It’s a Symplectic World

In the Hamiltonian formulation of classical mechanics a system with finitely many, say {n}, degrees of freedom is described by coordinates {q=(q_1,\dots,q_n)} determining the configuration of the system and conjugate momenta {p=(p_1,\dots,p_n)}. In mathematical terms, {q} serve as local coordinate functions on the configuration space {\mathcal{C}} which has the structure of a {n}-dimensional smooth manifold and {x=(q,p)} are local canonical coordinates on the cotangent bundle {\mathcal{P}=T^\ast\mathcal{C}} called the phase space. For example, a (mathematical) pendulum’s configuration is given by an angle {q} and its momentum by a real number {p}, thus having a circle as configuration space, {\mathcal{C}=\mathbb{S}^1} and a cylinder as phase space, {\mathcal{P}=\mathbb{S}^1\times\mathbb{R}}.

Pendulum and its phase space.
Pendulum and its phase space.

Dynamics is governed by a smooth real-valued function on the phase space {H\in C^\infty(\mathcal{P})} called the energy function or Hamiltonian and is prescribed by a system of ODE’s, the equations of motion, which read as

\displaystyle \dot q_i(t)=\frac{\partial H(q,p)}{\partial p_i},\quad \dot p_i(t)=-\frac{\partial H(q,p)}{\partial q_i},\quad i=1,\ldots,n. \ \ \ \ \ (1)

This can be written in the concise matrix-vector form

\displaystyle \dot x(t)=J\ \nabla H(x), \ \ \ \ \ (2)

where {\nabla=(\partial_q,\partial_p)} and {J} is the {2n\times 2n} matrix

\displaystyle J=\begin{pmatrix} 0_n&I_n\\-I_n&0_n \end{pmatrix}. \ \ \ \ \ (3)

The following properties of {J} can be easily checked

\displaystyle J^2=I_{2n},\quad J^{-1}=J^\top=-J,\quad \det(J)=+1. \ \ \ \ \ (4)

To obtain a particular solution initial values {x(t_0)=(q(t_0),p(t_0))=(q_0,p_0)} must be specified. Transformations of the phase space which preserve Hamilton’s equations (2) are clearly of high importance. They’re called canonical transformations. For example, when a system evolves for some time {t_1}, a canonical transformation is induced by taking every point as an initial value {x(t_0)} and mapping them into {x(t_0+t_1)}. Now let’s consider an arbitrary transformation of the phase space

\displaystyle y\colon\mathcal{P}\rightarrow\mathcal{P},\quad x\mapsto y(x). \ \ \ \ \ (5)

Then the time derivative of {y} along a solution {x(t)} reads as

\displaystyle \dot y=\frac{\partial y}{\partial x}\dot x =MJ\ \nabla_x H =MJM^\top \nabla_y H \ \ \ \ \ (6)

and hence

\displaystyle \dot y(t)=(MJM^\top)\nabla_y H, \ \ \ \ \ (7)

where {M_{i,j}=\partial y_i/\partial x_j} is the Jacobian of the transformation {y}. Now, it is clear that {y} is canonical iff {M} satisfies the following condition

\displaystyle MJM^\top=J. \ \ \ \ \ (8)

Such {2n\times 2n} matrices are called symplectic(*). Let’s take a closer look at the structure of such a matrix by splitting it up into four {n\times n} blocks

\displaystyle M=\begin{pmatrix}A&B\\C&D\end{pmatrix},\quad A,B,C,D\in\mathbb{R}^{n\times n}. \ \ \ \ \ (9)

Then condition (8) on {M} can be reformulated for the blocks as

\displaystyle {AB^\top=BA^\top,}\ \ \ \ \ (10)

\displaystyle {CD^\top=DC^\top,}\ \ \ \ \ (11)

\displaystyle {DA^\top=I_n+CB^\top.}\ \ \ \ \ (12)

What’s the determinant of a symplectic matrix? Well, since the determinant is multiplicative it is immediate from (8) that {\det(M)=\pm 1}. Actually, we can show that it’s {+1}.

Claim. Every symplectic matrix has determinant {+1}.

