# Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?

Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on manifolds.

I'm wondering why it's at all interesting to consider such things in the classical case. In more advanced theories of physics, manifolds become relevant, but in classical mechanics, everything is Euclidean and homogeneous, so this isn't really needed. There, I'm wondering what motivated people to develop the theory of Hamiltonian mechanics through the theory of symplectic manifolds.

One idea: There wasn't really and physical motivation, but someone noticed it, and it was mathematically nice. If this is the case, then how did people notice it?

However, maybe this theory was not developed until general relativity or quantum mechanics came about. That is, manifolds became relevant to more advanced physics, and Arnold thought it reasonable to teach students this manifold formalism in classical mechanics in order to better prepare them for more advanced physics. But, if this is true, no one thought of using symplectic manifolds purely for reasons of classical physics.

• It is not true that "in classical mechanics, everything is Euclidean". As soon as you add boundary conditions to your problem, the true phase space might be some weird manifold that's best described in non-Cartesian coordinates. The surface of a sphere, for example, is not Euclidean. Apr 8, 2011 at 14:51
• The word classical in the question and title is slightly misleading. Traditionally in physics classical refers to everything with $\hbar=0$, but @Davidac897 apparently wants to also exclude, e.g., general relativity, point particles moving in a (curved) manifold, and everything which is not formulated in an affine space in his definition of the word classical. Apr 8, 2011 at 16:15
• I am puzzled by this question. Classical mechanics is intrinsically symplectic, so symplectic manifolds are used because there's no way about it. You can pretend that you don't see this structure but it's always there. Also, it implies lots of useful theorems (as mentioned in Roy's answer) which are relevant e.g. for ergodic theory, statistical physics, etc. Not to mention connections with geometrical quantization (although for this pure Poisson structure is more relevant). Apr 8, 2011 at 19:12
• Just not to confuse people, geometrical quantization isn't related to quantization of geometry (which Roy also alludes to). Rather, it's the study of deformations of Poisson structure which leads to quantum groups and related concepts which have their main applications in condensed matter theory (at least AFAIK). Apr 8, 2011 at 19:15
• Sorry, by classical, I meant Newtonian mechanics (i.e. without relativity or quantum). Apr 10, 2011 at 2:18

The symplectic framework paves the way to classical mechanics on a Poisson manifold (i.e., with a Poisson bracket). See the book Mechanics and Symmetry by Marsden & Ratiu. https://cdn.preterhuman.net/texts/science_and_technology/physics/Introduction%20to%20Mechanics%20and%20Symmetry.pdf

The Poisson point of view unifies many classical systems that cannot be seen otherwise in a uniform way. For example, many fluid flow equations have a Hamiltonian description only when viewed on an infinite-dimensional Poisson manifold. See the survey article

PJ Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern Physics 70 (1998), 467. http://www.ph.utexas.edu/~morrison/98RMP_morrison.pdf

and the book

Beris and Edwards, Thermodynamics of flowing systems.

This question is somewhat of a physics history question about when and why. I dont have a specific reference as to who first introduced symplectic geometry into physics, but there are some broader indications as to what has happened.

The symplectic manifold idea is a formalisation (in terms of differential geometry) of the phase space idea. Looking up phase space in Wikipedia gives this useful reference and explanation:

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.

A second factor that would have been relevant in this case was Liouville's Theorem. This theorem asserts that the density of points in phase space is conserved. It is about volumes in phase space, thus suggesting a differential geometry treatment of phase space.

Once it became clear (I dont know whether this was Gibbs or later) that the symplectic form exists as a geometric object similar in some ways to a metric, and closely connected with the Poisson bracket and the rest of classical mechanics, then it would be natural to describe the whole of classical mechanics in this geometric way.

This could be another topic that was first noticed by mathematicians, but only noticed by physicists after the advent of quantum mechanics (1920s or 1930s) when explicit connections between the two theories (Classical and Quantum) were searched for. Names associated with this are Weyl and Moyal (in 1949). More on these early phase-space to QM ideas in this Wikipedia link.

It also needs to be remembered as we discuss post General Relativity and post Quantum Mechanics eras, that phase space formulations of GR have been sought as an aid to quantization. All this is a highly geometric approach to Quantum Gravity, associated with many names although Ashtekar worked on this overlap. Eventually this led to his (co-) formulation of Loop Quantum Gravity.

The action principle, stemming from Fermat and Maupartuis, and developed by Lagrange tells us that a classical system extremizes the action $S~=~\int L(q,{\dot q})dt$ so that the following equation is obeyed $$\frac{d}{dt}\frac{\partial L}{\partial{\dot q}}~-~\frac{\partial L}{\partial q}~=~0.$$ This defines a differential equation for each Lagrangian $L(q,{\dot q})$, which is equivalent to Newton’s second law. The action is a bare action plus a Hamiltonian under a Legendre transformation $Ldt~=~pdq~-~Hdt$, where this tells us that the Lagrangian is a form of constraint which sets the dynamics on some manifold of solution. This manifold is called a contact manifold.

The Hamiltonian can be showed with several steps of mathematics to obey the equations $${\dot q}~=~\frac{\partial H}{\partial p},~{\dot p}~=~-\frac{\partial H}{\partial q},$$ which are the Hamiltonian dynamical equations. In a general setting the evolution of quantity $z$ is given by ${\dot z}~=~\{z,~H\}$, which is a Poisson bracket. The antisymmetric structure of this theory can be encoded according to Hamiltonian flows determined by a symplectic matrix $\Omega_{ij}$ which is antisymmetric. If we let $z~=~(q_1,\dots, q_n,~p_1,\dots,p_n)$ be a $2n$ vector, then for $\Omega_{ij}$ a $2n\times 2n$ matrix $dz_i/dt~=~\Omega_{ij}z_j$ is a general form of dynamics in symplectic form.

This is an outline of how this is developed. Arnold’s book on Math Methods of Classical Mech is a good source for a deeper study of this.

• Right, by why do you need to say "in a general setting"? Aren't Hamilton's equations enough? In Newtonian mechanics, there really is no reason to think of these as flows, etc. So why and when did people start talking about this "general setting"? Apr 10, 2011 at 2:21
• @Davidac897: by general setting Lawrence means "without coordinates". By working in this abstract setting you gain lots of new tools, better view of the theory and proofs and calculations often simplify significantly. Are you familiar with canonical transformations and Hamilton-Jacobi equation? It can be very useful to work in general setting. So this answers your why. As for your when, that's hard to say. But it was probably in the beginning of the $20^{\rm th}$ century. Apr 10, 2011 at 6:40
• What is the expression for the matrix $\Omega_{ij}$? I cannot find it in Arnold text. Nov 18, 2017 at 20:02

Weirdly, no answer to this question mentioned the two fundamental reasons for using the machinery of differential geometry of manifolds in Hamiltonian Mechanics.

On the one side, even simple systems require a non-trivial mathematical (topological and differential) structure of the phase space. The most obvious example is the rigid body configuration space with a fixed point in $$3D$$. It is the set of all the orthogonal 3x3 matrices with determinant $$1$$ (aka the Lie group $$SO(3)$$). It is not a Euclidean space!

On the other side, starting with Poincaré's research on the three-body problem, geometrical approaches have been considered powerful methods to attack complex problems in Mechanics, often allowing at least qualitative results.

I would say that the number of results from the geometrical approach to Classical Mechanics over more than half-century of research has demonstrated beyond doubt the soundness of the original hopes.