How To Think About Dynamic Systems

Working with phase space

I’ve been thinking about how to reason about state in biological and physical systems. In classical mechanics, one way to describe dynamical systems is through the variables we can measure at a point in time.

For example, the measurable variables for a single particle are its position and velocity. If it’s moving in one dimension, the position is a single number, and the velocity is a single number, for a total of two variables.

Phase space is the space of where these state variables can live.

Take a pendulum. I can describe the system through the angle it makes with the vertical axis, and the velocity of this angle. If we plot the values against each other, and then let time pass, we can trace out the phase space of a pendulum and see some interesting shapes evolve.

In the case of a pendulum with no friction, we get a closed loop. The x axis is the angle, and the y axis is the angular velocity. The animation below shows how phase space is traced out as the system evolves.

It’s not all the time that we get closed loops though. If we add air resistance, we get a spiral since the pendulum converges to a stationary point.

There are already interesting questions about viewing mechanical systems through this lens. A lot of these are geometric. For example, is the phase space ‘bounded’ in its geometry? Does the phase space lie on some interesting surface or subset, and if so what’s the dimension of the subset? What do the ‘orbits’ of points look like in phase space? And is there a way I can tell if orbits overlap?

Trajectories in phase space have their own nomenclature. Attracting points in phase space are points that the system converges to if the initial conditions start in some ‘basin of attraction’. Chaotic systems often have a never ending trajectory that is bounded, but never overlaps on itself. These are called strange attractors.

The Van Der Pol oscillator is a system where, no matter where you start in phase space, the point converges to a fixed cycle geometry.

User:XaosBits at English Wikipedia, CC BY 2.5 <https://creativecommons.org/licenses/by/2.5>, via Wikimedia Commons

Then, we have the questions of dynamics. If I start on a point in phase space, what path is subsequently traced out?

In high school, we use Newton’s laws to calculate the equations of motion. To do this, we use Newton’s laws to figure out what the acceleration will be once you solve for the forces.

But there is another way to do this, called the Hamiltonian method. This method tells you where a point in phase space will go to as time evolves. To do this, we first need to calculate the Hamiltonian of a physical system, which is the sum of its kinetic and potential energy as a function of its phase space variables. If we label momentum as p, and position as q the Hamiltonian is given by

\(H(q, p) = T(p) + V(q)\)

where T(p) is the kinetic energy and V(q) is the potential energy.

With this Hamiltonian, we can calculate how each state variable will evolve according to Hamilton’s equations. The equations relate the time derivative of each state variable as a function of the Hamiltonian, and crucially gives us an avenue to see how the state of a system evolves.

\(\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}, \quad i = 1, \ldots, N\)

As an example, lets look at the motion of a spring. A spring’s Hamiltonian is

\(H = \frac{p^2}{2m} + \frac{1}{2}k q^2\)

where m is the mass, k is the spring constant, q is the displacement, and $p$ is the momentum. In this formulation, we treat position and momentum as its own variables, and then see what pops out. Hamilton’s equations for the spring give us the following equations

\(\dot{q} = \frac{p}{m}, \quad \dot{p} = -kq\)

Now if we treat both variables as a function of time, and then solve it, we get that the solution is

\(q(t) = q_0 \cos(\omega t) + \frac{p_0}{m\omega} \sin(\omega t)q(t) = q_0 \cos(\omega t) + \frac{p_0}{m\omega} \sin(\omega t)\)

\(p(t) = p_0 \cos(\omega t) - m\omega q_0 \sin(\omega t)\)

where omega = sqrt{k/m} is the natural frequency of the spring, and q_0 and p_0 are the initial displacement and momentum. Now, if we choose a point in phase space, we can let the equations play out and see what lines we trace. In the case of a spring, it traces out an ellipse.

There are also interesting modern treatments that use phase space to predict the dynamics of a system. One such way is the delayed embedding method, which is a way to ‘learn’ the phase space trajectory by taking many observations. You could also think about phase space in terms of any state you want. For example, in biological systems, these could be temperature, CO2 concentration, and oxygen concentration.