Differential Geometry of Rigid Bodies Collisions and Non-standard Billiards

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires boundary conditions that specify how orbits should be continued after collisions. A boundary condition is the assignment of a collision map at each tangent space on the boundary of $M$ that gives the post-collision state of the system as a function of the pre-collision state. Our main result is a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility, and the natural requirement that impulse forces only act at the point of contact of the colliding bodies. These assumptions can be stated in geometric language by making explicit a family of vector subbundles of the tangent bundle to the boundary of $M$: the diagonal, non-slipping, and impulse subbundles. Collision maps at a boundary configuration are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace. The space of such maps is naturally identified with the union of Grassmannians of $k$-dimensional subspaces of $\mathbb{R}^{n-1}$, $0\leq k\leq n-1$, each subspace specifying the directions of contact roughness. We then consider non-standard billiard systems, defined by fixing the position of one of the bodies and allowing boundary conditions different from specular reflection. We also make a few observations of a dynamical nature for simple examples of non-standard billiards and provide a sufficient condition for the billiard map on the space of boundary states to preserve the canonical (Liouville) measure on constant energy hypersurfaces.


Introduction
The classical theory of collisions of rigid bodies provides a very natural setting in which to explore the geometry and dynamics of mechanical systems on configuration manifolds with boundary. From this geometric perspective, the response of the system to collisions between its rigid moving parts is specified by assigning appropriate boundary conditions that tell how a trajectory should be continued once it reaches the boundary. For example, in the theory of billiard dynamical systems, a topic that may be defined very broadly as the study of Hamiltonian (more typically, geodesic flow) systems on Riemannian manifolds with boundary, one typically assumes that trajectories reflect off the boundary specularly-the simplest form of impact response compatible with the basic laws of mechanics such as energy conservation and time reversibility. Billiard systems with more general boundary conditions have to our knowledge been investigated very rarely. One pertinent example is [3], which is restricted to 2-dimensional billiards. (There is, of course, an extensive literature in engineering and applied physics about less idealized systems governed by impact interactions, but this literature is not concerned with the differential geometric issues that are the main focus here.) Our first goal is to classify boundary conditions for systems defined by two unconstrained rigid bodies in R n , n ≥ 1, under standard physical assumptions of energy conservation, linear and angular momentum conservation, time reversibility, linearity of response, and another condition to be defined shortly that extends momentum conservation and is typically made implicitly in textbooks. Collisions satisfying all of these properties will be called strict. They are formally represented by linear maps C q ∶ T q M → T q M , where M is the configuration manifold equipped with the kinetic energy Riemannian metric, q is a boundary configuration, and the tangent space T q M is the space of (pre-and post-) collision states. A boundary condition for the system then consists of the (differentiable, measurable, random, etc.) assignment of a strict collision map C q to each boundary configuration q ∈ ∂M .
In dimensions greater than 1, the collision map is not uniquely determined by the conditions of strict collision. It is well-known that the nature of the contact between the colliding rigid bodies also needs to be specified. The standard case in which C q is specular reflection corresponds to bodies having physically smooth surfaces.
Towards this classification we identify a family of subbundles of T (∂M ) arising naturally under the assumed physical laws and discuss some relationships among them. We then show examples of trajectories of non-standard (i.e., non-specular) boundary conditions and make a few observations about their dynamics based on numerical simulation. Although we leave for future work a more systematic analysis of rough billiard dynamics, we give here sufficient conditions for the non-standard billiard system to leave invariant the natural volume measure on a constant energy manifold (derived from the canonical symplectic form). The invariance of this billiard measure makes it possible to bring the tools of ergodic theory (see [4,8]) to the study of non-standard billiard systems, although we do not pursue this direction here.

