### Symplectic Geometry

For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum “physics is geometry” is a guiding principle of modern theoretical physics. Einstein’s General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:

$\displaystyle \mbox{Matter tells space how to curve. Space tells matter how to move.}$

Riemannian Geometry

For millennia the geometry of Euclid was the only show in town. In the early nineteenth century, non-Euclidean geometries that relax the parallel postulate were developed (independently) by Bolyai, Gauss and Lobachevsky. Later, the brilliant German mathematician Bernhart Riemann introduced ideas of sweeping power and originality, showing that Euclidean geometry is merely a special case of a vastly more general geometry of curved space.

Riemann showed that the entire structure of space, including its complex patterns of curvature, are encapsulated in a metric’, a simple expression for the distance between two points

$\displaystyle \mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu$

The space and its metric together are sufficient to erect an entire geometric edifice. Riemannian geometry provided Einstein with an ideal mathematical basis for his theory of gravitation.

Hamiltonian Mechanics

There is another, less well-known, geometry that is also deeply rooted in physics. It is called symplectic geometry. Symplectic geometry (SG) lies at the heart of mathematics and of physics. It is at the very foundation of classical mechanics. The behaviour of spinning tops, water waves, falling apples, planetary systems and galaxies can be described in terms of this geometry. SG was first introduced by Lagrange in 1808 in his study of solar system dynamics.

The Irish scientist William Rowan Hamilton is popularly known as the inventor of quaternions, a new species of numbers with exotic properties. But his work in mechanics is of far greater significance. Hamilton showed how a dynamical system can be described by a single mathematical expression involving the position and momentum.

The state of the system at any time is determined by a point in a multi-dimensional space called phase-space, and its evolution is described by Hamilton’s canonical equations, a system of exquisite elegance. Since Hamilton’s equations provide a gateway to quantum mechanics, the new geometry also underlies dynamics at an atomic scale. The symplectic structure is also intimately connected with Heisenberg’s uncertainty principle.

A particle moving in three dimensions is described by a point in six dimensions, three for position and three for momentum. So, the state of the system is given by a point ${(q_1,q_2,q_3,p_1,p_2,p_3)}$ in a six-dimensional phase space. Once the initial state is known, the trajectory can be calculated by solving the canonical equations.

Metric Geometry and Symplectic Geometry

To take the simplest example, let us consider two points and in the plane ${\mathbb{R}^2}$, represented by two vectors ${\mathbf{u}=(u_1,u_2)}$ and ${\mathbf{v}=(v_1,v_2)}$. The metric for Euclidean geometry is a map from ${\mathbb{R}^2\times\mathbb{R}^2}$ to ${\mathbb{R}}$, taking a pair of vectors as inputs and giving a real number:

$\displaystyle g(\mathbf{u},\mathbf{v}) = u_1 v_1 + u_2 v_2$

This immediately gives the length ${|\mathbf{u}|}$ of a vector

$\displaystyle |\mathbf{u}| = \sqrt{g(\mathbf{u},\mathbf{u})} = \sqrt{u_1^2+u_2^2} \,,$

and the angle ${\theta}$ between two vectors:

$\displaystyle \cos\theta = \frac{g(\mathbf{u},\mathbf{v})}{|\mathbf{u}|\cdot|\mathbf{v}|} \,.$

In symplectic geometry we also have a map from ${\mathbb{R}^2\times\mathbb{R}^2}$ to ${\mathbb{R}}$, taking a pair of vectors to a real number. The standard symplectic two-form is

$\displaystyle \Omega(\mathbf{u},\mathbf{v}) = u_1 v_2 - u_2 v_1$

This antisymmetric 2-form does not correspond to lengths and angles, but to areas: ${\Omega(\mathbf{u},\mathbf{v})}$ is the oriented area of the parallelogram spanned by the two vectors. Changing the order of the vectors changes the sign, hence the anti-symmetry of ${\Omega(\mathbf{u},\mathbf{v})}$. So, whereas Euclidean geometry is the geometry of lengths and angles, symplectic geometry is an areal geometry.

Euclidean geometry is easily extended to higher dimensions. The metric in ${n}$-dimensional Euclidean space arises from

$\displaystyle g(\mathbf{u},\mathbf{v}) = \sum_{i=1}^n u_i v_i \,,$

and lengths and angles follow as before. For symplectic geometry, things are trickier, since area is essentially two-dimensional. However, taking an even-dimensional space ${\mathbb{R}^{2m}}$ and two infinitesimal vector displacements ${\mathrm{d}\mathbf{u}}$ and ${\mathrm{d}\mathbf{v}}$, we can define ${\Omega(\mathrm{d}\mathbf{u},\mathrm{d}\mathbf{v})}$ to be the sum of the oriented areas of the shadows or projections of the parallelogram spanned by ${\mathrm{d}\mathbf{u}}$ and ${\mathrm{d}\mathbf{v}}$ onto ${m}$ coordinate planes. We write this as a wedge product:

$\displaystyle \Omega(du,dv) \equiv \mathrm{d}\mathbf{u}\wedge \mathrm{d}\mathbf{v} = \sum_{i=1}^m \mathrm{d}u_i\wedge \mathrm{d}v_i$

Riemann extended Euclidean space by allowing for curved spaces, with different properties at different places. Symplectic geometry can also be applied in curved spaces or manifolds. We consider a surface to be the integral of infinitesimal parallelograms and define oriented areas by integrating the shadows or projections of these elements.

Phase Fluid

Hamiltonian mechanics is essentially the symplectic geometry of phase space. Phase space comprises a system of 1-dimensional trajectories that fill it without ever intersecting each other. We can consider a cluster of points corresponding to a set of starting conditions. As time evolves, this cluster moves through phase space, like a parcel of fluid. Collections of points move around in phase space like parcels of fluid and this flow has a very special character. The parcels keep their initial volume, but much more than that: the flow preserves the areas of two-dimensional sheets of fluid as they move.

The property of being symplectic implies that the sum of the projections of a volume onto the ${N}$ coordinate-momentum planes or ${pq}$-planes is preserved for any closed curve in phase space as it moves under a Hamiltonian map:

$\displaystyle \sum_\mu \oint_{C_\mu(t)} p_\mu(t) \mathrm{d}q^\mu(t) = \sum_\mu \oint_{C_\mu(0)} p_\mu(0) \mathrm{d}q^\mu(0)$

The figure below, from V I Arnold’s book on Mathematical Methods in Classical Mechanics, illustrates this.

Corollary from V.I.Arnold’s book on Classical Mechanics.

The phase space is a symplectic manifold and the dynamics consists of an area-preserving time-dependent transformation of phase space. In other words, the flow is an evolving symplectic transformation or symplectomorphism. The mathematical framework of SG is not confined to mechanics. In addition to modelling planetary systems and galaxies, it applies to electornic circults, molecular systems and optical problems. It also provides a gateway to quantum mechanics.

Conclusion

Symplectic geometry is one of the most valuable products of the link between mathematics and physics. Its mathematical theory owes its existence to physics and, in turn, mathematical developments of SG have enriched physical theory. The word symplectic’ means intertwined or woven together. SG has the promise of weaving together key areas of mathematics and physics.

SG has been applied in many areas of mathematics: group theory, analysis, number theory and topology. Some commentators go so far as to say that SG could bring about a revolution in mathematics with impact comparable to the impact of introducing complex numbers.