Investigating total collisions of the Newtonian N-body problem on shape space
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec Investigating total collisions of theNewtonian N-body problem on shape space
Paula Reichert ∗ December 10, 2020Forthcoming in:
Foundations of Physics . Abstract
We analyze the points of total collision of the Newtonian gravitational system on shapespace (the relational configuration space of the system). While the Newtonian equationsof motion, formulated with respect to absolute space and time, are singular at the pointof total collision due to the singularity of the Newton potential at that point, this neednot be the case on shape space where absolute scale doesn’t exist. We investigate whether,adopting a relational description of the system, the shape degrees of freedom, which aremerely angles and their conjugate momenta, can be evolved through the points of totalcollision. Unfortunately, this is not the case. Even without scale, the equations of motionare singular at the points of total collision (and only there). This follows from the specialbehavior of the shape momenta. While this behavior induces the singularity, it at the sametime provides a purely shape-dynamical description of total collisions. By help of this, weare able to discern total-collision solutions from non-collision solutions on shape space, thatis, without reference to (external) scale. We can further use the shape-dynamical descriptionto show that total-collision solutions form a set of measure zero among all solutions. ∗ Mathematisches Institut, Ludwig-Maximilians-Universität München. E-mail: [email protected].
Introduction
Is the Big Bang the beginning of Everything? Or did the universe evolve through the Big Bangwhich would then provide a single instant in the eternal evolution of the world? In what follows,we want to analyze the points of total collision of the Newtonian N -body system – the Big Bangof the Newtonian universe – on shape space, the relational configuration space of the system.It is well-known that, in the usual description with respect to absolute space and time, theNewtonian solutions end at or begin at the point of total collision. This is due to the singularity ofthe Newton potential at that point. Being inversely proportional to the inter-particle distances,the Newton potential diverges at the point at which the inter-particle distances go to zero. As aconsequence, also the Newtonian vector field and the Newtonian equations of motion are singularat that point. This is why the Newtonian solutions, as described with respect to absolute space,cannot be continued through the point of total collision.In what follows we want to investigate whether the Newtonian solutions can be evolvedthrough the points of total collision on shape space. On shape space, absolute scale is no longerpart of the description, only angles (i.e. shape degrees of freedom) remain. Do these shapedegrees of freedom evolve (uniquely) through the points of collision? Unfortunately, this is notthe case. We will show why, although there exists a unique description of the Newtonian systemon shape space, a description, that is, which is free of scale, there is still a singularity at thepoints of total collision. This has to do with the special behavior of the shape momenta atthat point. While this means that, even on shape space, solutions cannot be continued throughthe points of total collision, we, at least, obtain a purely shape-dynamical description of totalcollisions, which allows us to specify the points of collision without reference to scale.Throughout the paper, we consider the Newtonian model of zero total energy E = 0 , zerolinear momentum P = 0 and zero angular momentum L = 0 (where E , P and L can be fixedbecause they are conserved quantities of motion), the so-called E = P = L = 0 Newtonianuniverse. In order to be able to compute everything explicitly, we later restrict the discussionto the three-body model. A similar, though more abstract discussion, however, can be madefor the N-body problem. In particular, it can be shown that, also for N particles, the shapeHamiltonian is a sum of squares of the shape momenta, the inverse of which enters the shapevelocities, which, at the points of total collision on shape phase space, diverge. Hence, also forthe relational N -body problem, there will be a singularity at the points of total collision.The discussion is essentially based on a reformulation of the Newtonian dynamics on shapespace, the relational configuration space of the system, where absolute position, orientation andsize are no longer part of the description. Instead, the system’s evolution is fully determined bythe evolution of shapes: triangle shapes in the case of three particles, where each triangle shapeis specified by two angles. For the three-particle Newtonian gravitational system with E = P = L = 0 , a relational formulation with scale, but without absolute position and orientation hasbeen obtained by Montgomery [2002]. A fully relational (Hamiltonian) formulation of this systemon scale-invariant shape phase space has eventually been obtained by Barbour, Koslowski, andMercati ([2013], [2015]).It is interesting that a shape-dynamical formulation of the Newtonian dynamics exists al-though the system is not scale-invariant in the first place. We show how this works and how all1emnants of scale are hidden in the time-dependence of the Hamiltonian vector field on shapespace. While the shape vector field is still singular at the points of total collision, we, at least,detect a purely shape-dynamical description of total collisions. This allows us to discern total-collision solutions from non-collision solutions without reference to scale. We can also use thisshape-dynamical description of total collisions to show that the set of total-collision solutionsforms a set of measure zero among all solutions.In what follows, we first analyze the E = P = L = 0 Newtonian gravitational system in itsformulation with respect to absolute space and time. We will study both the long-time behaviorand the behavior at the points of total collision. This will constitute section 1. In section 2, wederive the Newtonian dynamics on shape phase space, the relational phase space of the system.In section 3, we study the points of total collisions on shape space. We show that, due to thespecial behavior of the shape momenta, the shape degrees of freedom cannot be evolved throughthe points of total collision. Still, from the behavior of the shape momenta we obtain a purelyshape-dynamical description of total collisions. We conclude with a discussion of the result.
