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This report provides an introduction to geometric mechanics, which seeks to model the behavior of physical mechanical systems using differential geometric objects. In addition to its elegance as a method of representation, this formulation also admits the application of powerful analytical techniques from geometry as an aid to understanding these systems. In particular, it reveals the fundamental role that symplectic geometry plays in mechanics (something which is not at all obvious from the traditional Newtonian formulation), and in the case of systems exhibiting symmetry, leads to an elucidation of conservation and reduction laws which can be used to simplify the analysis of these systems. The contribution here is primarily one of exposition. Geometric mechanics was developed as an aid to understanding physics, and we have endeavored throughout to highlight the physical principles at work behind the mathematical formalism. In particular, we show quite explicitly the entire development of mechanics from first principles, beginning with Newton's laws of motion and culminating in the geometric reformulation of Lagrangian and Hamiltonian mechanics. Self-contained presentations of this entire range of material do not appear to be common in either the physics or the mathematics literature, but we feel very strongly that this is essential in order to understand how the more abstract mathematical developments that follow actually relate to the real world. We have also attempted to make many of the proofs contained herein more explicit than they appear in the standard references, both as an aid in understanding and simply to make them easier to follow, and several of them are original where we feel that their presentation in the literature was unacceptably opaque (this occurs primarily in the presentation of the geometric formulation of Lagrangian mechanics and the appendix on symplectic geometry). Finally, we point out that the fields of geometric mechanics and symplectic geometry are vast, and one could not hope to get more than a fragmentary glimpse of them in a single work, which necessiates some parsimony in the presentation of material. The subject matter covered herein was chosen because it is of particular interest from an applied or engineering perspective in addition to its mathematical appeal.
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