Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/334

It is evident that, when the system is considered from the point of view of general dynamics, the electric currents must be regarded as generalized velocities, and the quantities

$(L_{1}i_{1}+L_{12}i_{2})$ and $(L_{12}i_{1}+L_{2}i_{2})$

as momenta. The electromagnetic ponderomotive force on the rings tending to increase any coordinate x is ∂T/∂x. In the analogous hydrodynamical system, the fluid velocity corresponds to the magnetic force: and therefore the circulation through each ring (which is defined to be the integral ∫vds, taken round a path linked once with the ring) corresponds kinematically to the electric current; and the flux of fluid through each ring corresponds to the number of lines of magnetic force which pass through the aperture of the ring. But in the hydrodynamical problem the circulations play the part of generalized momenta; while the fluxes of fluid through the rings play the part of generalized velocities. The kinetic energy may indeed be expressed in the form

$K={\tfrac {1}{2}}(N_{1}\kappa _{1}^{2}+2N_{12}\kappa _{1}\kappa _{2}+N_{2}\kappa _{2}^{2})$,

where κ₁, κ₂, denote the circulations (so that κ₁ and κ₂ are proportional respectively to i₁, and i₂), and N₁, N₁₂, N₂, depend on the positions of the rings; but this is the Hamiltonian (as opposed to the Lagrangian) form of the energy-function,^[1] and the ponderomotive force on the rings tending to increase any coordinate x is - ∂K/∂x. Since ∂K/∂x is equal to ∂T/∂x, we see that the ponderomotive forces on the rings in any position in the hydrodynamical system are equal, but opposite, to the ponderomotive forces on the rings in the electric system.

The reason for the difference between the two cases may readily be understood, The rings cannot cut through the lines of magnetic force in the one system, but they can cut through the stream-lines in the other: consequently the flux of fluid through the rings is not invariable when the rings are moved, the invariants in the hydrodynamical system being the circulations.

1. Cf. Whittaker, Analytical Dynamics, § 109,