(18)
The interpretation is simplest when the co-ordinates q1, q2 are both of the same kind, e.g. both lines or both angles. We may then conveniently put p1 = p′2, and assert that the velocity of the first type due to an impulse of the second type is equal to the velocity of the second type due to an equal impulse of the first type. As an example, suppose we have a chain of straight links hinged each to the next, extended in a straight line, and free to move. A blow at right angles to the chain, at any point P, will produce a certain velocity at any other point Q; the theorem asserts that an equal velocity will be produced at P by an equal blow at Q. Again, an impulsive couple acting on any link A will produce a certain angular velocity in any other link B; an equal couple applied to B will produce an equal angular velocity in A. Also if an impulse F applied at P produce an angular velocity ω in a link A, a couple Fa applied to A will produce a linear velocity ωa at P. Historically, we may note that reciprocal relations in dynamics were first recognized by H.L.F. Helmholtz in the domain of acoustics; their use has been greatly extended by Lord Rayleigh.
The equations (13) determine the momenta p1, p2,... as linear functions of the velocities q˙1, q˙2,... Solving these, we can express q˙1, q˙2 ... as linear functions of p1, p2,... The resulting equations give us the velocities produced by any given Velocities in terms of momenta. system of impulses. Further, by substitution in (8), we can express the kinetic energy as a homogeneous quadratic function of the momenta p1, p2,... The kinetic energy, as so expressed, will be denoted by Τ`; thus
2Τ` = A`11p1² + A`22p2² + ... + 2A`12p-p2 + ...
(19)
where A`11, A`22,... A`12,... are certain coefficients depending on the configuration. They have been called by Maxwell the coefficients of mobility of the system. When the form (19) is given, the values of the velocities in terms of the momenta can be expressed in a remarkable form due to Sir W.R. Hamilton. The formula (15) may be written
p1q˙1 + p2q˙2 + ... = Τ + Τ`,
(20)
where Τ is supposed expressed as in (8), and Τ` as in (19). Hence if, for the moment, we denote by δ a variation affecting the velocities, and therefore the momenta, but not the configuration, we have
p1δq˙1 + q˙1δp + p2δq˙2 + q˙2δp2 + ... = δΤ + δΤ`