Throughout the paper, the term ‘Newtonian universe’ refers to the model of N particles movingthrough infinite, three-dimensional Euclidean space and attracting each other according to theNewtonian gravitational force law. In what follows, we consider the E = P = L = 0 Newtonianuniverse, i.e., the Newtonian model of zero total energy, E = 0 , zero total linear momentum, P = 0 , and zero total angular momentum, L = 0 . It has been an early result of the analysis of the gravitational N -body system that the long-timebehavior of the Newtonian universe is governed by the Lagrange-Jacobi equation, ¨ I = 4 E − V N . (2.1)This equation connects the second time derivative of the moment of inertia I to the total energy E and the potential energy V N of the system. The moment of inertia in the center-of-massframe (which we can use without loss of generality because the system is invariant under totaltranslations) is of a simple form, I = N X i =1 m i q i . (2.2)Note that, by definition, the moment of inertia I is an intrinsic measure of the size of the system.The Newton potential is defined to be V N = − N X i = j ; i,j =1 Gm i m j | q i − q j | (2.3)and E = T + V N is the total energy, with T = P Ni =1 p i / m i being the kinetic energy of thesystem. 2ince we consider the model of zero total energy, E = 0 , and since the potential energy isstrictly negative, V N < , the Lagrange-Jacobi equation tells us that the second time derivativeof I is strictly positive: ¨ I > . (2.4)This condition implies that the I -curve, if the solution exists for all times, is concave upwardswith a minimum I min at some moment t and with I increasing towards infinity in both timedirections away from that point, I → ∞ as t → ±∞ . In the case of a total collision, where theNewtonian solution ends at some point (the point of total collision) due to the singularity of thevector field at that point, the Lagrange-Jacobi equation implies that I min = 0 at t (expressingthe fact that the particles collide at some moment) while the I -curve is concave and goes toinfinity as t → ∞ or t → −∞ (which is the same since the system is time-symmetric). Let fromnow on, without loss of generality, t = 0 .Since the moment of inertia I is an intrinsic measure of the size of the system, we learn fromthe Lagrange-Jacobi equation that, in an eternal evolution, at one moment, the particles areclosest while they become spatially more and more separated in both time directions away fromthat point. If a Big Bang is part of such an eternal evolution – assume, for one moment, thatthe shape degrees of freedom can be evolved through the singularity of a total collision –, it hasto occur at the mid-point of the evolution, when the spacing between the particles is minimal, I = I min , and where, in the case of a total collision (a Big Bang of the Newtonian universe), I min = 0 . Only at that point, when I = I min , the Big Bang can be part of an eternal evolution(if at all) of the E = 0 Newtonian universe.From the Lagrange-Jacobi equation we further learn that, for the E = 0 Newtonian uni-verse, there exists a quantity which is strictly monotonic along the trajectories: the ‘dilationalmomentum’ D = N X i =1 q i · p i . (2.5)Expressed with respect to I = P i m i q i , we find that it is essentially its first time derivative, D = 1 / I . (2.6)Since ¨ I > , it follows that ˙ I is monotonically increasing along all trajectories. Hence, also D = 1 / I is monotonically increasing. Thus we can use D in order to specify the minimum ofthe I -curve. Since D ∼ ˙ I , it follows that I = I min if and only if D = 0 . (2.7)Given that D is monotonic, we can use it to parametrize the trajectories. Once we do that, wecall it ‘internal time’ and write τ = D . In fact, it will be this internal time parametrization whichallows us to reformulate the Newtonian dynamics, which is not invariant under scalings/dilations,on scale-invariant shape space (cf. Barbour et al. [2013]).Let me add one remark. The time-asymmetric processes of overall contraction and expansionof the system defines two gravitational arrows of time. Following Barbour et al. ([2013], [2015])we call ‘past’ the mid-point of the evolution when the particles are closest, τ = 0 , while we say3here are two ‘futures’ in both time directions away from it, as τ → ±∞ . If there is a totalcollision at the mid-point of the evolution (which Barbour et al. also call Janus point), we saythere is a Big Bang in the common past of the Newtonian universe. In order to study whether the shape degrees of freedom can be evolved through the Newtoniansingularity, we need to analyze the Newtonian dynamics at the points of total collision.In what follows, I introduce the standard definition of a total collision. We say that a totalcollision occurs if and only if the moment of inertia vanishes: I = 0 . (2.8)Total collisions have been discussed in the context of the three-body problem early in themathematical literature. The existence of solutions ending at, respectively starting at a totalcollision has been shown by Lagrange and Euler. Lagrange showed that, if there are threeparticles of equal masses forming an equilateral triangle and they are released with zero initialvelocity, they will collide. We will call this particular spatial configuration plus its reflectedversion (the reflected equilateral triangle) the two ‘Lagrange configurations’. Euler, in turn,showed that, if there are three particles of equal masses aligned with one particle centered betweenthe other two and they are released with zero initial velocity, they will collide. The respectiveconfigurations are called the ‘Euler configurations’. There are three Euler configurations, one foreach possibility to center one particle in-between the other two.Today, some general properties of total collisions are known. Sundman [1909] has shownthat, in order for a total collision to occur, the total angular momentum needs to be zero, L = 0 (if L = 0 , I is bounded away from zero by some positive constant: I ≥ I with I > ). LaterSaari [1984] has shown that, as the particles approach a total collision at time t = 0 , they forma central configuration and their position vectors q i ( t ) behave as t / . Consistently, Moeckel([1981], [2007]) identifies the central configurations as the ‘rest points’ of the collision manifold.This specifies the points of total collision on shape space. For three particles, there exist fivepoints of central configuration on shape space: the three Euler and two Lagrange points. Theseare the only points on shape space at which a total collision may occur.To gain some intuition about total collisions due the attractive gravitational force, let ussketch the asymptotic behavior of the particles as they approach a total collision. Saari [1984]shows that, as the particles approach a total collision at t = 0 , their position vectors behave as q i ( t ) = a i t α for some α , where a i is a vector constant. To be precise, he shows that there existpositive vector constants A i , B i such that, for sufficiently small t (in particular, for t → ), A i t α ≤ q i ( t ) ≤ B i t α . (2.9)From this result it follows that, in the limit t → , the colliding particles form a central con-figuration (see below). Moreover, we can even specify the α : in the limit t → , it is α = 2 / . Cf. Moeckel ([1981], [2007]) for a historical introduction. The following paragraphs follow Saari’s discussion of the matter in his paper on central configurations. q i ( t ) = a i t / . (2.10)To see this, reconsider the