Proof. Assuming that the block {A} is invertible {M} can be written as a product

\displaystyle M=\begin{pmatrix}A&0_n\\C&I_n\end{pmatrix} \begin{pmatrix}I_n&A^{-1}B\\ 0_n&D-CA^{-1}B\end{pmatrix}, \ \ \ \ \ (13)

thus its determinant is

\displaystyle \det(M)=\det(A)\det(D-CA^{-1}B)=\det(DA^\top-CA^{-1}BA^\top). \ \ \ \ \ (14)

where {\det(A^\top)=\det(A)} and the multiplicative property of the determinant were also utilized. Now, using eqs. (10) and (12) we get

\displaystyle \det(M)=\det(I_n+CB^\top-CB^\top)=\det(I_n)=+1. \ \ \ \ \ (15)

The proof is completed by noticing that matrices with {\det(A)\neq 0} form a dense subset of symplectic matrices and since the determinant is a continuous map we have {\det(M)=+1} for all symplectic matrices. {\square}

The above fact has a profound consequence. Canonical transformations preserve the phase space volume element! This is Liouville’s theorem. Furthermore, if the system has only bounded orbits we get Poincaré recurrence theorem which states that any neighbourhood of an initial value {x} is infinitely many intersected by the solution {x(t)} starting from {x} as {t} goes to infinity.

In a previous post a funny application of the recurrence theorem to the academic administrative system was described. Here we mention another three applications which are at least as interesting as the one mentioned before:

  1. Pick an arbitrary point on the unit circle, {x\in\mathbb{S}^1}, and consider the rotation {T_\alpha} around the origin through some angle {\alpha}. {T_\alpha} is clearly volume-preserving and the ‘orbit’ of any point is bounded. There are two different cases though:
    • {\alpha/\pi} is rational: then {T_\alpha^N(x)=x} for some {N}.
    • {\alpha/\pi} is irrational: then {\{T_\alpha^N(x):N\in\mathbb{N}\}} is dense in {\mathbb{S}^1}.

    Rational & irrational rotations.
    Rational & irrational rotations.
  2. Consider the first digits of the powers of {2} (in base ten). What are the frequencies of {1,2,\dots,9} among these digits? Any power {2^N} can be written as

    \displaystyle 2^N=D\times 10^m\quad\text{with some}\quad 1\leq D<10\quad\text{and}\quad m\in\{0,1,2,\dots\}. \ \ \ \ \ (16)

    The question is the probability distribution of {\lfloor D\rfloor}. Taking logarithm of both sides we get

    \displaystyle N\log_{10}2=m+\log_{10}D, \ \ \ \ \ (17)

    where {\log_{10}D\in[0,1)}. Multiplying (17) by {2\pi}, we can rephrase the question as follows: How often will the orbit {T_{2\pi\log_{10}2}(0)} intersect the intervals {[\log_{10}d,\log_{10}d+1)}, {d=1,2,\dots,9}. These clearly are the length of the intervals, therefore

    \displaystyle \text{frequency of }d\text{ as first digit}=\log_{10}(d+1)-\log_{10}d,\quad d=1,2,\dots,9. \ \ \ \ \ (18)

    \displaystyle \begin{tabular}{r|c|c|c|c|c|c|c|c|c}  Digit&1&2&3&4&5&6&7&8&9\\\hline  Frequency&0.301&0.176&0.125&0.097&0.079&0.067&0.058&0.051&0.045  \end{tabular} \ \ \ \ \ (19)

  3. A gas is prepared in one side of a two-compartment box. When the wall separating the compartments is removed the gas slowly, distributes equally among both halves of the box toward statistical equilibrium. As a consequence of Poincaré recurrence theorem there must be a time in the future when the gas will return back close to its initial condition with all molecules confined in one compartment. This page contains a simulation of randomly moving molecules in a box, thus Poincaré’s recurrence theorem applies. At t=0 the molecules are arranged in a nice square-shaped configuration. After 12800 (or 25600) steps a system returns to its initial state.

Here’s a video of pendulums of different lengths swinging around the same axis. One can clearly see that around 0:55 there are three sets of pendulums and in each set the pendulums are synchronized. Then at 1:23 there are two partitions and the recurrence time is ca. 2:45 minutes.

(*) The word ‘symplectic’ means complex and was coined by Weyl in 1939.

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