Notation, terminology, and standing assumptions
For the most part we consider the unconstrained motion of two rigid bodies, represented by the sets B 1 , B 2 ⊂ R n . We call these sets the bodies in reference configuration. Let G = SE(n) denote the Euclidean group of orientation preserving isometries of R n equipped with the standard inner product. The bodies are assumed to be connected n-dimensional submanifolds of R n with smooth boundaries. An interior configuration of the system consists of a pair (g 1 , g 2 ) ∈ G × G such that g 1 (B 1 ) and g 2 (B 2 ) are disjoint sets. The closure of the set of interior configurations, denoted M , will be called the configuration space of the system, and its boundary is the set of contact (or collision) configurations. M has dimension 2 dim(G) and the nature of the boundary ∂M will depend on geometric assumptions about the B j . We will soon state a sufficient, and fairly general for our needs, condition on the ∂B j for M to be a submanifold of G × G with smooth boundary.
The following notations are used fairly consistently throughout the paper. Points in B j are denoted b, b j . Elements of the Euclidean group, which is the semidirect product G = SO(n) ⋉ R n of the groups of rotations and translations, are written as pairs (A, a), where A ∈ SO(n) and a ∈ R n . Elements of the Lie algebra g = se(n) are written (Z, z), possibly with subscripts or superscripts, where Z ∈ so(n) and z ∈ R n . The outwardpointing unit normal vector to the boundary of B j at b ∈ ∂B j is denoted ν j (b). It is convenient to consider orthonormal frames σ at b ∈ ∂B j adapted to the bodies, in the following sense: σ ∶ R n → T b R n ≅ R n is an element of SO(n) such that σe n = −(−1) j ν j (b), where e n = (0, . . . , 0, 1) † is the last element of the standard basis {e 1 , . . . , e n } of R n and ' †' indicates matrix transpose. A configuration of the system of rigid bodies is given by a pair (g1, g2) of elements of the Euclidean group G. A boundary configuration can be parametrized by the tuple (b1, σ1, b2, σ2) where bj ∈ ∂Bj such that g1(b1) = g2(b2) and σj is an orthonormal frame at bj as will be explained in the text.
Because M is a submanifold of G × G, each tangent space can be canonically identified with T (e,e) (G × G) ≅ g ⊕ g by left-translation. Thus states of the system, defined as elements of T M , may be canonically identified with tuples (A 1 , a 1 , A 2 , a 2 , Z 1 , z 1 , Z 2 , z 2 ). We sometimes indicate the state by (q, ξ), where q = (g 1 , g 2 ) and ξ = (Z 1 , z 1 , Z 2 , z 2 ) ∈ g × g.
The position of material point b ∈ B j in the given state is then g j (b) = A j b + a j and its velocity is V (b) = A j (Z j b + z j ). The boundary configuration (g 1 , g 2 ) can also be parametrized, up to an overall rigid motion of the two bodies keeping their positions relative to each other unchanged, by (b 1 , σ 1 , b 2 , σ 2 ), where b j is in the boundary of B j and σ j is an adapted frame such that g 1 (b 1 ) = g 2 (b 2 ) and A 1 σ 1 = A 2 σ 2 . The tuple (b 1 , σ 1 h, b 2 , σ 2 h) corresponds to the same contact configuration, for all h ∈ H, where H ∶= SO(n − 1) = {A ∈ SO(n) ∶ Ae n = e n }.
These notions are illustrated in Figure 1.
Let Π b denote the orthogonal projection from R n to the tangent space to the boundary of B j at a boundary point b. The orthogonal projection to R n−1 = e ⊥ n will be denoted Π. The shape operator of the boundary of B j at the point b is the linear map defined by We say that S ∶ R n−1 → R n−1 is the shape operator S b in the adapted frame σ at b if σS = S b σ. The notation Ad σ S = σSσ −1 will be used often to indicate conjugation.
So as not to get distracted by regularity issues, we assume that the configuration manifold M has smooth boundary and that each boundary configuration corresponds to the bodies being in contact at a single common point. Proposition 2.1, which is a special case of Proposition 5.1, gives a sufficient condition for M to be nice in this respect. The hypotheses of the proposition will be assumed to hold throughout the paper.
Proposition 2.1. Suppose that S 1 + S 2 is nonsingular for every relative configuration (b 1 , σ 1 , b 2 , σ 2 ) of the rigid bodies, where S j ∶ R n−1 → R n−1 is the shape operator of the boundary of B j at b j in the adapted frame σ j . Then M is a smooth manifold of dimension 2 dimG with smooth boundary, and each boundary point q = (g 1 , g 2 ) represents a configuration with a unique point of contact. Moreover the map that associates to q ∈ ∂M the uniquely determined pair We call the b 1 , b 2 associated to q ∈ ∂M under the condition of Proposition 2.1 the contact points (in the reference configuration) associated to boundary point q.

The kinematic bundles
If a, b ∈ R n , let a ∧ b ∈ so(n) be the n-by-n matrix such that (a ∧ b) ij = a j b i − a i b j . If a, b are orthogonal unit vectors, a ∧ b is the infinitesimal generator of the one-parameter group in SO(n) that rotates the plane spanned by a and b and fixes pointwise the orthogonal complement of that plane. The boundary state of the two-body system consists of the boundary configuration q = (g 1 , g 2 ) ∈ G × G and velocities ξ = (Z 1 , z 1 , Z 2 , z 1 ) ∈ g × g.
We now define the kinematic bundles. Given q = (g 1 , g 2 ) ∈ ∂M , with g j = (A j , a j ) and associated contact points b 1 , b 2 , consider the following linear relations on the ξ ∈ T q (∂M ), where N j = ∂B j and ν j = ν j (b j ): As already noted, A j (Z j b j + z j ) is the velocity of the contact point b j in the given state. Observe that relation R 2 implies R 1 . It will be shown later that The physical interpretation of the kinematic bundles is as follows. A state satisfying R 1 has the property that the contact points have zero relative velocity in the normal direction to the plane of contact. Relation R 2 is satisfied exactly when the contact points are not moving at all relative to each other at the moment of contact. This means that the contact points do not slip past each other. Relation R 3 describes a state in which the tangent spaces to the bodies at the point of contact do not experience a relative rotation (on that tangent space). Thus it is a condition of non-twisting. Together R 2 and R 3 describe a state in which the bodies are rolling on each other. Definition 2.1 (Kinematic bundles). Let S, R, and S be the vector subbundles of T (∂M ) defined by Note that D ⊂ R ⊂ S. We refer to S as the non-slipping subbundle, R the rolling subbundle, and D the diagonal subbundle.
It will be shown that the diagonal subbundle D is the tangent bundle to the orbits of the action of G on M by left translations: g(g 1 , g 2 ) ∶= (gg 1 , gg 2 ). Later in we give a different definition of these subbundles, Definition 5.2, that makes their physical interpretation more clear. Then what is stated above as a definition is derived in Section 5.2.

