One of the principal ideas of geometric mechanics is reduction, which goes back to Jacobi's elimination of the node in the 3-body problem, but in its modern form is due to K. Meyer (1973) and independently J.E.
One of the important developments arising from the geometric approach to mechanics is the incorporation of the geometry into numerical methods.
[1] As a modern subject, geometric mechanics has its roots in four works written in the 1960s.
These were by Vladimir Arnold (1966), Stephen Smale (1970) and Jean-Marie Souriau (1970), and the first edition of Abraham and Marsden's Foundation of Mechanics (1967).
Smale's paper on Topology and Mechanics investigates the conserved quantities arising from Noether's theorem when a Lie group of symmetries acts on a mechanical system, and defines what is now called the momentum map (which Smale calls angular momentum), and he raises questions about the topology of the energy-momentum level surfaces and the effect on the dynamics.
In his book, Souriau also considers the conserved quantities arising from the action of a group of symmetries, but he concentrates more on the geometric structures involved (for example the equivariance properties of this momentum for a wide class of symmetries), and less on questions of dynamics.
These ideas, and particularly those of Smale were central in the second edition of Foundations of Mechanics (Abraham and Marsden, 1978).