Mathematics of Physics and Engineering

90 The Lagrange-Hamilton Method

corresponding time derivatives, so that the vector (q, q) uniquely deter-
mines the state of the system. We derived (2.3.27) under the assumption
Q — (Qii • • • i Qk) = —W, where V = V(q). Now assume that the gener-
alized force Q is no longer conservative, and has a non-conservative compo-
nent that depends only on q; many non-conservative forces, such as friction
and some electromagnetic forces, indeed depend only on the velocity. In
other words, we assume that

Q = Q(1) + Q(2), Q(1) = -Wi, Q(2) = jVV 2 (2.3.28)

for some scalar functions V\ = V\{q) and V% = V 2 (q). Then

Define V = Vi + V 2 so that dVi/dqj = dV/dqj, dV 2 /dqj = OV/dq,, and

If L = SK — V, then Lagrange's equations (2.3.27) follow after substi-
tuting (2.3.30) into (2.3.17). Alternatively, if we define the conservative
Lagrangian L\ = £K — Vi, then (2.3.17) and (2.3.30) imply

d dL\ dLi ^(2i

*^T-^ = «?

)

'

(2

3

31)

a modification of (2.3.17).

2.3.2 An Example of Lagrange's Method

Our goal in this section is to verify the Lagrange equations (2.3.17) for
a points mass that moves in space, but has only one degree of freedom.
The example below is adapted from the book Introduction to Analytical
Mechanics by N. M. J. Woodhouse, 1987.
Consider a smooth wire in the shape of an elliptical helix. Choose an
inertial frame with origin O and cartesian basis vectors (?, J, k), so that k
is along the axis of the helix. In this frame, the helix is the set of points
with coordinates (x, y, z) so that