Phase Space in Classical Mechanics

String Theory without the Baggage, Episode 1.

Hey Friends,

Today we begin our discussion with some broad ideas around the point particle. If you’d like a bit more context, check out Episode Zero:

String Theory without the Baggage

We’ll be publishing updates pretty much every day, for a little while anyway. Thanks for reading!

Sean

---

Traditionally, we model the space of possible positions of some dynamical object as R^3. Classically, the set of possible velocities - and therefore momenta - then also sweeps out a copy of R^3. Hence, the phase space of an individual, classical particle is R^6, to (co)tangent bundle associated to R^3:

\(\mathrm{T}^{\star}\mathbb{R}^{3} = \mathbb{R}^{6}.\)

The phase space of a physical system is the stage on which dynamics take place. We'll denote phase space by P. Physical trajectories amount to curves in phase space:

\(\gamma : \mathbb{R}\rightarrow \mathbb{P},\)

where curves like 𝛾 are traditionally parameterized by time.

The precise nature of those curves are determined by solutions to differential equations and are the basic objects of study in Classical Mechanics. These equations are determined by the Principle of Least Action.

To find these equations one must minimize the physical Action,

\(S(\gamma) = \int_{\gamma} dt\; L(x,\dot{x}) \quad \mathrm{or} \quad \int_{\gamma} dt\; (p\dot{x} - H(x,p)),\)

over the set of all possible paths through phase space, 𝛾. In this sense, the functions L or H define the physical system to be considered.

In this context p is the Legendre transform of dx/dt¹.

\(p = \frac{\partial L}{\partial\dot{x}},\quad \dot{x} = \frac{\partial H}{\partial p}.\)

So long as the integrand in the action S is convex² in the kinetic sector - that is p or dx/dt - then which function we specify - L or H - isn't particularly important.

For a classical particle, the action is traditionally specified in terms of a position-dependent³, potential energy function V:

\(\begin{equation}S_{\gamma} = \int_{\gamma}dt \, \frac{m}{2}\left( \dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2} \right) - V(x,y,z).\end{equation}\)

In this case the associated momentum is

\(\mathbf{p} = m\dot{\mathbf{x}}.\)

Additionally, the Principle of Least Action yields the equations consistent with Newton's second law:

\(m\ddot{\mathbf{x}} = \vec{\nabla} V.\)

In particular, in the absence of a meaningful potential gradient, velocity - and therefore momentum - remains unchanged.

Our aim is to generalize this discussion to the relativistic case, before we do that we need to address a complication with the relativistic phase space. Stay tuned for more!

1. In our equations, we will use a “dot” as a shorthand for the derivative with respect to time.

2. Convex means “has a positive second derivative”. For x > 0, x^2 is convex, while √x is not.

3. Because V also traditionally represents interactions, it may also be related to the velocities. This happens, for example in electrodynamics. Consistency requirements apply, but this often merely augments the definition of the momentum, p.