Geometric Mechanics is Classical Mechanics formulated in the language of modern Differential Geometry. Of course, while Lagrangian Mechanics, to a certain extent, retains the standard Differential Geometric form, Hamiltonian Mechanics alters it significantly enough to give it a special name, -Symplectic Geometry. Lagrangian Mechanics itself initially takes the form of Riemannian Geometry as was first clearly demonstrated by Hertz in his “Principles of Mechanics.” The departure from Riemannian to Symplectic takes place via the Legendre transform (or fibre derivative, in symplectic language) that maps the Lagrangian to the Hamiltonian. When the transform is regular (or rather, hyperregular), the map in a sense, takes the “velocities” to the “momenta.” Otherwise, there arise singular constraints and need to be treated quite differently.

John L. Synge was one of the first to cast Lagrangian Mechanics into a geometric framework. In his beautiful Handbuch der Physik encyclopedia article, he gave a treatment of Lagrangian Mechanics as Riemannian Geometry. But the modern, qualitative-quantitative form of Mechanics really took off with Henri Poincare’s introduction of the qualitative methods of Topological Analysis into Mechanics. His methods were characterised by the global geometric point of view. He treated a dynamical system as a vector field on the phase space of the system. A solution then corresponded to a smooth curve tangent at each of its points to the vector based at that point. This global point of view was capable of giving the complete information of the dynamical system. The manifold or bundle that arose could then be studied and the necessary structures like the metric, connections, almost-tangent structures, almost-complex structures as have now been developed, could be imposed. After Poincare, George Birkoff contributed immensely to the development of Dynamical systems, with the publication of his American Mathematical Society monograph, “Dynamical Systems,” one of the most illuminating books on the subject.

But there was another immense source of ideas and methods infiltrating into Mechanics. It was that of Elie Cartan. Elie Cartan’s work revolutionised Mechanics like nothing before. His powerful Exterior Calculus and classification of Lie Algebras and Lie Groups changed the face of Mechanics forever.

These twin streams of ideas and methods, from Poincare and Cartan were taken up by a new generation of mathematicians and the very fertile field of Geometric Mechanics was born. Presently, Geometric Mechanics has matured to take the shape of one of the most extensively developed and developing fields of Mathematics and Mathematical Physics.