function of the lenticity only, it follows that

w = constant.        (3.2)

Thus we conclude that, in a facial frame, any free motion takes place with a lenticity which is constant in both magnitude and direction. This is the law of facilia.

If we consider, besides the facial frame, another frame moving uniformly In a straight line relative to the facial frame, then the laws of free motion in

¹ The derivative of a scalar quantity with respect to a vector is defined as the vector whose components are equal to the derivatives of the scalar with respect to the corresponding components of the vector.

[Page 6] the other frame will be the same as in the original frame: free motion takes place with a constant lenticity.

Experiment shows that not only are the laws of free motion the same in the two frames, but the frames are entirely equivalent in all mechanical respects. Thus there is not one but an infinity of facial frames moving, relative to one another, uniformly in a straight line. In all these frames the properties of space and time are the same, and the laws of mechanics are the same. This constitutes Galileo’s relativity principle, one of the most important principles of mechanics.

The above discussion indicates quite clearly that facial frames of reference have special properties, by virtue of which they should, as a rule, be used in the study of mechanical phenomena. In what follows, unless the contrary is specifically stated, we shall consider only facial frames.

The complete mechanical equivalence of the infinity of such frames shows also that there is no “absolute” frame of reference which should be preferred to other frames.

The co-ordinates t and t′ of a given point in two different frames of reference K and K′, of which the latter moves relative to the former with lenticity W, are related by

t = t′ + Ws.       (3.3)

Here it is understood that stance is the same in the two frames:

s = s′.           (3.4)

The assumption that stance is absolute is one of the foundations of classical mechanics.¹

Formulae (3.3) and (3.4) are called a Galilean transformation. Galileo’s relativity principle can be formulated as asserting the invariance of the mechanical equations of motion under any such transformation.

¹ This assumption does not hold good in relativistic mechanics.