Newtonian gravitational potential (2.3): V N = − X i = j Gm i m j | q i − q j | . This potential forms part of the Newtonian law of gravitation which determines the acceleration ¨ q i of the i ’th particle (with mass m i ) as follows: m i ¨ q i = ∂V N ∂ q i = − X i = j Gm i m j ( q i − q j ) | q i − q j | . (2.11)This is a complicated equation, but sometimes it attains a simpler form. This is the case forcentral configurations. A configuration is called a ‘central configuration’ if, at some moment, thecenter-of-mass position vector q i of each particle is in line with its acceleration vector ¨ q i . Thatis, if and only if ∀ i = 1 , ..., N : λ q i = ¨ q i , (2.12)respectively, with (2.10): λ q i = 1 m i ∂V N ∂ q i . (2.13)Here λ = λ ( t ) is some common scalar factor of proportionality. Hence, if the particles form acentral configuration, the system mimics a central force problem.Using Saari’s result, namely that, in the limit t → , the position vectors behave as q i ( t ) = a i t α for some α , the Newtonian force law (2.10) turns into a i α ( α − t α − = − X i = j Gm j ( a i − a j ) | a i − a j | t − α . (2.14)This can be fulfilled if and only if α = 2 / . Consequently, q i ( t ) = a i t / and the system forms acentral configuration with λ ( t ) = α ( α − t − = − / t − , where the functional form of λ followsfrom differentiating q i ( t ) = a i t α twice with respect to t : ¨ q i = a i α ( α − t α − = λ ( t ) a i t α = λ ( t ) q i . (2.15)That is, in the limit t → (with a total collision at t = 0 ) the particles form a central configu-ration and ∀ i = 1 , ..., N there exist vector constants a i such that q i ( t ) = a i t / . From the long-time behavior of the E = 0 Newtonian universe we learned that, within an eternalevolution, a total collision can occur only at the mid-point t = 0 of the evolution, when the5pacing between the particles is minimal: I = I min . Of course, I min = 0 in the special case of atotal collision. We also learned that the mid-point t of the evolution is determined by D = 0 (since D ∝ ˙ I and ˙ I = 0 determines the minimum of the I -curve).From the behavior near total collisions we learned that a total collision can occur only if theparticles form a central configuration. For a system of three particles, there exist five centralconfigurations: two Lagrange and three Euler configurations. Within a relational description,these configurations are represented by five points on shape space: the two Lagrange and threeEuler points.In what follows, starting from the standard Hamiltonian description on absolute phase space,we will derive the formulation of the dynamics of the three-particle E = P = L = 0 Newtonianuniverse on the relational or shape phase space. We will show that, while on absolute phase spacethe vector field diverges at the point of zero spatial extension ( I min = 0 ) due to the singularityof the Newton potential, the shape vector field is non-singular at the Euler and Lagrange pointsat D = 0 ( I = I min ) for all but a measure-zero set of solutions. Unfortunately, this measure-zeroset of solutions is precisely the set of solutions for which a total collision occurs.While total collisions cannot be passed, not even on shape space, we obtain from the relationalanalysis a purely shape-dynamical description of total collisions. That is, we will be able tospecify the points of total collision not by reference to scale ( I = 0 ), but merely in terms ofshape degrees of freedom. If we consider a system with symmetries, like the E = P = L = 0 Newtonian universe whichis symmetric with respect to total translations and rotations, the dynamics doesn’t have to beformulated on ordinary phase space Γ . Instead, there exists a unique description of the systemand its dynamics on a lower-dimensional space, the ‘reduced phase space’ Γ .Mathematically, Γ and the dynamics on Γ are constructed from Γ and the dynamics on Γ by a method called symplectic reduction. For this paper, it suffices to know that the reducedphase space and the reduced Hamiltonian equations of motion are obtained by fixing both theconserved quantities of motion (here P and L ) and the connected gauge degrees of freedom (hereabsolute position and orientation) related to the symmetries of the system. Γ = T ∗ S R and shape phase space T ∗ S Within the Newtonian universe, it follows from the conservation of total linear momentum andtotal angular momentum, { P , H } = 0 and { L , H } = 0 where H = T + V N is the Hamiltonian ofthe system, P = P i p i total linear momentum and L = P i q i × p i total angular momentum, thatthe dynamics is invariant under spatial translations and rotations. This follows from Noether’stheorem. In the given case, where P = L = 0 , the dynamics is invariant under the full six-dimensional Euclidean group E (3) = R × SO (3) of spatial translations and rotations, where R See, for instance, Arnol’d [1989]. For the mathematical details, see also Marsden and Weinstein [1974] orIwai [1987]. See Montgomery [2002] and Barbour, Koslowski, and Mercati ([2013], [2015]) for the reduction of theNewtonian gravitational system. SO (3) the group of rotations. One can now construct the reduced phase space Γ (or at least one representative of it) fromphase space Γ ∼ = R N by setting P = P p i = 0 and L = P q i × p i = 0 (thereby fixing thesix conserved quantities of motion) and specifying the position and orientation of the system(thereby fixing the gauge degrees of freedom related to translational and rotational symmetry).To do the latter, we fix the center of mass to the origin, Q cm = P Ni =1 m i q i = 0 , and the three off-diagonal components I ij of the (symmetric) center-of-mass inertia tensor to the coordinate axes: I ij = [ P k =1 m k ( q k · q k I − q k ⊗ q k )] ij = 0 (with i < j ; i, j = 1 , , ). Let, to simplify notation, I L := ( I , I , I ) . That is, we set I L = 0 . Inserting these conditions into the equations ofmotion, we obtain the reduced Hamiltonian equations, that is, the equations of motion on Γ .Since we have twelve constraints, Γ is a space of N − dimensions.An equivalent way to obtain Γ is by first constructing the reduced configuration space S R –what we call ‘shape space with scale’ – and then determining its cotangent bundle T ∗ S R . Justlike phase space is the cotangent bundle of configuration space, Γ = T ∗ Q , reduced phase spaceis the cotangent bundle of reduced configuration space, Γ = T ∗ S R .The reduced configuration space S R is obtained from ordinary configuration space Q ∼ = R N by factoring out translations and rotations. To be precise, S R is the Riemannian quotient of Q with respect to the Euclidean group E (3) = R × SO (3) , S R = Q R × SO (3) . (3.1)We call S R shape space with scale in order to emphasize that what is left in the description ofthe system is shapes, that is, angles (or relative distances, respectively) and scale, that is, onequantity measuring the size of the system.For a system of three particles, S R is the ‘space of triangles’. Every point on S R determines adistinct triangle, specified by two angles ψ and φ , determining the shape of the triangle, and onescale factor R , specifying its size. Geometrically, S R is a cone over the two-sphere S (which canbe visualized as a collection of two-spheres around the origin with radius R ). Local coordinatesof S R are, e.g., spherical coordinates ψ , φ , and R .If we further quotient by dilations/scalings, we end up with ‘shape space’ S . Shape space S is the quotient of configuration space Q with respect to the