A Theorem on Ellipses, an Integrable System and a Theorem of Boltzmann
AA Theorem on Ellipses, an IntegrableSystem and a Theorem of Boltzmann
Giovanni Gallavotti and Ian Jauslin INFN-Roma1 & Universit`a “La Sapienza”, email: [email protected] Department of Physics, Princeton University, email: [email protected]
Abstract
We study a mechanical system that was considered by Boltzmann in1868 in the context of the derivation of the canonical and microcanonicalensembles. This system was introduced as an example of ergodic dynam-ics, which was central to Boltzmann’s derivation. It consists of a singleparticle in two dimensions, which is subjected to a gravitational attrac-tion to a fixed center. In addition, an infinite plane is fixed at some finitedistance from the center, which acts as a hard wall on which the particlecollides elastically. Finally, an extra centrifugal force is added. We willshow that, in the absence of this extra centrifugal force, there are twoindependent integrals of motion. Therefore the extra centrifugal force isnecessary for Boltzmann’s claim of ergodicity to hold.
Keywords:
Ergodicity, Chaotic hypothesis, Gibbs distributions, Boltzmann, Inte-grable systems
In 1868, Boltzmann [1868a] laid the foundations for our modern understand-ing of the behavior of many-particle systems by introducing the “microcanonicalensemble” (for more details on this history, see Gallavotti [2016]). The princi-pal idea behind this ensemble is that one can achieve a good understanding ofmany-particle systems by focusing not on the dynamics of each individual par-ticle, but on the statistical properties of the whole. More precisely, the state ofthe system becomes a random variable, chosen according to a probability distri-bution on phase space, which came to be called the “microcanonical ensemble”.An important assumption that was made implicitly by Boltzmann is that thedynamics of the system be ergodic. In this case, time-averages of the dynamicscan be rewritten as averages over phase space, and the qualitative properties ofthe dynamics can be formulated as statistical properties of the microcanonicalensemble.To support this assumption, Boltzmann presented a mechanical system thatvery same year (Boltzmann [1868b]) as an example of an ergodic system. Thissystem consists of a particle in two dimensions that is attracted to a fixed centervia a gravitational potential − α r . In addition, he added an extra centrifugalpotential g r . As was known since at least the times of Kepler, this systemis subjected to a central force, and is therefore integrable. In order to break1 a r X i v : . [ m a t h . D S ] A ug he integrability, Boltzmann added an extra ingredient: a rigid infinite planarwall, located a finite distance away from the center (see figure 1). Wheneverthe particle hits the wall, it undergoes an elastic collision and is reflected back.Boltzmann’s argument was, roughly, that in the absence of the wall, the dy-namics is quasi-periodic, so the particle should intersect the plane of the wallat points which should fill up a segment of the wall densely as the dynamicsevolves, and concluded that the region of phase space in which the energy isconstant must also be filled densely. As we will show, this is not the wholestory; following a conjectured integrability for g = 0, [Gallavotti, 2014, p.150],and first tests in [Gallavotti, 2016, p.225–228], we have found that, in the ab-sence of the centrifugal term g = 0, the dynamics (which has two degrees offreedom) still admits two constants of motion even in presence of the hard wall.This suggests that, if a suitable KAM analysis could be carried out, the systemwould not be ergodic for small values of g . Oh Figure 1: A trajectory. The large dot is the attraction center O , and the line isthe hard wall L . In between collisions, the trajectories are ellipses. The ellipsesare drawn in full, but the part that is not covered by the particle is dashed. Let us now specify the model formally, and state our main result more precisely.We fix the gravitational center to the origin of the x, y -plane and let L be theline y = h . The Hamiltonian for the system in between collisions is H = p x + p y − α r + g r (1.1)where α > , g ≥ , r = (cid:112) x + y and the particle moves following Hamilton’sequations as long as it stays away from the obstacle L . When an encounter with2 occurs the particle is reflected elastically and continues on.Boltzmann [1868b], considered this system on the hyper-surface A = p − αr + gr . The intersection of this hyper-surface with y = h is the region F A enclosed within the curves ± (cid:114) ( A − gx + h + α √ x + h ) , x min < x < x max (1.2)with x min and x max the roots of A = gx + h − α √ x + h . He argued that allmotions (with few exceptions) would cover densely the surfaces of constant A < α, g > g = 0.In this case, the motion between collisions takes place at constant energy A and constant angular momentum a , and traces out an ellipse. One of thefoci of the ellipse is located at the origin, and we will denote the angle thatthe aphelion of the ellipse makes with the x -axis by θ . Thus, the ellipse isentirely determined by the triplet ( A, a, θ ). When a