In the original, complete problems of the mechanics of ponderable bodies, the sa are linear, homogeneous functions of the qa whose coefficients are functions of the pa , in general, and one then has a system of linear equations:

Helmholtz – On the physical meaning of the principle of least action. 11 (3e)

which will represent the qb as linear, homogeneous functions of the sa when they are solved for those variables. That representation would not be possible if the determinant of the quantities s

were identically zero. However, the latter case cannot come about without the vis viva being zero for certain motions with finite velocities. Namely, since L is an essentially positive, homogeneous function of degree two of the qa , one will have:

If the aforementioned determinant were zero then all of the sa , and correspondingly L, as well, could be zero without the qa needing to be zero. The condition that the determinant of equations (3e ) is not identically zero can also be expressed as follows: No identity can exist between the quantities sa and pa , with the exception of the qa , and for that reason, the qa can always be represented as functions of the sa and the pa .

This relationship will not be changed if we set individual sa equal to constants, as in the case of hidden motions, or also set them equal to zero, as in the case of the eliminated pa .

The value of the remaining sa will not be changed by those variations. Since the same thing is also true for the electrical motions and reversible heat motions, to the extend that their physical laws have been ascertained up to now, there is, up to now, no physical motivation to consider the exceptional cases in which the determinant in equations (3e ) might be equal to zero, and for that reason the assumption will be made from now on that the determinant cannot be identically zero, except for at most special values of the pa . Once that condition has been established, the variational problem can be expressed in such a way that the equations that were singled out in the beginning of this paragraph, namely: (1) qa

dp dt a , will be assumed in it. As above, let H be a function of the pa and qa , and let the Pa be functions of time. One sets:

and demands that: (1e) δΦ1 = 0

must be true for arbitrary variations of the pa and qa , which are both to be treated as independent variables. One should have δpa = 0 at the times t0 and t1, while the dqa also remain arbitrary. The variation of qb yields:

which implies equations (1), since the determinant of the

should not vanish identically.

The variation of the pa is performed as above, and will yield the same result. If one denotes the function of the pa and qa that enters into (1d ) by:

(this is the energy, as will be shown in the next paragraph) then one will get: (1g)

I cite this form here, since we will encounter an analogous form in the conclusion, and both of them can be regarded as very exotic when one meets up with them in physical investigations before one knows for sure which quantities are to be referred to as pa , qa , and sa.

On the other hand, it is precisely these forms that are included in the complete statement of the problem.