seven-dimensional similarity group Sim (3) = R × SO (3) × R + , that is, S = Q R × SO (3) × R + . (3.2)We call S shape space (or shape sphere, since, geometrically, it is a two-sphere) in order toemphasize that what is left in the description of the system is shapes, that is, angles. Note thatwe have not quotiented by reflections. This is why S is the entire two-sphere S and not onlythe upper half-sphere (the lower half-sphere consists of the reflected triangle shapes). Note that only if P = L = 0 , the system is invariant under the full Euclidean group. If, e.g., L = 0 , the systemis only invariant with respect to one-dimensional rotations (around the axis pointing into the direction of L ) and,thus, only with respect to a subgroup of E (3) . Also only in the given case, if P = L = 0 , the reduced phase space Γ is isomorphic to (and, hence, can be identified with) the cotangent bundle of the reduced configuration space T ∗ S R which we construct below. For details, see the references in the preceding paragraph. S is the space of triangle shapes. Every point on S determines a distincttriangle shape, specified by two angles ψ and φ . Geometrically, S is represented by a unit two-sphere around the origin. Local coordinates of S are, e.g., polar coordinates ψ and φ . Just like T ∗ S R is the phase space related to shape space with scale, the cotangent bundle of shape space T ∗ S is the ‘shape phase space’. T ∗ S R Since the dynamics is invariant with respect to translations and rotations, we can formulate it onthe reduced phase space Γ = T ∗ S R . To obtain the reduced Hamiltonian equations of motion,we have to separate the absolute from the relational degrees of freedom. This is achieved by twoconvenient coordinate transformations. To formulate the dynamics on Γ , we need a canonicalset of local coordinates of T ∗ S R . In other words, we need a set of translationally and rotationallyinvariant coordinates plus their canonical conjugates.Let, from now on, N = 3 . In that case, Q ∼ = R and Γ = T ∗ Q ∼ = R and the componentsof the positions q i ∈ R ( i = 1 , , ) and momenta p i ∈ R ( i = 1 , , ) form a canonical set oflocal coordinates of Γ . In what follows, we first construct the translationally invariant Jacobicoordinates ρ i ∈ R ( i = 1 , ) from which, in turn, we obtain the translationally and rotationallyinvariant Hopf coordinates w = ( w , w , w ) . The three Hopf coordinates are local coordinatesof S R . Together with their canonical conjugates z = ( z , z , z ) they form a canonical set of localcoordinates of T ∗ S R .Starting from the canonical coordinates q i ∈ R ( i = 1 , , ) and p i ∈ R ( i = 1 , , ) andsetting Q cm = 0 and P = 0 , we obtain the six translationally invariant Jacobi coordinates: ρ = r m m m + m ( q − q ) , ρ = s m ( m + m ) m + m + m (cid:18) q − m q + m q m + m (cid:19) . (3.3)Their conjugate momenta are: κ = m p − m p p m m ( m + m ) , κ = s ( m + m )( m + m + m ) m (cid:18) p − m p + m p m + m (cid:19) . (3.4)From these, setting L = 0 and I L = 0 , we obtain the three rotationally invariant Hopfcoordinates (which have originally been proposed by Hopf in his discovery of the Hopf fibration,see Montgomery [2002]): w = | ρ | − | ρ | , w = ρ · ρ , w = ρ × ρ . (3.5)8heir canonical conjugates are: z = ρ · κ − ρ · κ | ρ | + | ρ | , z = ρ · κ + ρ · κ | ρ | + | ρ | , z = ρ × κ − ρ × κ | ρ | + | ρ | . (3.6)Together, the w , w , w and z , z , z form a canonical set of local coordinates of the reducedphase space T ∗ S R . On that space, the reduced Hamiltonian dynamics of the E = P = L = 0 Newtonian universe is formulated.Geometrically, each vector w = ( w , w , w ) determines a point on a two-sphere S || w || (0) around the origin where || w || = 1 / I. (3.7)Here I = P i m i q i is the moment of inertia of the three-particle system. Hence, while the radiusof the two-sphere determines the size of the triangle, the position on the sphere (specified bythe two angles ψ and φ ) determines its shape. Let the Hopf coordinates w , w , and w pointinto the x , y , and z direction, respectively. In that case, the collinear configurations lie on theequator while the equilateral triangle and its reflected version lie at the top and bottom of theshape sphere. T ∗ S R To study total collisions, we need to analyze the dynamics at the points of central configurationon shape space (the only points at which a total collision might occur). Let us specify thefive points of central configuration with respect to the Hopf coordinates. Let us, for means ofsimplicity, consider the equal-mass case: m = m = m = m . In that case,• the two Lagrange configurations (the equilateral triangle and its reflected version) arespecified by w = w = 0 , w = ±|| w || (3.8)• and the three Euler configurations (the three possible collinear configurations where oneparticle is centered in between the other two) are specified by w = 0 (3.9)and ( w , w ) ∈ (cid:26)(cid:18) || w || , (cid:19) , (cid:18) − || w || , √ || w || (cid:19) , (cid:18) − || w || , − √ || w || (cid:19)(cid:27) . (3.10)This is obtained directly from expressing the Lagrange and Euler configurations with respect tothe Jacobi and Hopf coordinates defined above.We see that the five central configurations determine five points on the two-sphere S || w || (where || w || = 1 / I ). Let, again, the Hopf coordinates w , w and w point into the x , y , and z direction, respectively. In that case, the two Lagrange points lie at the top and bottom of thesphere, while the three Euler points lie on the equator at equal distance from each other.9 .4 Reduced Hamiltonian dynamics on T ∗ S R The reduced Hamiltonian formulation of the E = P = L = 0 Newtonian universe is due toBarbour et al. [2013]. The reformulation of the Newton potential V N with respect to the Hopfcoordinates has been obtained before by Montgomery [2002].With respect to the Hopf coordinates w and their canonical conjugates z the reduced Hamil-tonian H on T ∗ S R can be expressed as follows (cf. Barbour et al. [2013]): H = T + V N = || w || · || z || + V S p || w || , (3.11)where V S = V S ( w ) . In accordance with Barbour et al., we call V S the ‘shape potential’. Explic-itly, the shape potential is of the following form (cf. Montgomery [2002]): V S = −√ X i
In order to discuss the Euler configurations, let us choose sphericalcoordinates
R, ψ, φ such that w = R sin ψ cos φ, w = R sin ψ sin φ, w = R cos ψ. (4.1)Note that, by definition of the spherical coordinates, R = || w || . (4.2)Since || w || = 1 / I according to (3.7), it follows that the radius of the shape sphere R is a measure of the size of the system. Now ψ = π/ specifies the equator of the shapesphere. Consequently, the Euler configurations lie on the equator of the shape sphere –since w = 0 from (3.9) holds if and only if ψ = π/ – while the Lagrange configurationslie at the top and bottom of the sphere – since w = w = 0 from (3.8) holds if and only if ψ = 0 or ψ = π .To be precise, the three Euler points are specified by ( ψ E , φ E ) ∈ (cid:26)(cid:18) π , (cid:19) , (cid:18) π , π (cid:19) , (cid:18) π , π (cid:19)(cid:27) . (4.3)This follows directly from (3.9) and (3.10) and the definition of ψ and φ from above.Connected to R, ψ, φ , there exist canonical conjugates p R , p ψ , p φ defined as follows: z = 1 R (cos φ ( Rp R sin ψ − p ψ cos ψ ) − p φ sin − ψ sin φ ) z = 1 R (sin φ ( Rp R sin ψ − p ψ cos ψ ) + p φ sin − ψ cos φ ) z = − p R cos ψ − R p ψ sin ψ. (4.4)Together, R, ψ, φ and p R , p ψ , p φ form a canonical basis of T ∗ S R .• Lagrange configurations.