collision occurs, A remainsunchanged, but a and θ change discontinuously to values ( a (cid:48) , θ (cid:48) ) = F ( a, θ ),and thus the Kepler ellipse of the trajectory changes. In addition, the semi-major axis a M of the ellipse is also fixed to a M = − α A (Kepler’s law): so thesuccessive ellipses have the same semi-major axis, while the eccentricity variesbecause at each collision the angular momentum changes: e = 1 + Aa α . Thus,the motion will take place on arcs of various ellipses E , which all share the samefocus and the same semi-major axis, but whose angle and eccentricity changesat each collision.Our main result is that the (canonical) map ( a (cid:48) , ϕ (cid:48) ) = F ( a, θ ), which mapsthe angular momentum and angle of the aphelion before a collision to theirvalues after the collision, admits a constant of motion. This follows from thefollowing geometric lemma about ellipses. Lemma 1:
Given an ellipse E with a focus at O that intersects L at a point P .Let Q denote the orthogonal projection of O onto L (see figure 2). The distance R between Q and the center of E depends solely on the semi-major axis a M , thedistance r from O to P , and cos(2 λ ) where λ is the angle between the tangentof the ellipse at P and L (to define the direction of the tangent, we parametrizethe ellipse in the counter-clockwise direction): R = (cid:114) r + 14 (2 a M − r ) + 12 r (2 a M − r ) cos(2 λ ) . (1.3)Proof: We switch to polar coordinates p = ( r cos ϕ, r sin ϕ ).3 PQh ϕ λ
Figure 2: The attractive center is O , hence it is the focus of the ellipse inabsence of centrifugal force g = 0. Q is the projection of O on the line L and P is a collision point. The arrow represents the velocity of the particle after thecollision.Let O (cid:48) denote the other focus of the ellipse, and C denote its center. Thefirst step is to compute the vector −−→ O (cid:48) P , which in polar coordinates is −−→ O (cid:48) P = ((2 a M − r ) cos ϕ (cid:48) , (2 a M − r ) sin ϕ (cid:48) ) (1.4)Let ψ := π + ϕ − λ denote the angle between the tangent of the ellipse at P andthe vector −−→ P O (see figure 3), and ψ (cid:48) := π + ϕ (cid:48) − λ denote the angle between thetangent of the ellipse at P and the vector −−→ P O (cid:48) . ϕ ϕ λψ ψO O PCQh
Figure 3: An ellipse with foci O and O (cid:48) and center C . The thick line is L , whichintersects the ellipse at P , and Q is the projection of O onto L . The dashedline is the tangent at P . λ is the angle between L and the tangent, ϕ is thepolar coordinate, ϕ (cid:48) is the angle between L and −−→ O (cid:48) P . ψ is the angle betweenthe tangent and −−→ P O , which is equal to the angle between the tangent and −−→
P O (cid:48) . R is the distance between Q and C .By the focus-to-focus reflection property of ellipses, we have ψ (cid:48) = π − ψ .Thus ϕ (cid:48) = 2 λ − π − ϕ and we find; 4 C QO' P t' t OO" C" Figure 4: Two ellipses, before and after a collision. The collision line L is theline at y = 1, P is the collision point; Q is the projection of O onto L ; the twoellipses E and E (cid:48) have a common focus O , and O, O (cid:48) are the foci of E , whereas O, O (cid:48)(cid:48) are the foci of E (cid:48) ; C and C (cid:48)(cid:48) are the centers of E and E (cid:48) respectively; theellipses are drawn completely although the trajectory is restricted to the partsabove y = h = 1. The distance from C (cid:48)(cid:48) to Q is the same as that from C to Q .The upper ellipse E contains the trajectory that starts at the collision point P following the other ellipse E (cid:48) which has undergone reflection. R = | Q − C | = 14 (cid:0) r + (2 a M − r ) + 2 r (2 a M − r ) cos(2 λ ) (cid:1) . (1.5)See figures 3 and 4. (cid:3) Theorem 1 : The quantity R = a + hαe sin θ ≡ α a M ( h + a M − R ) (1.6) where e is the eccentricity e = (cid:113) Aa α , is a constant of motion. Proof: During a collision, the value of λ changes from λ to π − λ , while r and a M stay the same. By lemma 1, this implies that the distance R between Q and the center of the ellipse is preserved during a collision. Furthermore, theposition of the center C of the ellipse is given by C = a M e (cos θ , sin θ ) so R = | Q − C | = a M e − a M eh sin θ + h . (1.7)Furthermore, the angular momentum is equal to a = a M α (1 − e ) so − R + h + a M = 2 a M α ( a + eαh sin θ ) (1.8)5s a conserved quantity. (cid:3) Remark:
Some useful inequalities are r max < a M ; x max = (cid:112) r max − h ; R ∈ (( a M − r ) , a M ); αh a M < R < (1 + a M h − (cid:16) a M h − rh (cid:17) ) αh a M (1.9)hence in the plane ( x, λ ) the rectangle ( − x max , x max ) × (0 , π ) (recall that x max is the largest x accessible at energy A ) is the surface of energy A and thetrajectories are the curves of constant R inside this rectangle. In the previous section, we exhibited a constant of motion, which, along with theconservation of energy, brings the number of independent conserved quantitiesto two. In a continuous Hamiltonian system, this would imply the existenceof action-angle variables, which are canonically conjugate to the position andmomentum of the particle, in terms of which the dynamics reduces to a linearevolution on a torus. In this case, the collision with the wall introduces somediscreteness into the problem, and the existence of the action angle variables isnot guaranteed by standard theorems. Indeed, in the presence of the collisions,we no longer have a Hamiltonian system, but rather a discrete symplectic map(or a non-differentiable Hamiltonian), which describes the change in the stateof the particle during a collision. In this section, we present some conjecturespertaining to the existence of action angle variables for this problem.The first step is to change to variables which are action-angle variables forthe motion in between collision. We choose the