The kinetic energy metric and the impulse subbundle
Suppose now that the bodies B j are assigned mass distributions represented by finite positive measures µ j supported on B j . Let m j ∶= µ j (B j ) be the mass of B j . We may assume without loss of generality that µ j has zero first moment: ∫ Bj b dµ j (b) = 0. This is to say that B j has center of mass at the origin of R n . The matrix of second moments of µ j is L j = (l rs ), with entries We call L j the inertia matrix of body B j . This matrix induces a map L j on so(n) that associates to Z ∈ so(n) the matrix L j (Z) = L j Z + ZL j ∈ so(n).
Definition 2.2 (Kinetic energy Riemannian metric). Given q ∈ M and u, v ∈ T q M , define the symmetric non-negative form on T q M by are the translates to g × g of u, v. When the above bilinear form is positive definite we call it the kinetic energy Riemannian metric on M . Denoting by ⋅ q the corresponding norm at q, we call 1 2 v 2 q the kinetic energy associate to state (q, v).
The kinetic energy function given in Definition 2.2 is easily shown (as indicated later) to come from integration with respect to the mass distribution measures of (one-half of) the Euclidean square norm of the velocity of material point b over the disjoint union of B 1 and B 2 . Thus Definition 2.2 agrees with the standard textbook definition of kinetic energy. It is also clear that the metric is invariant under the left-action of G on M . Note that the boundary of M is a G-invariant set.
For each u ∈ g we define vector field q ↦ũ q ∈ T q M , q = (g 1 , g 2 ), bỹ We call u ↦ũ the infinitesimal action derived from the left G-action on M andũ the vector field associated to u ∈ g.
is called the momentum map associated to the G-action on M .
The most straightforward way of introducing dynamics into the system is through Newton's second law. There are several equivalent forms of it as we note later. The following is particularly convenient for our needs. We first define a force field (possibly time dependent) as a bundle map F ∶ T M → T * M . Given a state (q,q), q = (g 1 , g 2 ), each component F j of F can be pulled-back to g * using right-translation R gj , so it makes sense to write This is Newton's second law written as a differential equation on the co-Lie algebra of G.
Other useful forms are mentioned later. One of them is indicated in the next proposition, in which we use the notation F # for the dual of F with respect to the left-invariant Riemannian metric and write Here we are using the differential of the left-translation map L gj .
where g j = (A j , a j ) and v c = A j z j is the velocity of the center of mass of body B j .
We assume that F results from the integrated effect of forces acting on the individual material points. That is, we assume that there exists a R n -valued measure ϕ j on B j parametrized by T M from which F is obtained by integration: is the velocity of the material point b in the state (q, u). Of special interest for us are the forces involved in the collision process. These impulsive forces are characterized by being very intense and of very short duration, applied on a single point-the point of contact in each body.
That the forces act on each body only at the point of contact greatly restricts the right-hand side of the equation of motion in Proposition 2.2. This is indicated in the next proposition. Proposition 2.3. We suppose that the force field F j acting on body B j is such that the force distribution measure ϕ is singular, concentrated at the point b j . Then the equations of motion of Proposition 2.2 reduce to For ideal impulsive forces (of infinite intensity and infinitesimal duration), momentum should change discontinuously. Integrating Equation 2.1 over a very short time interval [t − , t + ] around t produces a nearly discontinuous change in momentum while keeping the configuration essentially unchanged. We have informally It is not necessary for our needs to make more precise the limit process suggested by this expression. From it we obtain the form of the change in momentum after impact, which is given in the next proposition. Let q = (g 1 , g 2 ) ∈ ∂M be a collision configuration and denote by (Z ± 1 , z ± 1 , Z ± 2 , z ± 2 ) ∈ T q M the post-(+) and pre-(−) collision velocities of the two rigid bodies. Proposition 2.4 (Velocity change due to impulse at contact point). Given pre-collision velocity Under conservation of linear momentum m 1 A 1 u 1 + m 2 A 2 u 2 = 0 holds.
The proof of the above proposition is given in Section 4.4. The assumption that impulsive forces of one body on the other at the moment of impact are applied at the point of contact is a strong constraint. One can in principle conceive of force fields of relatively long range, acting throughout the bodies, that are briefly switched on at the moment of impact, then switched off as soon as the bodies lose contact. More realistically, the bodies could suffer a deformation around the region of impact, creating a small neighborhood of contact. Of course this goes beyond the rigid body model. Here it is assumed that these possibilities do not happen, and that any effect of one body on the other can only be transmitted through the single point of contact between them.
If L j is non-negative definite of rank at least n − 1, L j is invertible. With this in mind, Proposition 2.4 suggests the following definition.
Definition 2.4 (Impulse subbundle). The impulse subbundle of T M (over the base manifold ∂M ) is defined so that its fiber at q ∈ ∂M is the subspace We have now the following vector subbundles of i * (T M ), where i ∶ ∂M → M is the inclusion map: D ⊂ R ⊂ S ⊂ T (∂M ) and C. The latter subbundle is the only one that depends on the mass distributions.
Theorem 2.1. The impulse subspace C q is the orthogonal complement of the non-slipping subspace S q and contains the unit normal vector n q . Therefore, A physical interpretation of this orthogonal decomposition will be given shortly.

Collision maps
Let n q be the unit normal vector to ∂M pointing into M at a boundary configuration q.
Define the half-spaces We call any C q ∶ T − q M → T + q M a collision map at q. By a boundary condition we mean the assignment of such a map C q to each q ∈ ∂M . We only consider here linear collision maps; that is, C q extends to a linear map on T q M .

Definition 2.5 (Strict collision maps).
A collision map C q at q ∈ ∂M is strict if the following hold for all u, v ∈ T q M : 1. Conservation of energy: ⟨C q v, C q u⟩ q = ⟨v, u⟩ q . That is, C q is a linear isometry.
2. Conservation of momentum: P g (q, C q v) = P g (q, v).
4. Impulse at the point of contact: Proposition 2.5. Condition 2 of Definition 2.5 is equivalent to assuming that C q restricts to the identity map on D q . Condition 4 is equivalent to C q C q = C q and (C q − Id)C ⊥ q = 0. Thus energy conservation and impulse at a single contact point are together equivalent to C q being the identity on S q . In this sense, condition (4) of Definition 2.5 can be regarded as generalizing momentum conservation as we note in Proposition 2.5. In fact, conservation of momentum amounts to C q being the identity on D q , whereas 4 and Theorem 2.1 imply that C q is the identity on the bigger subspace S q . An intermediate condition is that C q restricts to the identity on the rolling subspace R q .
Corollary 2.1. Strict collision maps are the linear isometric involutions of T q M , q ∈ ∂M , that restrict to the identity map on the non-slipping subspace S q .
Collision maps have eigenvalues ±1. The map P ± ∶= I±Cq 2 is the orthogonal projection to the eigenspace associated to eigenvalue ±1.
Definition 2.6. The dimension of the eigenspace of C q associated to eigenvalue −1 will be called the roughness rank of C q . The image of the orthogonal projection P − (a subspace of C q ) will be called the roughness subspace at q.
The unit normal vector n q is always contained in the impulse subspace C q and it must necessarily be in the −1-eigenspace of C q .
Corollary 2.2. Let n be the dimension of the ambient Euclidean space. Identifying then the set of strict collision maps at any given boundary point is J 0 ∪ ⋅ ⋅ ⋅ ∪ J n−1 . Moreover, dim J k = k(n − k − 1) and k is the roughness rank at q. We call J k the Grassmannian of rough subspaces having roughness rank k.
It is easy to compute the dimensions of the Grassmannians J k for strict collision maps. They are given, up to dimension 5, by the following table: The table shows that in dimension 1 there is a unique strict collision map; in dimension 2 there are exactly 2 possibilities; and in dimension 3 there is one possibility of roughness rank 0 given by the standard reflection map, one possibility for maximal roughness rank 2, and a one-dimensional set of possibilities for roughness rank 1 parametrized by the lines through the origin in R 2 . For general n, the unique collision map of maximal roughness rank will be referred to as the completely rough reflection map.

