The theorem (16) leads to some important reciprocal relations. Thus,
let us suppose that the momenta p1, p2, ... all vanish with the
exception of p1, and similarly that the momenta p'1, p'2, ... all
vanish except p'2. We have then p1q`'1 = p'2q`2, or
q`2 : p1 = q`'1 : p'2 (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 [omega] in a link A, a couple Fa applied to A will
produce a linear velocity [omega]a 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.
Velocities in terms of momenta.
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
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
[Tau]'; thus