1. Equations of motion are not all independent, because coordinates are no
longer all indepe ndent
2. Forces are not known beforehand, and must be obtained from solution.
For holonomic constraints introduce generalized coordinates. Degrees of
freedom are reduced. Use independent variables, eliminate dependent coordi-
nates. This is called a transformation, going from one set of dependent variables
to another set of independent variables. Generalized coordinates are worthwhile
in problems even without constraints.
Examples of generalized coordinates:
1. Two angles expressing position on the sphere that a particle is constrained
to move on.
2. Two angles for a double pendulum moving in a plane.
3. Amplitudes in a Fourier expansion of r
j
.
4. Quanities with with dimensions of energy or angular momentum.
For nonholonomic constraints equations expressing the constraint cannot be
used to eliminate the dependent coordinates. Nonholonomic constraints are
HARDER TO SOLVE.
1.4 D’Alembert’s Principle and Lagrange’s Equations
Developed by D’Alembert, and thought of first by Bernoulli, the principle that:
X
i
(F
(a)
i
−
dp
i
dt
) · δr
i
= 0
This is valid for systems which virtual work of the forces of constraint van-
ishes, like rigid body systems, and no friction systems. This is the only restric-
tion on the nature of the constraints: workless in a virtual displacement. This
is again D’Alembert’s principle for the motion of a system, and what is good
about it is that the forces of constraint are not there. This is great news, but it
is not yet in a form that is useful for deriving equations of motion. Transform
this equation into an expression involving virtual displacements of the gener-
alized coordinates. The generalized coordinates are independent of each other
for holonomic constraints. Once we have the expression in terms of generalized
coordinates the coefficients of the δq
i
can be set se parately equal to zero. The
result is:
X
{[
d
dt
(
∂T
∂ ˙q
j
) −
∂T
∂q
j
] − Q
j
}δq
j
= 0
5
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