Non-standard billiard systems
We have so far considered systems consisting of two unconstrained rigid bodies. The results of this paper can be extended to situations in which one body or both are subject to holonomic and non-holonomic constraints. We will explore this extension more systematically elsewhere. Here we consider only the case in which body B 1 remains fixed in place whereas B 2 is unconstrained except for the condition that it cannot overlap with B 1 . The term billiard system will refer to a system of this kind where B 2 is a ball with rotationally symmetric mass distribution. The system will be called non-standard if the (strict) collision maps are not all specular reflection.  Let R denote the radius of B ∶= B 2 and m its mass. Due to rotational symmetry of the mass distribution of B, the matrix of inertia L is scalar, that is, L = λI. For example, a simple integral calculation shows that if B has uniform mass distribution, then λ = R 2 n+2 . (Recall that we have defined L as the matrix of second moments of the mass distribution measure divided by the total mass, so m does not appear in λ.) The (smooth) boundary of B 1 will be denoted N and the unit normal vector field on N pointing into the region of free motion of B will be denoted ν. Trajectories of the billiard system are sequences of states: Here g i is the contact configuration and ξ i the post-collision velocities in the body frame (reference configuration) at the ith collision.
To each contact state (g, ξ) = (A, a, Z, z) is associated a unique contact point b ∈ N and the post-collision velocity v = Az of the center of mass. The center of mass of B in configuration g is a and the velocity of any given material point When it is necessary to distinguish points in N and in B we write b ∈ N and b ○ ∈ B. The unit normal vector to B at b ○ will be written ν ○ (b ○ ) = b ○ R. The point of contact at the next collision, which only depends on b and v, will be denoted b ′ = T (b, v). See Figure 2.
1. From the current collision state (g, ξ) at time t one obtains the contact point b ∈ N and velocity v of the center of mass a of B where g = (A, a) = (A(t), a(t)). It should be kept in mind that ξ = (Z, z) describe post-collision velocities so v = Az points into the region of free motion of the ball.
2. Obtain the contact point b ′ = T (b, n) ∈ N and the time t ′ = t + τ of the next collision.
3. Obtain the next pre-collision state: This is the free (geodesic) motion between collisions. Observe that be the contact point on the ball in the reference configuration at the next collision and denote by Π ○ , Π ⊥ ○ the orthogonal projections to the tangent space to ∂B at b ○ and to , and T is a linear involution on T b○ (∂B) corresponding to a choice of collision map. For specular reflection T = I and for completely rough collisions T = −I.

Examples of non-standard billiards
We assume in all examples the uniform mass distribution on the ball B so λ = R 2 (n + 2), where R is the radius of B. Let first n = 2. In this case the only non-standard collision map corresponds to T = −I. Elements of the rotation group are parametrized by the angle of rotation θ and elements of the Lie algebra of SO(2) are written asθJ, where J is the rotation matrix by π 2 in the counterclockwise direction. Together with the standard coordinates (x, y) we obtain coordinates (θ, x, y) on SE(2). It will be convenient to make the coordinate change: This yields coordinates (x 0 , x 1 , x 2 ,ẋ 0 ,ẋ 1 ,ẋ 2 ) on the billiard state space. We also write v 0 =ẋ 0 and v = (ẋ 1 ,ẋ 2 ) † for the velocity of the center of mass of the disc.
The choice of coordinates is made so that the kinetic energy Riemannian metric becomes, up to multiplicative constant, the standard Euclidean metric. Then it can be derived from Equation 2.2 that the post-collision velocities (v + 0 , v + ) after collision at point of contact b ∈ N is the function of the pre-collision velocities (v − 0 , v − ) given by Thus the state updating equations for a 2-dimensional non-standard billiard system is as follows. If τ is the time of free flight between the two consecutive collisions and setting The geometric interpretation of those equations is explained in Figure 3. Note the role played by the angle β defined by cos β = 1 3, sin β = 2 √ 2 3. In [3] it is observed that β is the dihedral angle of a regular tetrahedron. In the figures to follow we only indicate the position of the center of the disc; we draw a smaller table whose boundary is at a distance R from the boundary of the original table and we imagine the center of the ball as a point mass bouncing off the boundary of this smaller region. The outgoing velocity is obtained by first reflecting v specularly on the plane spanned by e0 and Jν(v) to find v, then reflecting the latter specularly on the plane spanned by ν(b) and Jν(b), and finally rotating the resulting vector by β as indicated. As noted in [3], β is the dihedral angle of a regular tetrahedron.
Next we show examples of trajectories of systems with rough collisions. The examples are given here without much analysis. We leave the more systematic study of the dynamics of such systems for another article. As a first illustration, consider the case of a circular billiard table. The typical trajectory is shown in Figure 4 and a few more examples are shown in Figure 5.
The following proposition captures the main properties of trajectories of circular rough billiards readily observed in the     In the previous examples the boundary condition on M amounted to a constant (more precise, parallel) choice of C q . We wish to illustrate now boundary conditions for which the map q ↦ C q varies in a nontrivial way or is chosen randomly. Let the billiard system consist of a disc moving in an infinite strip bounded by two parallel lines. We suppose that one hemisphere of the boundary of the disc is rough and the other is smooth. In Figure 7 we show graphs of the position of the (center of) the disc along the longitudinal axis of the table as a function of the collision step. The time between two consecutive collisions is easily shown to be constant, so the step number is proportional to time. The three graphs describe the same trajectory at different time scales, as indicated in the legend of the figure. It is interesting to observe the apparent long range quasi-periodic behavior of trajectories. It is also interesting to note the differences between this example and the next shown in Figure 8. The setting is essentially the same, except that a point on the boundary of the disc is chosen to be rough or smooth randomly with equal probability. This is thus an example of a random boundary condition. The longitudinal motion now corresponds to a random walk, for which it is possible to prove a diffusion (Brownian motion) limit. We consider now a few examples in dimension 3. In all cases, a ball of uniform mass distribution moves between two parallel infinite plates. In dimension three, the roughness rank can be 0, 1, or 2. Standard specular reflection has roughness rank 0; we explore examples of roughness rank 1 and 2.  Figure 9 illustrates the case of roughness rank 2 collisions. The figure on the right shows the projection of the trajectory to the coordinate plane parallel to the plates. One notable property of the system is that trajectories are bounded. In dimension 2, a similar property was noted in [3].
When the roughness rank is one, the set of collision maps comprise a one-dimensional family. Figure 10 shows some combinations of boundary conditions. One clearly notices that whenever the roughness rank in at least one plate is not maximal, trajectories are no longer bounded. The boundary conditions for the systems of Figure 10 are of the following types: one plate has roughness rank 2 and the other has roughness rank 1 with constant rough direction (that is, constant map T in Equation 2.2); and both plates have roughness rank 1, with random rough direction for the bottom plate and either constant or independent random rough direction for the top plate. Specifically, we choose the directions given by angles 0, π 3, 2π 3 with equal probabilities. The legend of the figure shows which trajectory corresponds to which condition.

