Formulation of the principle

The function H′ thus enters in this case, which is free of the qb and pb , but also includes terms that are linear in the qa and originate in the values of the qb, and it completely replaces the original function in the definition of the equations of motions (2c).

Examples of this are the rotations of a top around its symmetry axis, when its direction, but not its angular velocity, can change around that axis, and furthermore, the motion of a system that is referred to a rectangular coordinate system that is rotating; e.g., the Earth.

Corresponding to this analogy that is given by the mechanics of ponderable bodies, we would meanwhile also like to refer to other cases of physical processes in which the function H includes terms that are linear in the velocities as cases with hidden motion, although at the moment there are cases in which the existence of such a hidden motion is not confirmed beyond a doubt, such as for the interaction between magnets and electrical currents.

It was adopted for the magnet by Ampère.

It also showed its influence in the electromagnetic rotation of the plane of polarization of light, as Sir W. Thomson remarked, even when no perceptible electric current acts upon it.

This case is distinguished from the one in which H includes the velocities only in terms of degree two essentially by the fact that the motion cannot go backwards under the same conditions unless the hidden motions are simultaneously reversed.

Other eliminations can bring about even more complicated forms for the function H, at least for restricted classes of motion; I have discussed such cases in my first treatise on the monocyclic motions (*).

We can choose the conditions of that elimination somewhat differently here. One assumes that a group of coordinates pc is present whose corresponding qc enter into the values of the vis viva only multiplied by each other, but not combined into products with the qb in this group, such that all

moreover, one assumes that the forces Pc always remain equal to zero. Under these circumstances, motions of the system are possible for which the pc remain continually constant, so the qc = 0. The equations of motion for this class of motions simplify due to the fact that when all qc = 0, one will also have all:

We thus get from (1c):

If equations (3), whose number is equal to that of the pc, succeed in expressing these quantities as functions of the pb and qb then one can eliminate the pc from H by means of the values thus-obtained, where, in general, H will be a complicated function of the qb that we would like to denote by H. From the principles of differential calculus, one will then have:

thus, due to equations (3), one will have, in turn:

(3b)

(*) This Journal, Bd. 97, pp. 120-122.

Helmholtz – On the physical meaning of the principle of least action. 10

The equations of motion (3a) then reduce to: (3c)

in which only the pb and qb appear, and generally H is no longer the sum of a function of the coordinates and a homogeneous function of degree two of the velocities. However, if the original H is such a function, and thus no hidden motions will have any influence, then equations (3) will be of degree two in the qb ; the value of the pc can then remain unchanged (even when it is multi-valued) when all of the qb simultaneously change their signs, from which, it will follow that the total motion can also go backwards in this case.

In the mechanics of ponderable masses, problems in which the function H includes the velocities qa in terms of first or higher degree can be referred to as incomplete problems, insofar as a part of the possible motions is excluded, and a part of the coordinates that are necessary for the determination of the position of the system does not enter into the function H, and certain forces are constantly set equal to zero, so they can no longer be determined arbitrarily.

Functional determinant of the moment of motion. For the sake of brevity, we would like to denote the quantities ∂H / ∂qa that enter into the previous derivatives by: (3d)

and call the sa moments of motion. For the motion of a free system that is referred to rectangular coordinates, they correspond to the product of the mass and velocity, whose differential quotient with respect to time is Newton’s measure of the corresponding force component:

In the cases that were summarized, the influence that the inertia of the moving masses exerts upon a well-defined type of motion was different from that of the position of the masses.

Thus, e.g., for a rotational motion of a solid body, the moment of motion is equal to the moment of inertia, multiplied by the angular velocity.

In that sense, the quantities sa now measure the influence that the inertia of the moving mass has, and its acceleration enlists a corresponding part of the force of motion, as equations (1c) show.