§ 5. Generalization of Hamilton’s differential equation.

Hamilton has taught us how to represent the function Φ that he defined, under somewhat restricting assumptions, to be:

as a function of time t = t1 – t0 and the values of the coordinates at the time t1 and t0 . We would like to denote the moments of motion for time t1 by pa and sa and the ones for time t0 , by pa and sa , resp. It is assumed that the changes in the pa during the time interval (t1 – t0) result from the laws of motion. The value of the integral that is denoted by Φ can then be calculated as a function of the pa , pa , and t, and for this kind of representation, we will have:

This entire conversion can also be performed under the extended assumptions that were made in § 1 (*). For the present purposes, it will suffice to do this under that assumption

(* )

C. G. J. Jacobi has already remarked that Pa = 0. Moreover, the function H might be an arbitrary function of the pa and qa , as long as it fulfills the continuity condition that were discussed above.

The first two systems of equations (10) are obtained by carrying out the partial integrations that convert the variation of Φ by the δqa into a variation by δpa .

The differential quotient with respect to time ∂Φ / ∂t that enters into the third of equations (10) will be obtained for unvaried values of the pa and pa when we look for the change in the value of Φ that occurs as a result of an actual motion for a lengthening of the time by dt. Thus, pa will increase by qa ⋅ dt, and on the other hand, equation (1a ) shows that the stated variation of Φ equals the final value of H ⋅ dt. Thus: H ⋅ dt = q t p     ∂Φ ∂Φ     + ⋅     ∂ ∂     ∑ a a ⋅ dt, or, from the first equations (10) and (4): t ∂Φ ∂ = E. The following relations between the quantities sa , sa , E now result from equations (10) when those quantities are represented as functions of the pa , pa , and t: (10b)

When these conditions are fulfilled:

(10d) E ⋅ dt − ∑ ∑ [ ] [ ] s dp d ⋅ + ⋅ a a a a a a s p = dΦ will be the complete differential of a function of the pa , pa , and t. Moreover, if the quantities E, sa , and sa that enter into the differential equation (10d ) are to correspond to the energy and moment of motion for a possible motion of the system that is not acted upon by external forces then they will not be completely independent of each other. In fact, as Hamilton has shown already, the equations of motion of the system can be represented by the system of equations: (10e) sa = const.

Since the sa are functions of the pa , of t, and of the pa , in general, the pa will become functions of time from this, and the pa and sa , which represent integration constants, can be determined, so the position of the system will be given for later instants. Now, if one substitutes the value of pa that is thus obtained in the value of E for a conservative system then it would be converted into a function of the pa and sa that cannot, however, be dependent upon time for any longer. If we revert to equations (10) then this will say that: (10f ) t ∂Φ ∂ = E ,   ∂Φ   ∂   a a p p , or that a first-order differential equation must exist between the differential quotients ∂Φ / ∂t and ∂Φ / ∂pa of the function Φ whose coefficients depend upon only the pa . However, we can likewise traverse the path of a system backwards from a certain final position, in which we will have to treat the values of the pa and sa as if they were constant. The equations: sa = const. will then yield the quantities pa as functions of t and the fixed values of sa and pa . When these values of the pa are substituted into the function E, it will prove to be a function of the sa and pa , from which, t must be absent. It follows from this that there must be a second differential equation for the function Φ: (10g)

between the differential quotients ∂Φ / ∂t and ∂Φ / ∂pa whose coefficients depend upon only the pa .

Hamilton gave a specific form to these two differential equations for the function Φ, since he considered the two components components of the electrokinetic potential to be given, and indeed in the older, more restricted, form, while here we seek only the most general character of those equations that simultaneously corresponds to the principle of the conservation of energy and that of least action.

This also comes down to saying that any pair of associated sa and sa should be values of the same function at the beginning and the end of the time interval t. If we apply the differential equation (10d ) to very small time intervals t then for actual motions, the quantities will have to be set to:

(10h) pa – pa = qa ⋅ t, and these qa will approach the values of the velocity all the more closely as t becomes smaller. Likewise, however, the difference (sa – sa ) must also approach zero with decreasing t.

If the differential equation (10d) and these auxiliary conditions are fulfilled then the variational problem will also be fulfilled. To that end, one needs only to hold the pa constant, and vary the pa in the way that they would change when one varies the time interval dt for an unperturbed traversal of the motion; thus: dpa = 0 and dpa = qa ⋅ dt. From (10d ), this will imply that:

in which, one must take E, sa , and qa under the integral sign to have the actual values that they have at time t, which is the start of the corresponding motion. That is the previous representation of the function Φ, and the differential equation (10d ) implies that this value of Φ must satisfy the minimum condition for the actual path of the system that is retraced. Namely, when we think of the path of the system from the position that is denoted by 0 to the one that is denoted by 2 as being divided by an intermediate, variable position, which we will denote by 1, then from (10i ):

Φ0, 2 = Φ0, 1 + Φ1, 2 .

If we now vary the coordinates of the intermediate position then, from (10d), we will have: δΦ0, 2 = − δΦ1, 2 = −∑[ ] s p ⋅δ a a a , and as a result: δΦ0, 2 = 0. As is easily seen, this can be carried over to an arbitrary subdivision of the path into arbitrarily many pieces, and that will imply that the integral Φ0, 2 will not vary when makes any sort of small changes to the intermediate positions. The minimal theorem depends upon the fulfillment of the differential equation (10d ), and for the extended form of H, as well as for the original, more restricted form that Lagrange and Hamilton started with. Helmholtz – On the physical meaning of the principle of least action. 32 The conditions that exist between the quantities that enter into the differential (10d ) that were discussed in this paragraph reduce to one equation that gives E as a function of the pa and sa when one employs C. G. J. Jacobi’s ( * ) conversion.