34 Classical mechanics
∑N
i=11
2
mir ̇i·(
dri
dt)
−C= 0
∑N
i=11
2
mir ̇i·dri−Cdt= 0 (1.9.7)so that
a 1 i=1
2
mir ̇i, a 1 t=C (1.9.8)(k= 1 since there is only a single constraint).
Assuming that the constraints can be expressed in the differentialform of eqn.
(1.9.4), we must also be able to express them in terms of path displacementsδqαin
order to incorporate them into the action principle. Unfortunately, doing so requires a
further restriction, since it is not possible to guarantee that a perturbed pathQ(t) +
δQ(t) satisfies the constraints. The latter will hold if the constraints are integrable, in
which case they are expressible in terms of path displacements as
∑^3 Nα=1akαδqα= 0. (1.9.9)The coefficientaktdoes not appear in eqn. (1.9.9) because there is no time displace-
ment. The equations of motion can then be obtained by adding eqn. (1.9.9) to eqn.
(1.8.6) with a set of Lagrange undetermined multipliers,λk, where there is one multi-
plier for each constraint, according to
δA=∫t 2t 1∑^3 N
α=1[
∂L
∂qα−
d
dt(
∂L
∂q ̇α)
+
∑Nck=1λkakα]
δqα(t) dt. (1.9.10)The equations of motion obtained by requiring thatδA= 0 are then
d
dt(
∂L
∂q ̇α)
−
∂L
∂qα=
∑Nck=1λkakα. (1.9.11)It may seem that we are still relying on the independence of the displacementsδqα,
but this is actually not the case. Suppose we choose the first 3N−Nccoordinates to be
independent. Then, these coordinates can be evolved using eqns.(1.9.11). However, we
can chooseλksuch that eqns. (1.9.11) apply to the remainingNccoordinates as well.
In this case, eqns. (1.9.11) hold for all 3Ncoordinates provided they are solved subject
to the constraint conditions. The latter can be expressed as a setofNcdifferential
equations of the form
∑^3 N
α=1akαq ̇α+akt= 0. (1.9.12)Eqns. (1.9.11) together with eqn. (1.9.12) constitute a set of 3N+Ncequations for the
3 N+Ncunknowns,q 1 ,...,q 3 N,λ 1 ,...,λNc. This is the most common approach used in
numerical solutions of classical-mechanical problems.