The Conservation of Energy

§ 2. Relationship to the principle of the constancy of energy

If one multiplies the equations of motion (1c), in sequence, by qa and adds them then one will get:

If we set:

as we have done up to now, then we can write: (4a)

The sum that enters into this is the work that the force Pa does on the environment in the time interval dt, and that will then imply that the quantity E continually increases or decreases according to whether that force does positive or negative work, respectively. It will then follow from this that E denotes the energy supply of the system, expressed in terms of its coordinates pa and velocities qa

It emerges from this that the principle of least action, when it is taken in the form of § 1, always includes the principle of the constancy of energy.

On the other hand, the principle of least action is not necessarily true in all conceivable cases that are subject to the law of the constancy of energy. One can make many supplements to the system of equations (1c ) that do not at all affect the derivation of equations (4a ), but they will probably cancel the summation in the variational formula. For example, one adds a term (ϕ ⋅ qb ) to those equations (1c ) that have the index a in their terms, and a term (− ϕ ⋅ qb ) to the ones that have the index b, in which ϕ is any function of the coordinates. If, in order to derive the energy equation, we then multiply the former equation by qa and the latter one by qb then the extra terms will drop away, and the constancy of energy will not be affected. By contrast, the corresponding variation:

can be considered to be the complete variation of a function of the pa and qa under the integral only when ϕ depends upon the variables (qb ⋅ pb ) and (qb ⋅ pa ).

If the function ϕ that enters into the supplementary terms is independent of the velocities then the corresponding motion will not be reversible. However, we can make ϕ into a linear function of the velocities; the entire motion can then go backwards, as well.

Since one can install such terms in any arbitrarily-chosen pair of equations from the system (1c), a great multitude of cases are conceivable in which the law of the constancy of energy is valid, but not that of least action.

It follows that when the latter principle is true, it will express a special character of the conservative natural forces that are present that is not already present as a result of their being defined to be conservative forces. Illuminating this idea more clearly will be the objective of the investigation that follows.

Explanatory example:

Since it will be repeatedly desirable to cite examples in the sequel in which the significance of the theorems that are obtained becomes intuitive, I shall allow myself to cite some suitable spaces here to the extent that it is necessary to characterize them, and to which I can refer briefly, not only for the contents of this and the following paragraphs, but also later on.

I. Example of top.

Let the top be a rotating body on a gimbal mounting. The outer ring a might rotate around a vertical axis, and we let α denote the angle of rotation, when measured from a well-defined vertical plane in space. The second ring b rotates inside of the first one around a horizontal axis, and I shall let β denote the angle between the planes of the rings a and b. The rotational axis of the top lies in the ring b at right angles to the rotational axis between a and b. Let the angle between a reference meridian on the top and the plane of b be γ, let the moment of inertia of the top around its rotational axis be A, and let the moment of inertia around one of its equatorial axes by B; that of the ring itself will be neglected. The vis viva on the top will then be:

That will yield the forces A, B, C that tend to increase the angles α, β, γ, respectively:

(4a)

The force A is simply a rotational moment that makes the circle a rotate, and likewise, B makes b rotate. However, C rotates the circle relative to b, so it must have its pivot at b.

If the force C is absent then (4c) will imply that: (4d)

That will yield the value of H′, according to (2b):

the values of the forces, when derived in accordance with (4a ), (4b ), and (4c ): (4f )

The first constant term in the value of H′ can be omitted, since it enters into just the arbitrary constant of the value of E; the last term the linear one that is absent from the value of L.