In order to discuss the Lagrange configurations, we choose13pherical coordinates R ′ , ψ ′ , φ ′ (with R ′ = R ) such that w = R ′ cos ψ ′ , w = R ′ sin ψ ′ cos φ ′ , w = R ′ sin ψ ′ sin φ ′ . (4.5)Note that this new choice of coordinates represents a permutation of the w , w , and w (a rotation of the axes such that w becomes w and so on). Now ψ ′ = π/ denotes theequator of the shape sphere. It follows that now the two Lagrange configurations lie onthe equator of the shape sphere, since w = 0 (3.8) holds if and only if ψ ′ = π/ . Now theEuler configurations lie on a meridian, the intersection of the shape sphere and the ‘new’ w = 0 plane.To be precise, the two Lagrange points are specified by ( ψ ′ L , φ ′ L ) ∈ (cid:26)(cid:18) π , π (cid:19) , (cid:18) π , π (cid:19)(cid:27) . (4.6)This follows directly from (3.8) and the definition of ψ ′ and φ ′ from above.Again, connected to R ′ , ψ ′ , φ ′ , there exist canonical conjugates p ′ R , p ′ ψ , p ′ φ defined as follows: z = − p ′ R cos ψ ′ − R ′ p ′ ψ sin ψ ′ z = 1 R ′ (cos φ ′ ( R ′ p ′ R sin ψ ′ − p ′ ψ cos ψ ′ ) − p ′ φ sin − ψ ′ sin φ ′ ) z = 1 R ′ (sin φ ′ ( R ′ p ′ R sin ψ ′ − p ′ ψ cos ψ ′ ) + p ′ φ sin − ψ ′ cos φ ′ ) . (4.7)Also R ′ , ψ ′ , φ ′ and p ′ R , p ′ ψ , p ′ φ form a canonical basis of T ∗ S R .Unfortunately, we need two different coordinate system ( R, ψ, φ and R ′ , ψ ′ , φ ′ , respectively) totreat each the Euler and Lagrange configurations. This is due to the fact that the physical vectorfield on T ∗ S is singular at the top and bottom of the shape sphere (due to a coordinate-singularityat that point, see Appendix A) and, once we put the Euler or the Lagrange configurations on theequator, at least one of the other central configurations lies at the top or bottom of the shapesphere (see the equations (3.8)–(3.10) which determine the Euler and Lagrange points). T ∗ S R and T ∗ S in terms of the new coordinates Since the kinetic term T of the Hamiltonian is invariant under permutations of the w , w , w and z , z , z coordinates, we can write down the physical vector field on T ∗ S R (respectively,the equations of motion) in such a way that it does not distinguish between the two differentchoices of spherical coordinates (which represent precisely two different permutations of the w and z coordinates). This is possible as long as we don’t write down the explicit form of the shapepotential V S (respectively V ′ S ) which is not invariant under the given permutations. Of course, we might come up with a more complicated definition of spherical coordinates, which allows totreat all the central configurations at once. Given that the vector field is non-singular everywhere except at thetop and bottom of the shape sphere, we would need a spherical coordinate system which is tilted with respect tothe Hopfian system. But then we wouldn’t be able to solve the equations of motion analytically, we wouldn’t evenbe able to write down the shape potential in a nice form, that’s why we better keep the two coordinate systemswe have introduced above. H from (3.11) with respect to the unprimed spherical coordinates specified in(4.1) and (4.4), we find that H = p ψ + sin − ψp φ + R p R R + V S ( ψ, φ ) √ R . (4.8)Analogously, with respect to the primed spherical coordinates from (4.5) and (4.7), H can bewritten as H = p ψ ′ + sin − ψ ′ p φ ′ + ( R ′ ) ( p ′ R ) R ′ + V ′ S ( ψ ′ , φ ′ ) √ R ′ (4.9)with V ′ S ( ψ ′ , φ ′ ) = V S ( ψ, φ ) . (The explicit expression of V S ( ψ, φ ) and V ′ S ( ψ ′ , φ ′ ) is given in Ap-pendix B).The respective Hamiltonian equations of motion on T ∗ S R are d ψ d t = 2 p ψ R , d p ψ d t = 2 sin − ψ cos ψp φ R − ∂V S /∂ψ √ R , d φ d t = 2 sin − ψp φ R , d p φ d t = − ∂V S /∂φ √ R , d R d t = 2 R · p R , d p R d t = p ψ + sin − ψp φ − R p R R + 12 V S ( ψ, φ ) R / . (4.10)and, analogously, for the primed coordinates. Only the functional form of the shape potentialdepends on the choice of coordinates, i.e., V ′ S ( ψ ′ , φ ′ ) = V S ( ψ, φ ) .Note that, of course, since scale is still part of the description, the equations of motiondiverge at R = 1 / I = 0 (the point of total collision) representing the singularity of the Newtonpotential at that point. Since the physical vector field is singular at R = 0 , the solutions cannotbe continued through the point of total collision.However, now that we have separated scale and shape degrees of freedom, we can write downthe equations of motion on scale-invariant T ∗ S solely in terms of the shape variables (and internaltime τ = D ). Recall that the equations on T ∗ S are generated by the internal Hamiltonian H from (3.17), the canonical conjugate of τ = D . Recall also that H is expressed on T ∗ S (the H = D = 0 hypersurface) in terms of the internal, i.e. shape variables by help of the constantenergy constraint H = 0 .Expressed with respect to the spherical coordinates (4.1), H = log p / I = log √ R, (4.11)where we used that R = || w || = 1 / I . This becomes a function of the shape coordinates giventhat H = 0 , that is, √ R (cid:12)(cid:12) H =0 = p R ( ψ, φ, p ψ , p φ ) . With H from (4.8), the constant energycondition reads H = p ψ + sin − ψp φ + R p R R + V S ( ψ, φ ) √ R = 0 . (4.12)Solving this equation for R and using that, when expressed with respect to the spherical co-ordinates (4.1) and (4.4), the dilational momentum D from (3.15), namely D = 2 w · z , turns15nto D = 2 R · p R , (4.13)we obtain: √ R (cid:12)(cid:12) H =0 = p ψ + sin − ψp φ + 1 / τ − V S ( ψ, φ ) . (4.14)Here we have set τ = D . Inserting this into (4.11), we obtain the internal Hamiltonian H governing the motion on T ∗ S with respect to internal time τ : H = log p ψ + sin − ψp φ + 1 / τ − V S ( ψ, φ ) (4.15)Note that this Hamiltonian is time-dependent ( τ -dependent). This is the last remnant of scale,incorporating the scale-dependence of the system, in the otherwise scale-free description on shapephase space T ∗ S .We can now write down the Hamiltonian equations of motion on T ∗ S , d ψ/ d τ = ∂ H /∂p ψ , d p ψ / d τ = − ∂ H /∂ψ , and so on: d ψ d τ = 2 p ψ p ψ + sin − ψp φ + τ , d p ψ d τ = 2 sin − ψ cos ψp φ p ψ + sin − ψp φ + τ + ∂ log( − V S ) ∂ψ , d φ d τ = 2 sin − ψp φ p ψ + sin − ψp φ + τ , d p φ d τ = ∂ log( − V S ) ∂φ . (4.16)Again, for the primed variables, the equations of motion are analogous (only that ψ = ψ ′ , φ = φ ′ , p ψ = p ′ ψ , p φ = p ′ φ , and V S = V ′ S ). In what follows, we want to derive the equations of motion on shape phase space T ∗ S at theJanus point ( τ = 0 ) at the central configurations.Remember that we chose spherical coordinates such that, for each choice, the central config-urations under consideration lie on the equator of the shape sphere. In other words, it suffices toanalyze the vector field (4.16) at the equator, that is, at ψ = π/ ( ψ ′ = π/ ). Since sin π/ and cos π/ , we obtain the equations d ψ d τ = 2 p ψ p ψ + p φ + τ , d p ψ d τ = (cid:20) ∂ log( − V S ) ∂ψ (cid:21) ψ = π , d φ d τ = 2 p φ p ψ + p φ + τ , d p φ d τ = (cid:20) ∂ log( − V S ) ∂φ (cid:21) ψ = π (4.17)and, analogously, for the primed coordinates.To evaluate this vector field at the central configurations ( ψ E , φ E ) and ( ψ ′ L , φ ′ L ) , it re-mains to determine the partial derivatives ∂ log( − V S ) /∂ψ and ∂ log( − V S ) /∂φ (respectively, ∂ log( − V ′ S ) /∂ψ ′ and ∂ log( − V ′ S ) /∂φ ′ ) at these points. Since, for the two different choices ofspherical coordinates, the functional form of the shape potential differs, we have to analyze eachthe Euler and Lagrange configurations separately. The respective computation of the partial16erivatives of the logarithm of the shape potential is given in Appendix B.In Appendix B, it is shown that all partial derivatives vanish identically (see (5.3) – (5.6)).Consequently, for the Euler configurations ( ψ E , φ E ) , (cid:20) ∂ log( − V S ) ∂ψ (cid:21) ( ψ E ,φ E ) = 0 , (cid:20) ∂ log( − V S ) ∂φ (cid:21) ( ψ E ,φ E ) = 0 . (4.18)and for the Lagrange configurations ( ψ ′ L , φ ′ L ) , (cid:20) ∂ log( − V ′ S ) ∂ψ ′ (cid:21) ( ψ ′ L ,φ ′ L ) = 0 , (cid:20) ∂ log( − V ′ S ) ∂φ ′ (cid:21) ( ψ ′ L ,φ ′ L ) = 0 . (4.19)The fact that, at the Euler and Lagrange points, all partial derivatives of log( − V S ) vanishtells us that the central configurations are stationary points of the (negative) shape potential.This follows from the fact that ∂ log( − V S ) /∂x = V − S ∂V S /∂x and < V S ( ψ E , φ E ) < ∞ ( E/L , p ψ (0) , p φ (0)) . Among these mid-point data, those with shapemomenta p ψ (0) = p φ (0) = 0 form a set of measure zero (see section 4.4). Being stationary points, each of the central configurations is either a minimum or a maximum or a saddle pointof the shape potential. If we look at the second derivatives or, more precisely, at the eigenvalues of the Hessianmatrix with entries ∂ V S /∂ ψ , ∂ V S /∂ψ∂φ , ∂ V S /∂φ∂ψ and ∂ V S /∂ φ , we find that the Lagrange configurationsare maxima, while the Euler configurations are saddle points of the shape potential. p ψ (0) = 0 and p φ (0) = 0 (where it suffices that oneof the shape momenta is non-zero) determine solutions of the form that the system passes acentral configuration at the Janus point, τ = 0 , with some non-zero minimal moment of inertia, I min = 0 . To see that the set of solutions with p ψ (0) = p φ (0) = 0 forms a set of measure zero among allsolutions, consider the canonical volume measure (the reduced Liouville measure) µ on T ∗ S , µ = d ψ d φ d p ψ d p φ . (4.22)This measure is derived in Appendix C. It is obtained from the original Liouville measure Q Ni =1 d q i d p i on Γ by fixing the momentum constraints and factoring out the gauge volume.It is stationary, that is, conserved under the internal Hamiltonian time evolution. Hence, it isthe appropriate measure for the statistical analysis of the system.Let, in what follows, Γ E,L ⊂ T ∗ S denote the set of solutions which, at τ = 0 , pass a centralconfiguration, Γ E,L = { ( ψ, φ, p ψ , p φ ) ∈ T ∗ S| ( ψ, φ ) = ( ψ E/L , φ E/L ) } , and let Γ S ⊂ Γ E,L further denote the subset of points where the vector field (4.21) is singular,i.e., Γ S = { ( ψ, φ, p ψ , p φ ) ∈ T ∗ S| ( ψ, φ ) = ( ψ E/L , φ E/L ) , p ψ = p φ = 0 } . In what follows, let us determine the conditional measure µ (Γ S | Γ E,L ) of the set of solutionswhich, at τ = 0 , pass a central configuration with p ψ = p φ = 0 . According to the measure µ on T ∗ S , µ (Γ S | Γ E,L ) = µ (Γ S ) µ (Γ E,L ) = R T ∗ S δ ( ψ − ψ E/L ) δ ( φ − φ E,L ) δ ( p ψ ) δ ( p φ ) dψdφdp ψ dp φ R T ∗ S δ ( ψ − ψ E/L ) δ ( φ − φ E,L )d ψ d φ d p ψ d p φ = 1 R R d p ψ d p φ = 0 . (4.23)We have just shown that, among all solutions which pass a central configuration at τ = 0 ,those with p ψ = p φ = 0 (which are precisely those which pass a total collision, see section 4.5)form a set of measure zero. Even more, since Γ E/L ⊂ T ∗ S (and T ∗ S ⊂ Γ ), they form a set ofmeasure zero on all of T ∗ S (and Γ ), that is, among all solutions. In order to see why, at the moment of total collision, the shape momenta are necessarily zero,we have to go back to the constant-energy condition (4.12). Using the relation τ = 2 R · p R (see(4.13)), (4.12) can be rewritten as H = p ψ + sin − ψp φ + 1 / τ R + V S ( ψ, φ ) √ R = 0 . 18f we multiply this constraint with R , we obtain p ψ + sin − ψp φ + 1 / τ + V S ( ψ, φ ) √ R = 0 . (4.24)Now study this constraint in the limit R → and ψ → ψ E,L , φ → φ E,L (the limit of atotal collision). We know that, at the central configurations, ψ E/L = π/ (hence, sin ψ E/L = 1 ).Moreover, | V S ( ψ E/L , φ E/L ) | < ∞ (to see this, insert ( ψ E/L , φ E/L ) into (5.1), respectively (5.2),or note that the shape potential diverges only at the points of binary collision). It follows that,in the limit ψ → ψ E,L , φ → φ E,L , the constraint equation turns into p ψ + p φ + 1 / τ + V S ( ψ E/L , φ E/L ) √ R = 0 , (4.25)where | V S ( ψ E/L , φ E/L ) | < ∞ .In the limit R → , (4.25) holds if and only if, in that limit, p ψ , p φ and τ converge to zero. Inother words, in case a total collision occurs, necessarily τ = 0 (which we knew) and p ψ = p φ = 0 .The other way round, if τ = 0 and p ψ = p φ = 0 , it follows from (4.25) that the particles have tocollide, R = 0 .Note that we get this result almost alone from the limit R → , that is, without demandingthat the solutions converge to a central configuration. All we need is that | V S ( ψ, φ ) | < ∞ (respectively, | V ′ S ( ψ ′ , φ ′ ) | < ∞ ). If the shape potential is bounded, (4.24) holds in the limit R → if and only if, in that limit, τ → and p ψ → , p φ → . We have thus obtained a purely shape-dynamical description of total collisions. Recall thatin (2.8) we defined a total collision via the vanishing moment of inertia: I = 2 R = 0 . Now we are able to