Advanced Dynamics
Emphasizing learning through problem solving, this text analyzes in detail the strengths and weaknesses of various approaches to dynamics, and describes techniques that will improve computational efficiency considerably.
TABLE OF CONTENTS Advanced Dynamics
Chapter 5: Equations of Motion: Integral Approach
Integral principles and, in particular, Hamilton's principle, have long occupied a prominent position in analytical mechanics. Hamilton's principle, first announced in 1834, presents a variational principle as the basis for the dynamical description of a holonomic system. This approach tends to view the motion as a whole and involves a search for the path in configuration space which yields a stationary value for a certain integral. As a result, one obtains the differential equations of motion.
The requirement of stationarity does not apply to nonholonomic systems. Nevertheless, one can use integral methods to obtain the equations of motion for nonholonomic systems. Here we use the integral of the variation rather than the variation of the integral. In this chapter, we shall discuss the derivation and application of these methods, particularly with respect to nonholonomic systems.
5.1 Hamilton's Principle
Holonomic System
Consider a dynamical system whose motion satisfies Lagrange's principle, namely,
| (5.1) |  |
There are n generalized coordinates and the ?qs satisfy the instantaneous constraints. The kinetic energy T( q, 
, t) is written for the unconstrained system, and is assumed to have at least two continuous derivatives in each of its arguments. Q i is the generalized applied force associated with q i.
Now integrate (5.1) with respect to time over the fixed interval t 1 to t 2. Using integration by parts, we find that
| (5.2) |  |
Hence, we obtain
| (5.3) |  |
The ?qs satisfy the m
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