§ 7. Introducing the moment of motion as an independent variable, in place of velocity.

The differential equation (10d) gives many opportunities for the transformation of values by a different choice of independent variables, which Hamilton (* ) has already partially employed.

However, since he assumed that the vis viva was a homogeneous function of second degree in the velocities there, here, I shall allow myself to carry out those of these conversions under which the action of external variable forces does not

(* ) Philosph. Transact., 1835, pt. I, pp. 98-100. – See also Jacobi, loc. cit., Lect. XIX. Helmholtz – On the physical meaning of the principle of least action. 35 need to be excluded for the general form of the problem. One will get them when one replaces the velocities qa in the values of H (E, resp.) with the moments of motion sa . We have considered the kinetic potential H to be a function of the pa and qa . Thus: (12)

We have denoted:

It then follows from this that:

If the determinant of the quantities ∂sa / ∂qb is not equal to zero then we can introduce the pa and sa as variables, in place of the pa and qa , and, from (12a ), that will yield:

In this, H is assumed to be a function of the pa and the qa for the partial differentiations, but E is assumed to be a function of the pa and sa

That will imply the value of force that was given in (1c ): (12b)

The corresponding variational problem will take on a somewhat different form from the one that Hamilton gave to it:

In this, Pa are regarded as functions of only time, and E, as a function of the pa and sa

One varies the pa and the sa independently of each other and demands that the δsa = 0 at the limits of the time interval. The condition: δΨ = 0 will then give the two systems of equations of motion (12b ) with no other auxiliary equations.

In this manner of representation, we do not at all generally need to know the kinetic energy, but we must know the quantities sa that we can derive from the qa for the general form of the E only by means of H. We will obtain the corresponding form of the differential equation (10d ) for the case in which the P are equal to zero when we add to both sides of that equation, namely:

That will give: (12c)

Thus, if E, the pa , and the pa are represented as functions of time t and the sa and sa then:

The middle equation of this system can be employed, again, like (10b), in order to define a second reciprocity theorem, by which, a displacement δp1 is performed at the start of the time interval t, while all other pa and all sa remain unchanged; let s2 be changed by δs2 after the time interval t.

Under the reverse motion, only p2 will be changed by δp2, and Helmholtz – On the physical meaning of the principle of least action. 37 the moment s1 will be changed by δs1 after the time interval t. One will then have, once more: (12g ) δp1 : δs2 = δp2 : δs1 , assuming that the determinant of the equations:

is not equal to zero. If it is then the two positions will be reciprocal foci of the motion. Berlin, April 1886.