Symplectic geometry -- and how it relates to the Hamiltonian

Last week, we introduced Hamiltonian mechanics -- another perspective on classical mechanics. This week, we want to recast Hamiltonian mechanics in a more geometric language. This geometric language ends up being very interesting and useful even beyond its original applications in physics, so even if you didn't read last week's article, you still might enjoy this one!

Phase spaces

In mathematics, one often deals with strange shapes (manifolds). This is because it is often helpful to think about the phase space of some physical system; this is a geometric object representing all possible states of the system.

As a simple example, imagine you have a single particle in 1 dimension. The basic hypothesis in mechanics is that if you know the particle's position and momentum (or velocity) then you can predict its future position. This basic principle holds in all approaches to mechanics: Newton's, Lagrange's, and Hamilton's.

The phase space of this physical system (of a particle in 1-dimension) is a two dimensional space, where the coordinates on your two-dimensional space capture the position and momentum. This concept can be a bit tricky to grasp at first, so a reader new to phase spaces might want to ponder the widget below. In the left panel, you'll see our particle moving around on a 1-dimensional line; in the right panel, you'll see the corresponding motion in phase space. Remember: the particle is only moving in 1-dimension -- the right picture is somewhat `artificial,' in that it is only something we draw to help us understand the 1-dimensional motion of the particle depicted on the left. In the widget below, the particle has mass 1 kilogram, so it's momentum is just 1 kilogram times its velocity.

Notice how our particle always stays on an ellipse. This is because of conservation of energy: the kinetic energy plus the potential energy is always constant.

The motion shown above is the motion of a harmonic oscillator -- in other words, the motion of a particle attached to an oscillating spring. The kinetic energy is, as always, \[\frac{1}{2}mv^2,\] where \(m\) is the mass and \(v\) is velocity. The potential energy is \(\frac{1}{2}kx^2,\) where \(x\) is the position and \(k\) is the spring constant, which depends on how stiff your spring is.

Using that momentum \(p\) is \(mv,\) we can rewrite kinetic energy as \(\frac{1}{2m}p^2.\) Therefore conservation of energy tells us that \[\frac{1}{2m} \cdot p^2 + \frac{k}{2} \cdot x^2 = E,\] for some constant \(E\) (the total energy of the system). But remember: \(m\) and \(k\) are just some constants, and if you graph \[Ap^2 + Bx^2 = C\] in the \((p, x)\) coordinate plane, you'll always get an ellipse! (of radius \(\sqrt{\frac CA}\) in the \(p\)-direction and radius \(\sqrt{\frac CB}\) in the \(x\)-direction).

Perhaps this example convinces you of the utility of the phase space: the conservation of energy can be re-discovered in phase space as the observation that our particle always remains confined to an ellipse!

Incorporating force

Newton told us that, to understand the motion of a particle, you need to know three things: its current position, its current velocity, and what forces are acting on it.

As we explained last week, Hamilton realized that there's a slightly different way to understand motion: record the current position and momentum, and then enforce a conservation law. Remarkably, enforcing the conservation law is enough to understand how the motion will evolve over time.

In honor of Hamilton's achievement, the total energy of a physical system is called the Hamiltonian. When our phase space is a manifold \(M,\) the Hamiltonian is a function \[H \colon M \to \mathbb{R}.\] In other words, the Hamiltonian takes as input a point on phase space, and outputs a real number.

In the harmonic oscillator example from above, our manifold \(M\) was the 2-dimensional space of pairs \((x, p)\) (position \(x\) and momentum \(p\)), and our Hamiltonian was \[H(x, p) = \frac{1}{2m}p^2 + \frac{k}{2}x^2.\] Observe that all the 'physics' goes into constructing \(M\) and choosing the Hamiltonian: we needed to know that \(kx^2/2\) was the potential energy of our system in order to define \(H.\)

From here, Hamilton gives us a way to turn \(H\) into a differential equation -- solving this differential equation tells us how our particle will evolve over time. But before discussing how we can turn \(H\) into a differential equation, let's zoom out for a minute, and discuss ODEs (ordinary differential equations) on manifolds in general.

ODEs on manifolds

A vector field on a manifold \(M\) is a choice of tangent vector at each point of \(M.\) The geometric way to talk about ODEs is to talk about flows along vector fields: these are paths where, at every point in the path, the velocity is given by the vector field.

This definition of flow might seem complicated, but we're going to go through it more slowly in a second. For now, look at this widget showing a vector field on a sphere; the widget also animated a few particles flowing along this vector field. Below this widget on the sphere, we have a second widget, going more slowly through an example of flowing along a vector field.

Turning Hamiltonians into vector fields

In the last section, we saw how vector fields can be used to geometrize ordinary differential equations. So, what are Hamilton's equations of motion in terms of vector fields?

That is, suppose we have a phase space \(M\) of a physical system, and a Hamiltonian \(H\colon M \to \mathbb{R}\) on that physical system. How can we turn the Hamiltonian \(H\) into some vector field \(X_H\) that describes the motion of a particle whose energy is given by \(H\)?

The first observation is that the vector field \(X_H\) should depend only on the 1-form \(dH,\) and not on the function \(H.\) This is because the Hamiltonian is the sum of kinetic energy and potential energy -- but potential energy is only meaningful up to a constant. For example, gravitational potential energy is equal to \(mgh,\) where \(m\) is your mass, \(g\) is the acceleration due to gravity (some constant), and \(h\) is your height. But height isn't a meaningful concept -- height above sea level? height above the ground? height above the center of the Earth?

