TABLE OF CONTENTS Motivated by the developments of Joseph Louis Lagrange (1736 1813) and others in the late 1700s (the underpinnings of Chapter 5), and also by a desire to establish new insights that generalize these results, Sir William Rowan Hamilton (1805 1865) vastly extended the generalized approach to analytical dynamics. [1], [2] Hamilton's innovations during the 1830s provide new insights and methods for addressing: 1) rigorous analytical integrals of general motion and 2) new canonical forms of the equations of motion, and associated canonical coordinate transformations.
These innovations are presented in this chapter, along with examples that illustrate typical applications. We mention that the excellent texts by Pars, [3] Brouwer, [4] and Goldstein [5] inspired some of our developments here, and also provide important sources for further pursuit of this material.
[1]Lagrange, J. L., M canique Analytique, Paris, 1788.
[2]Hamilton, W. R., "On a General Method in Dynamics," Collected Papers of W. R. Hamilton, Vol. II, Cambridge Univ. Press, 1940, pp. 103 211.
[3]Pars, L. A., A Treatise on Analytical Dynamics, Wiley, New York, 1965, Chaps. 22 25.
[4]Brouwer, D., and Clemence, G., Methods of Celestial Mechanics, Academic Press, New York, 1961, Chap. 17.
[5]Goldstein, H., Classical Mechanics, Addison-Wesley, 1950.
We begin this discussion by introducing the Hamiltonian function 
, which is closely related to the Lagrangian 
The Hamiltonian is defined in terms of 
as follows:
Introducing the following definition for the conjugate (or...