invariant measure
A fundamental property of the dynamics of standard billiard systems is the existence of a canonical invariant measure on constant energy surfaces, sometimes referred to as the Liouville measure. We give here a sufficient condition for the same measure to be invariant under non-standard collisions.
Let S denote the boundary of the configuration manifold of the two-bodies system. As before, we assume that S is smooth. We fix a value E of the kinetic energy and denote Define the contact form θ on T M to be the 1-form such that θ v (ξ) = ⟨v, dπ v ξ⟩ q , where π is the base point projection from T M to M (and we indicate the element of T M by v rather than (q, v) in subscripts). It is well-known that dθ defines a symplectic form on T M . It can also be shown that the restriction of dθ to N E defines a symplectic form on N E ∖ T S. (See, for example, [5].) The billiard map T on N E associates the post-collision state of the system at the time of a collision to the post-collision state at the next collision. There are well-known issues about this map, even for standard billiards in dimension 2, that make the precise specification of its domain difficult to describe. See, for example, [4]. Here we assume that the domain of T consists of a "large" open set of full Lebesgue in N E and omit any further reference to it since this issue of domains is not specific to our rough billiards. The next result is shown by a local argument and considerations of domain do not play a role.
billi ard table r e t u r n s i d e Theorem 2.2. Suppose that the field of collision maps q ∈ S ↦ C q is piecewise smooth and parallel (where it is smooth) with respect to the Levi-Civita connection associated to the kinetic energy Riemannian metric. Let Ω = dθ ∧ ⋯ ∧ dθ be the form (of degree 2n − 2, where n is the dimension of the ambient Euclidean space) derived from the canonical symplectic form dθ on N E ∖ T S. Then Ω is, up to sign, invariant under the billiard map. The theorem will be proved in Section 7. Corollary 2.3 is due to the following observation. The boundary of M is a flat surface with the Euclidean metric and the vectors e 0 , Jν(b) shown on the right-hand side of Figure 3 constitute a parallel frame. The orthogonal line distributions C q and S q are also parallel as the angle between each of them and e 0 is constant. But these are the eigenspaces of C q for the eigenvalues −1 and 1, respectively. It follows that the field of rough collision maps is parallel.
We will leave for a future paper a more detailed investigation of invariant measures of non-standard billiards. Here we simply illustrate Theorem 2.2 with a numerical observation concerning the motion of a disc in a rectangular table with rough collisions.
The geometric set-up is shown in Figure 11. The configuration manifold in this case is a parallelepiped R in dimension 3 and the canonical billiard measure on the manifold N of unit length vectors with base points on the boundary of R has density proportional to where φ is the angle the vector v makes with the normal vector to the boundary.
It can be shown that Theorem 2.2 applies to this case. As an experiment to illustrate invariance of the billiard measure for rough collisions we sample initial conditions on the face x 2 = 0 with the uniform distribution for the (x 0 , x 1 ) positions and initial velocity v having probability density proportional to cos φ relative to the uniform probability on the unit hemisphere. If the return states to the face x 2 = 0 are distributed according to the same measure, then the angle φ for the return velocity must be distributed relative to Lebesgue measure on [0, π 2] with density sin(2φ), which is the marginal density function for the angle distribution with respect to the Lebesgue measure dφ on [0, π 2], under the assumption that the billiard measure is invariant. This is indeed the case as shown in Figure 12. A large number (10 5 ) of initial conditions starting from one side of the rectangle are sampled from the normalized billiard measure restricted to that side. For each trajectory, the return state to that side is computed and the angle relative to the normal (to the side of the angle-position parallelepiped corresponding to that side of the rectangle) is recorded. The distribution of values is shown in the above histogram. The superimposed line is the graph of sin(2φ).

The Euclidean group and its Lie algebra
The proofs of the main statements made above will be given after we establish some basic material. Despite the classical nature of the subject, standard textbook treatments of collisions of rigid bodies are not adequate for our needs while the more differential geometric texts in mechanics mostly do not treat this topic. Thus we find it necessary to develop the subject more or less from scratch. In this section we review general facts about the Lie theory and Riemannian geometry of the Euclidean group with left-invariant metrics that will be needed thoughout the paper.

