# Applications of symplectic geometry to classical mechanics

It is claimed that classical mechanics motivates introduction of symplectic manifolds. This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space.

I am wondering whether symplectic geometry has applications to classical mechanics. Was this connection useful for classical mechanics? Were methods of symplectic geometry relevant for it via, say, the above theorem?

• Do you know about Foundations of Mechanics by Abraham/Marsden? This book seemed to be fairly well known when I was in college, especially when the greatly expended 1980 2nd edition was published. – Dave L Renfro Oct 19 at 19:57
• I find this question puzzling, since symplectic geometry comes from Hamiltonian mechanics, which was developed specifically for studying classical mechanics. And Hamiltonian mechanics has been extremely useful for applications to classical mechanics. – Deane Yang Oct 20 at 2:26
• @DeaneYang : What you say just means that there is a connection between the two subjects. But I am trying to find an application. I.e. what concrete question in mechanics was solved with the use of symplectic geometry? – makt Oct 20 at 4:33
• Physicists solve mechanics problems using canonical transformations, which are same as symplectomorphisms of open sets in $\mathbb{R}^{2n}$. – Deane Yang Oct 20 at 4:39
• Here is the wikipedia page on canonical transformations. Most of it uses the definitions and notations of classical physics, but there is a section at the end giving the "modern mathematical description". There are two classical mechanics books listed at the end. They, as well as any other classical mechanics textbook, show how to use the Hamiltonian formalism to solve mechanics problems. en.wikipedia.org/wiki/Canonical_transformation – Deane Yang Oct 20 at 14:39

V.I. Arnold's Mathematical Methods of Classical Mechanics is entirely based on the ideas and methods of symplectic geometry, such as the Birkhoff normal form, the Kolmogorov- Arnold-Moser theorem on the persistence of invariant tori, and the intersection theory of Lagrangian submanifolds.

Alan Weinstein points out that the first symplectic submanifold was already introduced "avant la lettre" by Lagrange in 1808. Weinstein goes on to state what he calls the symplectic creed that Everything is a Lagrangian submanifold --- how that underlies classical mechanics is discussed in this MO question.

• My feeling is that saying that Arnold's book is "entirely based on the ideas and methods of symplectic geometry" is an exaggeration. The first 6 chapaters deal with classical mechanics without symplectic geometry at all. Then symplectic manifolds are introduced in order to discuss Hamiltonian formalism. This might be convenient but probably not strictly necessary since in many other books Hamiltonian formalism is discussed without the language of symplectic geometry (see e.g. vol. 1 of Landau-Lifshitz). – makt Oct 20 at 9:34

The list will be long, very long indeed. But to start:

1. Questions about dynamics of Hamiltonian systems are at the heart of symplectic topology, symplectic capacities are precisely introduced for that purpose to understand the difference between a mere volume-preserving flow and a Hamiltonian flow. This includes questions about closed orbits etc. Here the books of Hofer-Zehner or McDuff-Salomon are a good start.

2. Even if interested only in mechanics in $$\mathbb{R}^{2n}$$: as soon as it comes to symmetries (and mechanics deals a lot with symmetries) one inevitably ends up with concepts of phase space reduction. The reduced phase space of the isotropic harmonic oscillator (could there be something more relevant for mechanics ?) is $$\mathbb{CP}^n$$ with Fubini-Study Kähler structure. Quite a complicated geometry already. In classical textbooks you discuss the Kepler problem by fixing the conserved quantities (angular momentum, etc) to certain values. This is just a phase space reduction in disguise. The geometry becomes less-dimensional but more complicated by doing so. Coadjoint orbits are symplectic and needed for descriptions of symmetries in a similar fashion. Without geometric insight, their structure is hard to grasp, I guess. The aforementioned textbook of Abraham and Marsden as well as many others provide here a good first reading. In fact, up to some mild topological assumptions any symplectic manifold arises as reduced phase space from $$\mathbb{R}^{2n}$$ according to a theorem of Gotay and Tuynman. From that perspective, symplectic geometry is mechanics with symmetries.

3. If trying to understand Hamilton-Jacobi theory, it is pretty hard to get anywhere without the geometric notion of a Lagrangean submanifold. This was perhaps one of the main motivations for Weinstein's Lagrangean creed.

4. Mechanical systems with constraints require a good understanding of the geometry of the constraints. This brings you into the realm of symplectic geometry where coisotropic submanifolds (aka first class constraints in mechanics) are at home.

5. When restricting the configuration space of a mechanics system (think of the rigid body) then you are actually talking about the cotangent bundle of the config space as (momentum) phase space. This is perhaps one of the very starting points where symplectic geometry takes of.

6. Going beyond classical mechanics, one perhaps is interested in quantum mechanics: here symplectic geometry provides a very suitable platform to ask all kind of questions. It is the starting point to try geometric quantization, deformation quantization and alike.

7. Maybe more exotic, but I really like that: integrable systems can have quite subtle and non-trivial monodromies. There is a very nice book (and many papers) of Cushman and Bates on this. The mechanical systems are really simple in the sense that you find them in all physics textbooks. But the geometry is hidden and highly non-trivial as it involves really a global point of view to uncover it.

8. From a more practical point of view, non-holonomic mechanics is of great importance to all kind of engineering problems (robotics, cars, whatevery). Here a geometry point off view really help and is a large area of research. Also mechanical control theory is not only about fiddling around with ode's but there is a lot of (symplectic) geometry necessary to fully understand things. The textbooks of Bloch as well as Bullo and Lewis might give you a first hint why this is so.

9. As a last nice application of (mostly linear) symplectic geometry one should not forget optics! This is of course not mechanics, but optics has a very interesting symplectic core, beautifully outlined in the textbook by Guillemin and Sternberg.

Well, I could go on, but the margin is to small to contain all the information, as usual ;) Of course, for many things one can just keep working in local coordinates and ignore the true geometric features. But one will miss a lot of things on the way.

• Thank you. This is an impressive list. I am not a specialist, but my feeling is that it requires a little broader scope of classical mechanics than I am used to. But may be this is what people mean when they discuss applications of symplectic geometry to it. – makt Oct 20 at 15:35
• @makt : Already under the point 1 of the list, there are recent applications to finding periodic orbits in the three body problem (papers by O. van Koert, U. Frauenfelder, et al.). I would say this qualifies as "classical mechanics". Historically the so-called "Arnold conjecture" in sympletic topology originated in the desire to generalize Poincaré's last geometric theorem (proved by G.D. Birkhoff see irma.math.unistra.fr/~maudin/Arnold.pdf for a very interesting account of this history), again largely motivated by celestial mechanics. – BS. Oct 26 at 16:16

There is a "symplectic structure" on the set of body motions.

During the years 1960–1970, Jean-Marie Souriau, proved that under very general assumptions, the set of all possible solutions of a classical mechanical system, involving material points interacting by very general forces, has a smooth manifold structure (not always Hausdorff) and is endowed with a natural symplectic form. He called it the manifold of motions of the mechanic

J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1969.

J.-M. Souriau, La structure symplectique de la mécanique décrite par Lagrange en 1811, Mathématiques et sciences humaines, tome 94 (1986), pages 45–54. Numérisé par Numdam, http://www.numdam.org.