Delaunay variables, whose anglesare the argument of the aphelion θ defined above, the mean anomaly M , andwhose actions are the angular momentum a , and another momentum usuallydenoted by L and related to the semi-major axis a M and to the energy E = A : L := − (cid:114) α a M , a M := − α A , A := p + a r − αr ≡ − α L (2.1)It is well known that this change of variables is canonical. In between collisions,the dynamics of the particle in the variables ( M, θ ; L, a ) is, simply,˙ M = α L , ˙ θ = 0 , ˙ L = 0 , ˙ a = 0 . (2.2)These variables are thus action-angle variables in between collisions, but whena collision occurs, θ and a will change.The following conjecture states that there exists an action-angle variableduring the collisions. 6 onjecture 1: There exists a variable γ and an integer k such that, every k collisions, the change in γ is γ (cid:48) = γ + ω ( L, R ) (2.3) in which case γ is an angle that rotates on a circle of radius depending on L, R .The function ω ( L, R ) has a non zero derivative with respect to R at constant L ,i.e. the motion on the energy surface is quasi periodic and anisochronous. We will now sketch a construction of this variable γ , which we obtain usinga generating function F ( L, R, M, θ ).First of all, by theorem 1, the angular momentum a ( θ ) is a solution of a = R − hα sin θ (cid:114) − a L (2.4)that is, if ε = ± , a = R − h α L sin θ + ε (cid:114) h α L sin θ + h α sin θ − Rα h L sin θ (2.5)and a = η √ a , so that there may be four possibilities for the value of a denoted a = a ε,η ( θ , R, L ) with ε = ± , η = ± . The choice of the signs ε = ±
1, and η must be examined carefully.We then define the generating function F ( L, R, M, θ ) = LM + (cid:90) θ a ( L, R, ψ ) dψ (2.6)which yields the following canonical transformation: γ = ∂ R (cid:90) θ a ε,η ( L, R, ψ ) dψM (cid:48) = M + ∂ L (cid:90) θ a ε,η ( L, R, ψ ) dψ (2.7)It is natural, if Boltzmann’s system is integrable (at g = 0), that the newvariables are its action angle variables and M (cid:48) , γ rotate uniformly in spite of thecollisions.However, in this case, the signs ε and η may change from one collision tothe next, complicating the situation. A careful numerical study of the systemhas led us to the following conjecture (see figure 5). Conjecture 2: If R > hα (which is the case in which the circle, of radius R ,of the centers encloses the focus O ), when the motion collides for the n -th time, .66609090.6660910 γ n Figure 5: A plot of the increment in γ between the n -th and the n + 2-ndcollision as a function of n . The blue ‘+’ signs correspond to even n , and thered ‘ × ’ to odd n . The variation of ∆ γ is as small as 1 part per million, thussupporting conjecture 2. the angular momentum is proportional to ( − n , and, thus, (cid:15) = ( − n . Thesign η is fixed to + . The increment ∆ γ in γ between the n -th and the n + 2 -thcollision is independent of n .Remark: The change of variables over the variables a, θ to R, γ at fixed L is remarkably essentially the same as the one ( a priori unrelated) to find action-angle variable for the auxiliary Hamiltonian R = R ( a, θ ). This might remaintrue even when R < hα : interpretable as a kind of auxiliary pendulum motion.At the time of publication, it has been brought to our attention that G.Felder has proved that the orbits are all either periodic or quasi-periodic, whichwould be implied from conjecture 1.
In this brief note, we have shown that the system considered by Boltzmann in1868, in the case g = 0, admits two independent constants of motion. Thisindicates that it should be possible to compute action angle variables for thissystem, which is not entirely trivial because of the discontinuous nature of thecollision process. If such a construction could be brought to its conclusion, thenit would show that the trajectories are either periodic or quasi-periodic, a factwhich is consistent with the numerical simulations we have run.This is not a contradiction of Boltzmann’s claim that this model is ergodic,since Boltzmann considered the model at g (cid:54) = 0. However, we expect that a8AM-type argument can be set up for this model, to show that the systemcannot be ergodic, even if g >
0, provided g is sufficiently small. However itmay still have invariant regions of positive volume where the motion is ergodic. Acknowledgements : The authors thank G. Felder for giving us the impetusto write this note up in its current form, and to publish it. I.J. gratefullyacknowledges support from NSF grants 31128155 and 1802170.
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