Generalities
The isometry group of (R n , ⋅), where '⋅' indicates the standard inner product, is the Lie group of all the affine maps of the form x ↦ Ax + a for A ∈ O(n) and a ∈ R n under composition of maps. The closed subgroup of orientation preserving isometries, in which A ∈ SO(n), is the Euclidean group in dimension n, denoted SE(n). The latter is isomorphic to the semidirect product SO(n) ⋉ R n with multiplication operation It is also isomorphic to a subgroup of the general linear group GL(n + 1, R) under the correspondence The Lie algebras of SO(n) and SE(n) will be denoted so(n) and se(n). The former consists of all the skew-symmetric matrices in the linear space M (n, R) of n × n real matrices and se(n), when SE(n) is viewed as a subgroup of GL(n + 1, R), consists of the matrices where X ∈ so(n) and x is any vector in R n . Indicating the matrix by the pair (X, x), the Lie bracket is written One-parameter subgroups of SE(n) have the form It is useful to introduce the wedge product, the bilinear operation that associates to a pair of vectors a, b in R n the skew-symmetric matrix a ∧ b ∈ so(n) whose (i, j)-entry is The following elementary properties of the wedge product will be used. The transpose of a matrix will be indicated by U † . Proposition 3.1. Let a, b, u be (column) vectors in R n , A ∈ SO(n) and Z ∈ so(n). Then 6. Let V be the span of orthogonal unit vectors a, b ∈ R n . Then (a ∧ b) 2 = −I and R(θ) ∶= exp(θa ∧ b) = (cos θ)I + (sin θ)a ∧ b ∈ SO(n).
Thus R(θ) is the identity on V ⊥ , and a rotation on V .
8. For a ∈ R 3 set ω(a)b ∶= a × b-the cross-product by a on the left. Then a ∧ b = ω(a × b) and Aω(a)A −1 = ω(Aa).
Proof. All properties are proved by straightforward calculations.

Left-invariant Riemannian metrics on SE(n)
Let ⟨⋅, ⋅⟩ be a left-invariant Riemannian metric on the connected Lie group G with Lie algebra g. Let ∇ be the associated Levi-Civita connection. Define B ∶ g × g → g by If X, Y, Z are left-invariant vector fields on G such that X e = u, Y e = v, Z e = w, then from A left-invariant vector field X is a geodesic vector field if and only if 0 = ∇ X X = −B(X, X). It is not difficult to show that if the metric is bi-invariant then B(u, u) = 0 for all u ∈ g. We adopt the notation: Proposition 3.2. Let g(t) be any smooth curve in G and X a vector field along g(t), not necessarily left-invariant. Define z(t) ∶= g(t) −1ġ (t) and w(t) ∶= g(t) −1 X g(t) . Then In particular, g(t) is a geodesic if and only ifż = B(z, z).
Proof. Let e 1 , . . . , e n be a basis of g and E 1 , . . . , E n the respective left-invariant vector fields on G. We write X = ∑ f j E j , g ′ (t) = ∑ h j (t)E j (g(t)). Then, using Equation 3.2, , from which we obtain the desired expression after left-multiplying by g(t) −1 .
Let the Lie algebra of G be g = s ⊕ r, where r is an ideal and s is a Lie subalgebra. Let ⟨⋅, ⋅⟩ be a left-invariant Riemannian metric on G and ∇ the corresponding Levi-Civita connection. We suppose that s and r are orthogonal subspaces. Proof. All properties follow from the definition of B, Expression 3.1 for the Levi-Civita connection, and the assumption that the subalgebra s and the ideal r are orthogonal.
Let S and R be the subgroups of G having lie algebras s and r, respectively. Then G is the semi-direct product G = S ⋉ R, where R is a normal subgroup of G. We now assume that R is a vector subgroup, hence abelian, and that S is a compact subgroup acting on R by linear transformations, S ⊂ GL(R), preserving an inner product ⟨⋅, ⋅⟩ on R. That is, S is a subgroup of the orthogonal group O(R, ⟨⋅, ⋅⟩). Elements of G will be denoted (A, a) where A ∈ S and a ∈ R. Indicating the action of S on R by Aa, the multiplication in G takes the form (A 1 , a 1 )(A 2 , a 2 ) = (A 1 A 2 , A 1 a 2 + a 1 ).

Newtonian mechanics of rigid bodies
We give here some alternative expressions of Newton's equation of motion. The approach, if not the notations, is essentially that of [1]. Other useful references are [2] and [7].

Momentum of a tangent vector and the momentum map
If M is a Riemannian manifold with metric ⟨⋅, ⋅⟩ and v ∈ T q M , we denote P(q, v) ∶= ⟨v, ⋅⟩ q ∈ T * q M and call this covector the momentum associated to the (velocity) vector v at (configuration) q. The pair (q, v) will be called a state of the system. We often indicate states by (q,q), dotting the quantities of which time derivative is taken. The momentum map P g ∶ T M → g * was defined in Section 2.3.
If M = G is endowed with a left-invariant Riemannian metric, the momentum map for the left-action of G is given by P g (g,ġ)(u) = ⟨ġ, (dR g ) e u⟩ g . Because the Riemannian metric is left-invariant, In this case we also write P g (v) ∶= ⟨v, ⋅⟩ e ∈ g * for v ∈ g. Then where Ad * g α = α ○ Ad g and Ad g is the differential of the map L g ○ R g −1 . Proposition 4.1. Given a smooth curve g(t) in G and setting z(t) ∶= g(t) −1ġ (t), then In particular, g(t) is a geodesic if and only if momentum P g (g,ġ) is constant.
Proof. First note that d dt It follows from the definitions that The expression on the far right is now P g (ż − B(z, z)) ○ Ad g −1 evaluated at u.
Then ⟨⋅, ⋅⟩ 0 is an Ad G -invariant non-degenerate positive bilinear form on so(n) and the associated left-invariant Riemannian metric on G is bi-invariant. Thus for any leftinvariant Riemannian metric ⟨⋅, ⋅⟩ there must exist a linear map L ∶ g → g, symmetric and positive definite with respect to ⟨⋅, ⋅⟩ 0 , such that ⟨Z 1 , Z 2 ⟩ = 1 2 ⟨L(Z 1 ), Z 2 ⟩ 0 . We are interested in such L that arises from a symmetric matrix L ∈ M (n, R) according to the definition L(Z) ∶= ZL + LZ, in which case If u 1 , . . . , u n is a basis of R n of eigenvectors of L, Lu i = λ i u i , then the u i ∧ u j comprise a basis of so(n) such that L(u i ∧ u j ) = (λ i + λ j )u i ∧ u j . If L is symmetric and non-negative definite of rank at least n − 1, then L is an isomorphism and ⟨Z 1 , Z 2 ⟩ ∶= 1 2 Tr(L(Z 1 )Z † 2 ) is a left-invariant Riemannian metric on SO(n). The tensor B for this metric is Proof. Let λ 1 , . . . , λ l be the distinct eigenvalues and V 1 , . . . , V l the respective eigenspaces of L. Let π j ∶ R n → V j denote the orthogonal projections. Then π j L = Lπ j = λ j π j . It suffices to show that L has trivial kernel. Thus suppose L(Z) = 0. Then for all i, j, But λ i + λ j > 0 by the assumptions on L so all blocks π i Zπ j are zero, hence Z = 0. The expression for B follows from Tr

