Gauss’s principle 35Note that, even if a system is subject to a set oftime-independentholonomic
constraints (akt= 0), the Hamiltonian is still conserved. In order to see this, note that
eqns. (1.9.11) and (1.9.12) can be cast in Hamiltonian form as
q ̇α=∂H
∂pαp ̇α=−∂H
∂qα−
∑Nck=1λkakα∑^3 N
α=1akα∂H
∂pα= 0. (1.9.13)
Computing the time-derivative of the Hamiltonian, we obtain
dH
dt=
∑^3 N
α=1[
∂H
∂qα
q ̇α+∂H
∂pα
p ̇α]
=
∑^3 N
α=1[
∂H
∂qα∂H
∂pα−
∂H
∂pα(
∂H
∂qα+
∑Nck=1λkakα)]
=
∑Nck=1λk∑^3 N
α=1∂H
∂pαakα= 0. (1.9.14)
From this, it is clear that no work is done on a system by the imposition of holonomic
constraints.
1.10 Gauss’s principle of least constraint
The constrained equations of motion (1.9.11) and (1.9.12) constitute a complete set of
equations for the motion subject to theNcconstraint conditions. Let us study these
equations in more detail. To keep the notation simple, let us consider just a single
particle in three dimensions described by a Cartesian position vectorr(t) subject to
a single constraintσ(r) = 0. According to eqns. (1.9.11) and (1.9.12), the constrained
equations of motion take the form
m ̈r=F(r) +λ∇σ∇σ·r ̇= 0. (1.10.1)These equations will generate classical trajectories of the system for different initial
conditions{r(0),r ̇(0)}provided the conditionσ(r(0)) = 0 is satisfied. If this condition
is true, then the trajectory will obeyσ(r(t)) = 0. Conversely, for eachrvisited along