The forces on moving systems

§ 4. The general characteristics of the forces on moving systems.

The forces that act upon systems at rest from the outside, which satisfy the law of the constancy of energy, exhibit certain legitimate relations with each other that are expressed in the equations:

and that when these equations are fulfilled, the value of the potential energy can be found.

One likewise finds that similar relations that are implied immediately by Lagrange’s expressions for the forces can be presented for moving systems that are subject to the law of minimum kinetic energy. They are thus not merely to be regarded as functions of the coordinates pa, as in the systems at rest, but also as functions of the velocities qa and the accelerations:

(9)

Equation (1c ):

(1c)

immediately yields:

A. Forces and accelerations.

When represented in this form, the forces are linear functions of the accelerations. The coefficient of the q′ b in the value of the force Pa can thus be written:

(9a)

i.e.: If the acceleration q′ b makes the force Pa larger by a certain amount then the same increase in the acceleration q′ a will make the force Pb larger by the same amount.

Whether such an influence is present in a given case or not will depend upon whether the quantities

are non-zero or equal to zero, respectively. The stated quantities are zero, for example, for the motions of a completely-free system of ponderable masses when they are referred to rectangular coordinates. Every individual force component affects the acceleration only in the direction of the coordinate to which it is refers. For the top in example I of § 2, we have:

in which α″, β″, γ″ denote the accelerations of the angles α, β, γ, resp.

In example II for the electrodynamic effects, one has:

The former equation says: Since the ponderomotive force of the circular current does not depend upon the acceleration of the current, the induced electromotive force can also not depend upon the acceleration of the current conductor (but possibly upon the velocities, in both cases). The latter equation says that when for a given position and form of the circular currents b and c, a rise in the force Eb that acts upon b cause an increase in Jc by electromagnetic induction, the same rise in the force Ec will produce the same effect on Jb .

This reciprocal relationship is not present in example III for the thermodynamic effects, since the vis viva L of heavy masses does not depend upon the temperature, and therefore the product ϑ ⋅ qa will not enter into the value of (F – L) = H.