Kinetic energy metrics on SE(n) for rigid bodies in R n
The left-invariant metrics on G = SE(n) of interest here are derived from mass distributions on the rigid body. Let B ⊂ R n denote the body in its reference configuration.
We call Φ ∶ G × B → R n the position map and use the alternative For now (until we consider collisions shortly) B may be any measurable set with a finite (positive) measure µ defining its mass distribution. Recall from Section 2.3 that m ∶= µ(B) is the mass of the body and the first moment of µ is 0. When considering the motion of several bodies, we assume that the center of mass of each of them in the standard configuration is at 0. Elements of g = so(n) × R n will be written in the form ξ = (Z, z). Let L g and R g denote left and right-multiplication by g. We will very often use the identification G × g ≅ T G given by (g, ξ) ↦ (dL g ) e ξ. Each v ∈ T g G gives rise to the map V v ∶ B → R n defined by V v (b) = (dΦ b ) q v, which is the velocity of b in state (g, v). The kinetic energy Riemannian metric on G is defined so that the inner product of u, v ∈ T g G is given by Proof. To see this, note first that Here we have used the form of the exponentiation in SE(n) given in Section 3.1. Therefore, as A leaves invariant the standard inner product in R n , and so ⟨(dL g ) e ξ, (dL g ) e η⟩ g = ⟨ξ, η⟩ e .
Recall the inertia matrix L introduced in Section 2.3.
Proposition 4.4. The matrix L associated to mass distribution µ satisfies: 1. For arbitrary n × n matrices Z 1 and Z 2 , 2. If A ∈ SO(n), the inertia matrix of the rotated body gB is L A ∶= ALA † .
Proof. These are obtained by elementary calculations.
Let L(Z) = LZ + ZL where, from now on, L is an inertia matrix.
Corollary 4.1. The kinetic energy Riemannian metric can be written in the form where u, v ∈ T g G and their left-translates to g are indicated by (Z u , z u ) and (Z v , z v ).
Proposition 4.5 (Tensor B for se(n)). Let SE(n) be given the left-invariant Riemannian metric associated to the inertia matrix L. Then The claimed identity now follows from the definition on B.
Proposition 4.6. Give G = SE(n) the left-invariant Riemannian metric defined by a mass distribution on the rigid body B with inertia matrix L and mass m. Let ξ = (W, w) ∈ g and (g, v) ∈ T G where g = (A, a) and v = (dL g ) e (Z, z). Then Here x c = a is the position of the center of mass of the body in configuration g and v c ∶= Az is the velocity of the center of mass for the given state (g, v).
Proof. This is a straightforward computation based on the definition of P g and the expression of the Riemannian metric given in Corollary 4.1.
Let ⟨⋅, ⋅⟩ g be the left-invariant inner product on g given by Then, with the notation of Proposition 4.6,

Singular force fields and impulses
Let M be the configuration manifold of a mechanical system with the kinetic energy Riemannian metric and material body B with mass distribution measure µ.
A force field on M is a bundle map F ∶ T M → T * M possibly depending on time, although we omit explicit reference to the time variable. So if v ∈ T q M , then F (q, v) ∈ T * q M . Forces acting on B typically arise from a R n -valued (possibly time dependent) measure ϕ on B, parametrized by T M , called the force distribution. From such a measure we define the force field F (q, v) ∈ T * q M such that for each u ∈ T q M , We are interested in cases where ϕ q,v is singular, supported on a single point of B.
Proof. These are consequences of Propositions 3.2 and 4.1.
Then, using the notation of 4.2, 2 follows from these remarks, keeping in mind that v c = Az and x c = a. Let the force be applied on a single point Q = Q(t, g,ġ) ∈ R n , so that ϕ is supported on Q. Let u = (dR g )ξ ∈ T q M , ξ = (W, w) ∈ g. Note that we are using right-translation here since we wish to evaluate the last of the equivalent equations of Proposition 4.7 on the Lie algebra element ξ. Then there exists I = I(t, g,ġ) depending only of F such that This gives It follows from the expression 4.4 of F that Writing f c ∶= Ay, this is equivalent to f c = I and Ad A L(Y ) = (Q − x c ) ∧ f c . Therefore, the equation of motion becomes Our informal discussion of the idea of impulse from earlier in the paper and the above remarks now give the expression ((Q − x c ) ∧ I c , I c ) ∈ g for the change in momentum due to singular forces applied to Q. This gives the following. Proposition 4.8 (Change in momentum due to impulsive forces). If the rigid body with mass m, inertia matrix L and associated Lie algebra map L, is subject to an impulsive force concentrated at point Q ∈ R n at a given time, then momentum changes discontinuously according to for some vector I c ∈ R n depending on the state of the body. As before, x c = a indicates the center of mass of the body in configuration g = (A, a), v ± = Az ± are the velocities of the center of mass immediately prior to and after the application of the impulse, and (g, (Z ± , z ± )) are the pre-and post-impulse states of the body.