In order to make sense of height, you have to make an arbitrary choice of where to measure from; depending on what choice you make, this leads to some ambiguity in the meaning of potential energy, and hence creates an ambiguity in the total energy \(H.\) The 1-form \(dH\) (which is a fancy version of the derivative) eliminates this ambiguity.

Rephrasing our question, we now ask: given a 1-form \(dH,\) how can we turn it into a vector field \(X_H\)?

Jacobi always said "Man muss immer umkehren" -- if you're stuck on a math problem, you should invert. So, let's go backwards for a second: starting with a vector field \(V,\) how might we turn it into a 1-form?

This question is easier to answer. If you're an expert, we'll just say that linear maps \(TM \to T^*M\) are the same as 2-forms on \(M.\) Breaking this down more concretely, remember that a 1-form \(\alpha\) on \(M\) is some gadget which takes as input a tangent vector \(W\) and outputs a number.

Recall that a 2-form \(\omega\) on \(M\) was a certain object that took as input two tangent vectors, and outputted a number. If we fix a specific 2-form \(\omega,\) then we can turn any vector field \(V\) into a 1-form \(\alpha\): just define \[\alpha(W) := \omega(V, W).\]

But this was the opposite of what we wanted to do: we started with a vector field and outputted a 1-form, whereas we really want to start with the 1-form \(dH\) and output a vector field \(X_H.\) Fortunately, the operation described above is sometimes invertible, allowing us to go backwards: whenever \(\omega\) is a symplectic 2-form (this means \(\omega\) is 'closed and non-degenerate', but if you don't know differential geometry you should ignore this and just think 'symplectic means we can invert'), we can invert this correspondence, in the sense that every 1-form \(\alpha\) arises from a unique vector field \(V.\) We then define \(X_H\) as the unique vector field making \[dH(W) = \omega(X_H, W)\] true.

In other words, if we equip our phase space \(M\) with a symplectic 2-form \(\omega,\) then we can turn Hamiltonian functions into vector fields. Flowing along those vector fields then tells us how our physical system will evolve!

An example

The above discussion was a little abstract, so let's go through it in the concrete example of the harmonic oscillator.

Remember that our phase space was the 2-dimensional space with coordinates \((x, p)\) (position and momentum), and our Hamiltonian was \[H(x, p) = \frac{1}{2m}p^2 + \frac{k}{2}x^2,\] where \(m\) and \(k\) were arbitrary constants (mass and the spring stiffness).

To turn this into a vector field \(X_H,\) we need one more piece of information: a symplectic form on \(M.\) What should our symplectic form be? To find the right symplectic form in this example, we're going to cheat a little by using Hamilton's equations to figure out what \(X_H\) is, and then reverse engineer \(\omega\) afterwards; we'll say a bit more about the general situation later.

Hamilton's equations tell us that, whatever the final motion is, it should obey \[x'(t) = \frac{\partial H}{\partial p},\] \[p'(t) = -\frac{\partial H}{\partial x}.\]

In other words, if we have a curve \(t \mapsto (x(t), p(t))\) in our phase space that corresponds to the motion of some physical system, then the derivative of that curve at time \(t\) should be the vector \[\left(\frac{\partial H}{\partial p}, -\frac{\partial H}{\partial x}\right).\]

This means the vector field \(X_H\) should be \[X_H = \left(\frac{\partial H}{\partial p}, -\frac{\partial H}{\partial x}\right),\] because the derivative of the flow of a particle along a vector field is supposed to be the vector field itself.

With a little computational skill in differential geometry, from this expression for \(X_H,\) and the expression \[dH = \frac{\partial H}{\partial p} dp + \frac{\partial H}{\partial x} dx,\] one can derive that the symplectic form \(\omega\) is the 2-form \[\omega = dx \wedge dp.\]

In general, the symplectic form on phase space will look something like this; but if you have a 3-dimensional particle, or a system of ten 2-dimensional particles, etc., you might have more positions and momentums to keep track of, making your symplectic form look like \(\sum_i dx_i \wedge dp_i,\) where you sum over every possible direction.

This gives a geometric way to formulate Hamilitonian mechanics: you do some physics to design a phase space \(M\), symplectic form \(\omega,\) and Hamiltonian \(H\); then to solve your physics problem, you do some differential geometry to turn \(H\) into a vector field \(X_H,\) and find how particles flow along \(X_H.\)

Symplectic manifolds in pure mathematics

In physics, one has these phase spaces \(M\) -- shapes whose points represent the possible states of some physical system. In mathematics, one often finds moduli spaces; a moduli space is just some geometric shape whose points represent something (this is a pretty broad definition!). For example, the circle is the moduli space of all directions: each point on a circle represents a different direction, and all the possible directions (in 2-dimensional space) are points on a circle!

Remarkably, it turns out that the moduli spaces arising in pure mathematics often admit symplectic structures, just as the phase spaces of physics do! This allows one to use ideas from physics to study shapes arising in pure mathematics. These symplectic structures are particularly important in geometric representation theory, which has come to play an increasingly large role in the Langlands program. The idea of geometric representation theory is to turn questions in representation theory -- a branch of abstract algebra -- into geometry problems, by looking at complicated moduli spaces related to your original question. These complicated moduli spaces tend to have symplectic structures, and using tools from symplectic geometry we can better understand these moduli spaces, and hence solve our original representation theory question.