Applications of Symplectic Geometry

[djohannsen]

I am a mathematics graduate student who is doing research in symplectic geometry (specifically symplectic toric orbifolds, symplectic reduction, Hamiltonian actions of tori in the symplectic category, etc). I often have tried to convince others of the importance of symplectic geometry, so I come here seeking some help. I know, of course, that the language of classical mechanics is really that of symplectic geometry. I've also heard tell that there is a branch of optics that uses the language and machinery of symplectic geometry. However, I am hoping to come up with a list of very specific (dare I say engineering'ish) applications. Can anyone suggest some very specific applications of symplectic geometry to physics and engineering? I would be grateful for any replies.



Dave

 

[Valerie Park]

Symplectic geometry has been used in various areas of physics, including classical and quantum mechanics, optics, and electromagnetism. One of the most interesting applications is in the area of optics, where symplectic geometry provides a natural language for describing the propagation of light through optical systems. In particular, it is used to describe the behavior of optical systems with multiple lenses, as well as the behavior of wavefronts in systems with curved mirrors. It is also used to model the behavior of laser beams in nonlinear media, such as crystals and plasma. In addition, symplectic geometry is used in the study of dynamical systems, which are mathematical models of physical systems that evolve over time. It can be used to study the stability of orbits of planets, moon, and other celestial bodies, as well as the motion of particles in electric and gravitational fields. Symplectic geometry is also used in the study of Hamiltonian systems, which are mathematical models of physical systems that have energy. Such systems include particle systems, plasmas, fluids, and many other types of systems. The use of symplectic geometry allows us to understand the dynamics of these systems in terms of conservation laws, such as the conservation of energy. Finally, symplectic geometry is used in the study of control theory. Control theory is concerned with the design of controllers for physical systems, such as robots, aircraft, and space vehicles. In this context, symplectic geometry can be used to represent the state of a system and its evolution over time, as well as to design controllers that are able to optimally regulate a system's behavior.