[Concept] : Conservation Law

If the laws of forces acting on a system’s particles and the state of the system at a certain initial moment are known, the motion equations can help predict the subsequent behaviour of the system, i.e. find its state at any moment of time. However, an analysis of a system’s behavior by the use of the motion equations requires so much effort, that a comprehensive solution seems to be practically impossible.

Under these circumstances the following question natural­ly comes up: are there any general principles following from Newton’s laws that would help avoid these difficulties by opening up some new approaches to the solution of the problem.

It appears that such principles exist. They are called conservation laws.

As it was mentioned, the state of a system varies in the course of time as that system moves. However, there are some quantities, state functions, which possess the very important and remarkable property of retaining their values constant with time. Among these constant quantities, energy, momentum and angular momentum play the most significant role. These three quantities have the important general property of additivity: their value for a system com­posed of parts whose interaction is negligible is equal to the sum of the corresponding values for the individual con­stituent parts (incidentally, in the case of momentum and angular momentum additivity holds true even in the pres­ence of interaction). It is additivity that makes these three quantities extremely important.

By way of explanation, the energy conservation law is associated with uniformity of time, while the laws of conservation of momentum and angular momentum with uniformity and isotropy of space respectively. This implies that the conservation laws listed above can be derived from Newton’s second law. We shall not, however, discuss this problem in more detail.

The laws of conservation of energy, momentum and angu­lar momentum fall into the category of the most fundamental principles of physics, whose significance cannot be overes­timated. These laws have become even more significant since it was discovered that they go beyond the scope of mechanics and represent universal laws of nature. In any case, no phenomena have been observed so far which do not obey these laws. They are among the few most general laws underlying contemporary physics.

The conservation laws do not depend on either the paths of particles or the nature of acting forces. Consequently, they allow us to draw some general and essential conclusions about the properties of various mechanical processes without resorting to their detailed analysis by means of motion equations. For example, as soon as it turns out that a cer­tain process is in conflict with the conservation laws, one can be sure that such a process is impossible and it is no use trying to accomplish it.

Since the conservation laws do not depend on acting forces, they may be employed even when the forces are not known. In these cases the conservation laws are the only and indispensable instrument.

Even when the forces are known precisely, the con­servation laws can help substantially to solve many problems of motion of particles. Although all these problems can be solved with the use of motion equations (and the conserva­tion laws provide no additional information in this case), the utilization of the conservation laws very often allows the solution to be obtained in the most straightforward and elegant fashion, obviating cumbersome and tedious calcula­tions. Therefore, whenever new problems are ventured, the following order of priorities should be established: first, one after another conservation laws are applied and only having made sure that they are inadequate, the solution is sought through the use of motion equations.

We shall begin examining the conservation laws with the energy conservation law, having introduced the notion of energy via the notion of work.

Let a particle travel some arbitrary path under the action of the force F. In the general case the force F may vary during the motion, both in magnitude and direction. Let us consider the elementary displacement , during which the force F can be assumed constant and along the direction of displacement.

The action of the force F over the displacement dr is characterized by a quantity equal to the scalar product  and called the elementary work of the force F over the dis­placement . It can also be presented in another form:

where a is the angle between the vectors F and ,  is the elementary path, and Fs is the projection of the vector F on the vector  (Fig. 37).

Thus, the elementary work of the force F over the dis­placement  is

The quantity is algebraic: depending on the angle between the vectors F and , or on the sign of the projection F, of the vector F on the vector , it can be either positive or negative, or, in particular, equal to zero (when either of force and displacement vanishes or when force is perpendicular to the displacement).

Integrating the expression over all elementary sections of the arbitrary  path from point 1 to point 2, we find the work of the force F over the given path.

The expression can be graphically illustrated. Let us plot F, as a function of the particle position along the path. The elementary work  is seen to be numerically equal to the area of the shaded portion enclosed by the graph and the x-axis. Here the area of the figure lying over the x axis is taken with the plus sign (it corre­sponds to positive work) while the area of the figure lying under the x axis is taken with the minus sign (it corresponds to negative work).

Let us consider a few examples involving calculations of work.

The work of the elastic force , where r is the radius vector of the particle A relative to a stationary point. Let us displace the particle A experiencing the action of that force along an arbitrary path from point 1 to point 2. Assume that point 1 and 2 are at distances a and b from the stationary point above respectively. We shall first find the elementary work performed by the force F over the elementary displacement :

Elementary work for displacing the particle by a small distance  :

Now integrating the above equation from a to b on both sides with respect to r gives:

Similarly, the reader is expected to try for columbic and frictional force. Columbic force is inversely proportional to the square of the radius vector and frictional force is a constant with respect to radius vector.