Page 220 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Lagrange's

Equation

Joseph L. C. Lagrange (1736-1813) developed a general treatment of dynamical
systems formulated from the scalar quantities of kinetic energy T, potential energy
U, and work W. Lagrange’s equations are in terms of generalized coordinates, and
preliminary to discussing these equations, we must have clearly in mind the basic
concepts of coordinates and their classification.

7.1 GENERALIZED COORDINATES

Generalized coordinates are any set of independent coordinates equal in number to
the degrees of freedom of the system. Thus, the equations of motion of the
previous chapter were formulated in terms of generalized coordinates.
In more complex systems, it is often convenient to describe the system in
terms of coordinates, some of which may not be independent. Such coordinates
may be related to each other by constraint equations.
Constraints. Motions of bodies are not always free, and are often con­
strained to move in a predetermined manner. As a simple example, the position of
the spherical pendulm of Fig. 7.1-1 can be completely defined by the two indepen­
dent coordinates if/ and (/>. Hence, if/ and cf) are generalized coordinates, and the
spherical pendulum represents a system of two degrees of freedom.
The position of the spherical pendulum can also be described by the three
rectangular coordinates, x, y, z, which exceed the degrees of freedom of the
system by 1. Coordinates x, y, z are, however, not independent, because they are
related by the constraint equation:
-h y^ + z^ = 0 (7.1-1)
One of the coordinates can be eliminated by the preceding equation, thereby
reducing the number of necessary coordinates to 2.

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