84 Systems of Point Masses
If a « 0 and 6 is so small that sin# « #, then equation (2.2.45) becomes
{21/3)0+ g0 = 0. (2.2.46)Comparing this with a similar approximation for the simple rigid pendu-
lum (2.1.10), page 42, we conclude that a thin uniform stick of length £
suspended at one end oscillates at about the same frequency as a simple
rigid pendulum of length (2/3)1
The objective of the above example was to illustrate how the Euler
equations work. Because of the simple nature of the problem, the system
of three equations (2.2.35) degenerates to one equation (2.2.45). In fact, an
alternative derivation of (2.2.45) is possible by avoiding (2.2.35) altogether;
the details are in Problem 2.4, page 417.
Note that if the frame O is fixed on the Earth, then this frame is not
inertial, and the Coriolis force will act on the pendulum, but a good pin
joint can minimize the effects of this force.
For two more examples of rigid body motion, see Problems 2.7 and 2.8
starting on page 419.2.3 The Lagrange-Hamilton Method
So far, we used the Newton-Euler method to analyze motion using forces
and the three laws of Newton (it was L. Euler who, around 1737, gave a
precise mathematical description of the method). An alternative method
using energy and work was introduced in 1788 by the French mathe-
matician JOSEPH-LOUIS LAGRANGE (1736-1813) and further developed in
1833 by the Irish mathematician Sir WILLIAM ROWAN HAMILTON (1805-
1865). This Lagrange-Hamilton method is sometimes more efficient than the
Newton-Euler method, especially to study systems with constraints. An ex-
ample of a constraint is the rigidity condition, ensuring that the distance
between any two points is constant.
In what follows, we provide a brief description of the Lagrange-Hamilton
method. The reader is assumed to be familiar with the basic tools of multi-
variable calculus, in particular, the chain rule and line integration.