Consider a mechanical system of

N point-like particles whose motion is determined by a Hamilton function

H. If the system is confined to a bounded region of phase space

${\mathbb{R}}_{\mathbf{q},\mathbf{p}}^{6N}$, Poincaré’s recurrence theorem tells us that any initial pattern of positions and velocities (specified within a given error) will recur, independently of any permutation in the numbering of the particles of the system. The recurrence time is however generally extremely long, (see [

11] for a recent analysis of recurrence time), except of course for periodic (or quasi-periodic) systems. Of course, the boundedness condition is essential: a free particle in an infinite Universe will never return to its initial position. Suppose indeed that the system, represented by a phase point

$(\mathbf{q},\mathbf{p})=({\mathbf{q}}_{1},...,{\mathbf{q}}_{N},{\mathbf{p}}_{1},...,{\mathbf{p}}_{N})$, with

${\mathbf{q}}_{i}=({x}_{i},{y}_{i},{z}_{i})$ and

${\mathbf{p}}_{i}=({p}_{{x}_{i}},{p}_{{y}_{i}},{p}_{{z}_{i}})$, is confined to a “universe”

$\mathcal{U}$. We are not asking for an exact return of

$(\mathbf{q},\mathbf{p})$ but we content ourselves with the return of some (arbitrarily) small neighbourhood

$\Omega $ of that point. Then, an upper bound for the first return time of that neighbourhood has a magnitude of order

$T\approx \mathrm{Vol}\left(\mathcal{U}\right)/\mathrm{Vol}(\Omega )$. This number is usually very large. Let us now focus on a subsystem, identified with a point

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})=({\mathbf{q}}_{1},...,{\mathbf{q}}_{n},{\mathbf{p}}_{1},...,{\mathbf{p}}_{n})$ with

$n<N$. Assume first that the total Hamiltonian function is of the type

where

$({\mathbf{q}}^{\prime \prime},{\mathbf{p}}^{\prime \prime})=({\mathbf{q}}_{n+1},...,{\mathbf{q}}_{N},{\mathbf{p}}_{n+1},...,{\mathbf{p}}_{N})$. Due to the absence of interaction between the two subsystems

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$ and

$({\mathbf{q}}^{\prime \prime},{\mathbf{p}}^{\prime \prime})$, their motions are independent; the time-evolution of

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$ is thus governed solely by its own private Hamiltonian

${H}^{\prime}$; the equations of motions are

and their solutions only dependent on the initial values

${\mathbf{q}}^{\prime}\left(0\right)$ and

${\mathbf{p}}^{\prime}\left(0\right)$. The corresponding universe

${\mathcal{U}}^{\prime}$ consists of the set of all points

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$ such that

$({\mathbf{q}}^{\prime},{\mathbf{q}}^{\prime \prime},{\mathbf{p}}^{\prime},{\mathbf{p}}^{\prime \prime})$ is in

$\mathcal{U}$ for some

${\mathbf{q}}^{\prime \prime},{\mathbf{p}}^{\prime \prime}$; it is thus the projection of

$\mathcal{U}$ on the reduced phase space

${\mathbb{R}}_{{\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime}}^{6n}$ and, accordingly, the corresponding neighbourhood

${\Omega}^{\prime}$ is the projection of

$\Omega $ on

${\mathbb{R}}_{{\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime}}^{6n}$. Let us now compare the return time

${T}^{\prime}\approx \mathrm{Vol}\left({\mathcal{U}}^{\prime}\right)/\mathrm{Vol}\left({\Omega}^{\prime}\right)$ for the system

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$ with that of

$(\mathbf{q},\mathbf{p})$. To fix the ideas, we choose for

$\mathcal{U}$ a hypercube with sides of length

L and for

$\Omega $ a hypercube with sides of length

$\epsilon \ll L$. It follows that

$T\approx {(L/\epsilon )}^{6N}$ and that

${T}^{\prime}\approx {(L/\epsilon )}^{6n}$ so that the ratio

$T/{T}^{\prime}$ between both return times is of order

${(L/\epsilon )}^{6(N-n)}$. Consider next the general case, where the subsystems interact; we can no longer separate the variables that

${\mathbf{q}}_{i}$ and

${\mathbf{p}}_{i}$; this is the case if for instance,

although everything will hold for an arbitrary function of the variables

${\mathbf{q}}_{j},{\mathbf{p}}_{j}$. We consider again the subsystem

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$; its motion will now depend on the global behaviour of the system

$(\mathbf{q},\mathbf{p})$, since the solutions of the corresponding Hamilton equations

now depend on the initial values of

all variables

${\mathbf{q}}_{j},{\mathbf{p}}_{j}$, not only the

n first. It follows that the motion of the subsystem

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$ is not governed by a Hamiltonian; this can be easily seen by finding the explicit solutions for simple systems. Let us illustrate this using Sharov’s argument [

12]. Suppose Equation (

4) represents the time-evolution of a bona fide Hamiltonian system. Denoting by

${\mathbf{q}}_{1}^{0},...,{\mathbf{q}}_{n}^{0};{\mathbf{p}}_{1}^{0},...,{\mathbf{p}}_{n}^{0}$ any set of initial conditions we have

where

$\mathcal{J}\left(t\right)$ is the Jacobian of the transformation from the initial conditions to

$({\mathbf{q}}^{\prime},{\mathbf{p}}^{\prime})$. If the system Equation (

4) is Hamiltonian, then this transformation must be canonical, so we should have

$\left|\mathcal{J}\right(t\left)\right|=1$. However, we have, as Sharov [

12] showed,

$d\mathcal{J}\left(t\right)/dt\ne 0$, hence

$\mathcal{J}\left(t\right)\ne \mathcal{J}\left(0\right)=1$. In fact, the principle of the symplectic camel which we discuss below implies, without any calculation at all, that

$\left|\mathcal{J}\right(t\left)\right|\ge 1$. Thus:

What about the return time? A first educated guess is that since the subsystem interacts (perhaps very strongly) with the rest of the system this interaction will influence the return time which will become much longer than in the interaction-free case, perhaps even of the order $T\approx {(L/\epsilon )}^{6N}$, at which the total system returns.