define it merely in terms of τ and the shape variables, that is, withoutreference to external scale. According to this new definition, a total collision occurs if and onlyif, at the moment of minimal extension, τ = 0 , the particles form a central configuration and allshape momenta are zero, that is, if and only if, at τ = 0 , ( ψ, φ ) = ( ψ E/L , φ E/L ) and p ψ = p φ = 0 . (4.26)Importantly, this definition is free of external scale. It is a definition in purely relational, i.e.shape-dynamical, terms. To obtain this stronger result, we would have to deal with the binary collision points ( ψ ij , φ ij ) , the (only)points where the shape potential diverges (inspect (5.1) and (5.2)), separately. Unfortunately, we cannot excludethe binary-collision points a priori . Assume, for instance, that two particles collide just as fast as R → , whilethe third particle collides with the two of them only as fast as √ R → . In that case, a total collision occursprecisely at a binary collision point. Then − V S ∼ / √ R and the last term in (4.24) would become a constant inthe limit R → . Since that term is negative ( V S is negative), it could be encountered by a positive term, a sumof squares of non-zero p ψ , p φ and τ . In that case, we would no longer be able to conclude that p ψ = p φ = τ = 0 in the limit R → . Hence, to obtain the stronger result, namely that R = 0 if and only if p ψ = p φ = τ = 0 (without assuming the particles to form a central configuration), we would have to show that a behavior as theone described above, with a total collision at a binary collision point, cannot happen. Conclusion This paper has shown that, even though a purely relational formulation of the E = P = L = 0 Newtonian universe of three bodies exists – a formulation, that is, of the Newtonian gravitationalsystem on shape phase space – and even though this implies that absolute scale is no longer partof the (unique) description of the system, the Newtonian singularity at the points of total collisionremains.The relational analysis has shown that, at the points of total collision, the equations of motionare singular due to the particular behavior of the shape momenta, which are necessarily zero atthese points. While this implies that the singularity remains, it also has one positive effect. Itallows us to define total collisions in purely shape-dynamical terms, that is, without referenceto (external) absolute scale. Instead, we are able to distinguish total-collision solutions fromnon-collision solutions on shape phase space. This shape-dynamical description we also use toshow that solutions featuring a total collision form a set of measure zero among all solutions.The general form of the analysis suggests that all results, which have been obtained for thethree-body model, should hold for the N -body model as well. That is, also for N particles, weexpect the shape equations of motion to be singular at the points at (and only at) which theshape momenta are all zero. And again, total collisions on N − dimensional shape phasespace will be points of central configuration with vanishing shape momenta at the moment ofminimal extension of the system. Of course, for N particles, we won’t be able to make theexplicit computations, but instead will have to give arguments of a more general and abstractform. But this is merely a technical issue and should be possible in principle.Although the result is negative in the sense that, even on shape space, the singularity remains,the original idea – to glue together two solutions from absolute space via the unique evolutionof certain relational (that is, shape) variables – need not be dismissed altogether. Although theshape variables we presented didn’t serve for that purpose, one might still hope to find some othervariables, singular variables, which allow for an evolution though the points of total collision.But this will involve a much harder study of the singularity, a topic for future research.20 ppendix Appendix A: Singularity of the physical vector field at ψ = 0 We show that the vector field (4.16) is singular at ψ = 0 and ψ = π (top and bottom of theshape sphere). An analogous result holds for the primed coordinates (at ψ ′ = 0 and ψ ′ = π ).Consider the equations of motion on T ∗ S (4.16). The corresponding physical vector field issingular at ψ = 0 and ψ = π if and only if at least one of the equations diverges.Consider the equation for p ψ , d p ψ d τ = 2 sin − ψ cos ψp φ p ψ + sin − ψp φ + τ + ∂ log( − V S ) ∂ψ . In the limit ψ → and ψ → π , it is lim ψ → /π (cid:20) − ψ cos ψp φ p ψ + sin − ψp φ + τ (cid:21) = lim ψ → /π (cid:20) − − ψ cos ψp φ − − ψp φ − − ψp φ (cid:21) = lim ψ → /π (cid:20) ψ sin ψ + sin ψ (cid:21) = ∞ . This shows the assertion.Note that this singularity is not a physical singularity, but a coordinate singularity. It canalso be read off the Jacobian, that is, the determinant of the transformation matrix from the w to the spherical coordinates, det d ( w , w , w ) d ( R, ψ, φ ) = R sin ψ, which is zero at R = 0 , ψ = 0 , and ψ = π . Appendix B: Partial derivatives of the shape potential V S From (3.12) we know that the shape potential V S = V S ( w ) can be written as V S = −√ X i The binary collision vectors always lie on the w = 0 plane.For the unprimed coordinates (4.1) and (4.4), this means that they lie on the equator ofthe shape sphere. The equator is specified by the angle ψ = π/ , hence, ψ ij = π/ . Ifthe particles have equal masses, the three binary collision points are further specified by φ ij ∈ { π, π, π } (where, again, the i and j refer to the collision particles). Hence, for the21iven coordinates, w · b ij = || w || sin ψ cos( φ − φ ij ) . Here the ( φ − φ ij ) are the angles between the b ij and the projection of w onto the w = 0 plane and the term || w || sin ψ is the component of w which is parallel to the w = 0 plane.The shape potential can then be written as V S = −√ X i Given the primed coordinates (4.5) and (4.7), the w = 0 plane is specified by φ ′ ij = 0 , respectively φ ′ ij = π . (In this case, the intersection of the shapesphere and the w = 0 plane is a meridian, not the equator.) The three binary collisionvectors are further specified by three angles ψ ′ ij (with i < j and i, j = 1 , , ). In the equal-mass case, the respective collision vectors are b ij = ( φ ′ ij , ψ ′ ij ) ∈ { (0 , π ) , (0 , π ) , ( π, π ) } .We now have w · b ij = || w ||| cos φ ′ | cos( ψ ′ ± ψ ′ ij ) . In this setting, || w || · | cos φ ′ | is the component of w parallel to the w = 0 plane and ( ψ ′ ± ψ ′ ij ) is the angle between the projection of w onto the w = 0 plane and the binarycollision vector (where the plusminus takes into account that ψ runs from 0 to π and notfrom 0 to π , hence, in order to correctly measure the angle between the vectors w and b ij we need to adjust the sign appropriately, where we have to take ‘ − ’ if both vectors areon the same side of the w = 0 plane and ‘ + ’ in case they are on opposite sites).The shape potential can now be written as V ′ S = −√ X i Given the first choice of coordinates, the three Euler configurationsare specified by ( ψ E , φ E ) ∈ { ( π , , ( π , π ) , ( π , π ) } . This is again the equal mass case.Recall that, in that case, the binary collision vectors were specified by ψ ij = π/ (whichwe don’t need here since ψ ij doesn’t appear in the V S ) and φ ij ∈ { π, π, π } . Then, with22 S from (5.1) and given ( ψ E , φ E ) , we obtain: (cid:20) ∂ log( − V S ) ∂ψ (cid:21) ψ E ,φ E = (cid:20) Gm / X i Given the second choice of coordinates, the two Lagrangeconfigurations are specified by ( ψ ′ L , φ ′ L ) ∈ { ( π , π ) , ( π , π ) } . In this case, remember thatthe binary collision vectors were specified by ( ψ ′ ij , φ ′ ij ) ∈ { ( π, , ( π, , ( π, π ) } . Then,with V ′ S given by (5.2), we have: (cid:20) ∂ log( − V ′ S ) ∂ψ ′ (cid:21) ψ ′ L ,φ ′ L = (cid:20) − Gm / X i 2) + cos(5 / π )) (cid:21) = 0 . (5.6)To obtain the last equation, we used that cos(1 / π ) = − cos(5 / π ) and cos( π/ 2) = 0 .23 ppendix C: Measure-zero set of solutions Since the internal vector field of the E = P = L = 0 Newtonian universe on T ∗ S is a Hamilto-nian vector field, it follows that shape phase space volume is conserved under the internal timeevolution. In other words, the volume measure on T ∗ S is stationary.To obtain the volume measure on T ∗ S , we use a formula of Faddeev [1969] for the reducedvolume measure/the Liouville measure on the reduced phase space. Faddeev shows that, ifthe reduced phase space is obtained from some original phase space Γ by imposing constraints H a = 0 , χ b = 0 with { H a , χ b } 6 = 0 , then the volume measure (Liouville measure) on the reducedphase space is µ = | det { H a , χ b }| Y a,b Y i δ ( H a ) δ ( χ b )d q i d p i . (5.7)Here q i , p i are the canonical coordinates on Γ .In our case, Γ ∼ = R with canonical coordinates q i , p i ∈ R and the pairs of conjugateconstraints are { P , Q cm } , { L , I L } , and { D, H } (see section 3.1). Inserting this in (5.7), we findthat the reduced volume measure d µ on T ∗ S is given by µ = | det { ( P , L , D ) , ( Q cm , I L , H ) }| Y i =1 δ ( P ) δ ( L ) δ ( D ) δ ( Q cm ) δ ( I L ) δ ( H )d q i d p i . (5.8)To abbreviate the notation, let us define A := { ( P , L , D ) , ( Q cm , I L , H ) } . Explicitly, A is thematrix A = { P , Q cm } { L , Q cm } { D, Q cm }{ P , I L } { L , I L } { D, I L }{ P , H } { L , H } { D, H } . Now we can use that P and L are conserved quantities of motion, i.e., { P , H } = { L , H } = 0 . Using this, it is easy to compute the determinant of A : | det A | = | det { ( P , L , D ) , ( Q cm , I L , H ) }| = |{ P , Q cm }{ L , I L }{ D, H }| = |{ D, H }| , (5.9)where the last equation follows from the fact that { P , Q cm } = { L , I L } = 1 . Inserting (5.9) in (5.8) and using the definition of the Jacobi and Hopf coordinates fromsection (3.2) as well as the definition of the spherical coordinates (where everything is analogousboth in the primed and unprimed coordinates), we find that Z | det { ( P , L , D ) , ( Q cm , I L , H ) }| Y i =1 δ ( D ) δ ( H ) δ ( P ) δ ( Q cm ) δ ( L ) δ ( I L )d q i d p i = Z |{ D, H }| δ ( D ) δ ( H ) δ ( P ) δ ( Q cm ) δ ( L ) δ ( I L )d P d Q cm d L d I L d R d ψ d φ d p R d p ψ d p φ = Z |{ D, H } ∗ | δ ( D ) δ ( H )d R d ψ d φ d p R d p ψ d p φ . (5.10)Here { D, H } ∗ denotes the Poisson bracket on T ∗ S R , i.e., the Poisson bracket evaluated on theconstraint surface determined by P = L = Q cm = I L = 0 . To compute the integral (5.10), we24se that δ ( n ) ( f ( x )) = 1 | det ∂ f i /∂ x j | δ ( n ) ( x − x ) , where the x ’s are the zeros of f ( x ) . Here f = ( H, D ) T , x = ( R, p R ) T and x = ( R ∗ , T with x solving the constraints H = 0 and D = 0 . That is, δ ( H ) δ ( D ) = 1 |{ H, D } ∗ | δ ( R − R ∗ ) δ ( p R − . (5.11)Inserting this into (5.10), we obtain Z |{ D, H } ∗ | δ ( D ) δ ( H )d R d ψ d φ d p R d p ψ d p φ = Z δ ( R − R ∗ ) δ ( p R )d R d ψ d φdp R d p ψ d p φ = Z d ψ d φ d p ψ d p φ . (5.12)Hence, the Liouville measure µ on T ∗ S is given by µ = d ψ d φ d p ψ d p φ . (5.13)It is a consequence of Faddeev’s construction that µ is gauge-invariant. Moreover, since thereduced internal dynamics (the dynamics on T ∗ S ) is Hamiltonian, it is stationary. That is,volume is conserved under the internal time evolution.To see this consider the Lie derivative L X H of the measure along the Hamiltonian vector field X H on T ∗ S , the vector field connected to H from (4.15). The Lie derivative vanishes, L X H µ = 0 . (5.14)Hence, µ is conserved under the time evolution connected to the Hamiltonian vector field X H on T ∗ S .That the Lie derivative vanishes can be proven as follows. Note that µ = 1 / ω ∧ ω ) , where ω = d ψ ∧ d p ψ + d φ ∧ d p φ is a symplectic two-form on T ∗ S . Since it is symplectic, it is closed, d ω = 0 . Using this and the fact that the vector field is Hamiltonian, i.e., ω ( X H , · ) = d H , weobtain: L X H µ = 12 ( L X H ω ∧ ω + ω ∧ L X H ω )= 12 (cid:2) (d ω )( X H , · , · ) + d( ω ( X H , · )) (cid:3) ∧ ω + 12 ω ∧ (cid:2) (d ω )( X H , · , · ) + d( ω ( X H , · )) (cid:3) = 12 (cid:2) ◦ d H (cid:3) + 12 (cid:2) d ◦ d H + 0 (cid:3) = 0 . (5.15)Since the Lie derivative along the internal Hamiltonian vector field vanishes, the measure isconserved under the internal time evolution. Consequently, it is the appropriate measure for thestatistical analysis on T ∗ S . 25 eferences [1] Arnol’d, V. I. (1989): Mathematical Methods of Classical Mechanics. Springer.[2] Barbour, J., Koslowski, T., and Mercati, F. (2013): “A gravitational origin of the arrowsof time.” ArXiv: 1310.5167 [gr-qc].[3] Barbour, J., Koslowski, T., and Mercati, F. 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