Several bodies and momentum conservation
If the mechanical system consists of several unconstrained rigid bodies, B 1 , . . . , B k (in reference configuration) subject to forces F j (q, v), j = 1, . . . k, the configuration manifold M is a subset of the product G × ⋯ × G, with one copy of G = SE(n) for each body. We consider for now only motion in the interior of M .
We say that forces are internal to the system if they are somehow due to the influence of the bodies on each other. More specifically, we use term 'internal' when F j = ∑ i≠j F ij and the F ij = F ij (q, v)-the force body i exerts on body j in state (q, v)-satisfies the property of action-reaction: F ij = −F ji . If the forces are derived from, possibly singular, then the action-reaction property, expressed in terms of ϕ, means that for almost every b, b ′ (with respect to ϕ). Newton's equation applied to body j is then and the total momentum P(q,q) = ∑ j P j (q,q) is conserved: Another way to interpret the notion of forces internal to the system is to assume that the total work the F i do along a rigid motion of the entire system, that is, the work along a path in M of the form γ(t) ∶= e tξ q = (e tξ g 1 , . . . , e tξ g k ) is zero. The total work is then and, again, the total momentum (now in the sense of the momentum map on g) is constant. In this sense, conservation of momentum follows, as expected, from a symmetry property.
Of particular interest here are two bodies that interact through impulses applied to a common point of collision Q. Then for each body, indicated by the index i = 1, 2, where the impulse vectors satisfy I c,1 + I c,2 = 0 by conservation of momentum. Proposition 2.4 is now a consequence of this observation and of Proposition 4.8.

Kinematics of two rigid bodies
The configuration manifold of a system of several (unconstrained) rigid bodies in R n is a submanifold with boundary of the product of copies of the Euclidean group SE(n), one copy for each body. The Riemannian metric is then the product of the Riemannian metrics for each single body. Here we focus on the boundary of the configuration manifold of two bodies and certain structures therein. Let B 1 and B 2 be submanifolds of R n of dimension n having smooth boundary and equipped with mass distribution measures µ 1 and µ 2 with masses m j ∶= µ j (B j ) < ∞ and zero first moment. The bodies need not be bounded. The configuration manifold M is by definition the closure of where G = SE(n). We further assume that each collision configuration q = (g 1 , g 2 ) ∈ ∂M is such that Let N j be the boundary of B j and let ν j be the outward-pointing unit normal vector field on N j . By a (positive) adapted orthonormal frame at b ∈ N j of sign ∈ {+, −} we mean a positive orthogonal map σ ∶ R n → T b R n ≅ R n such that σe n = ν j (b). Here e n is the last vector of the standard basis (e 1 , . . . , e n ) of R n . Hence σ is an element of SO(n) mapping R n isometrically to T b N j . If σ is an adapted frame and h ∈ H ∶= SO(n − 1), then σh is also an adapted frame with the same base point as σ. In this way, H acts freely and transitively by right multiplication on the set of adapted frames at any given point of N j . Figure 13: Interpretation of the map Ψ. The transformation g j sends body Bj from its standard configuration to the configuration that takes the adapted frame σj to the standard frame in R n , and the point bj into the line through the origin along en a distance s 2 from the origin.
We denote by F j the principal H-bundle of adapted (positive) orthonormal frames over N j of sign . Elements of F j will be written (b, σ), where σ is in the fiber F j (b). The extended configuration manifold is the product M e ∶= F + 1 × F − 2 × G × [0, ∞). We can now define the map Ψ ∶ M e → G × G by Ψ(b 1 , σ 1 , b 2 , σ 2 , g, s) = (g 1 , g 2 ) where The geometric interpretation of Ψ is shown in Figure 13. Note that points on the boundary of M correspond under Ψ to points in M e with coordinate s = 0. The groups G and H naturally act on M e on left and right, respectively: The quotient M e H is easily seen to be a smooth manifold and the projection M e → M e H is a principal H-bundle. It is also immediate from the definitions that for all ξ ∈ M e , where the action of G on G×G is defined by g(g 1 , g 2 ) = (gg 1 , gg 2 ). Therefore, Ψ induces a G-equivariant map Equivariance means Ψ(gq) = gΨ(q). The G-action on M e admits a smooth cross-section:  It is natural to expect that under reasonable assumptions the restriction of Ψ to a neighborhood of the boundary of M e H will be a diffeomorphism onto a neighborhood of the boundary of M , thus providing a useful parametrization for the purpose of understanding collisions. Figure 14 shows some of the situations that must be avoided. We give shortly a few sets of sufficient conditions for Ψ to be a local diffeomorphism, but our immediate goal is to explore M e , M e H, and their boundaries a little further.

The tangent bundle of ∂M e and ∂(M e H)
Recall that the shape operator of a hypersurface N ⊂ R n with unit normal vector field ν at a point b ∈ N is the linear map Levi-Civita connection for the standard Euclidean metric in R n . We write ∇ v X ∶= Π b D v X for a tangent vector field X on N , where Π b is the orthogonal projection from R n to T b N . This is the Levi-Civita connection on N for the induced metric. Let (b, σ) be a point in the adapted frame bundle F over N and (v, ζ) a tangent vector to F at (b, σ). Then, by differentiating ν(γ(t)) = −(−1) σ(t)e n , where (γ(t), σ(t)) is a smooth curve representing (v, ζ) at (b, σ) = (γ(0), σ(0)), we obtain where V ∶= σ −1 ζ can be regarded as an element of so(n) just as σ is viewed as an element of SO(n). The tangent bundle of F for any smooth hypersurface N has now the following description. Let (b, σ) ∈ F . Then As before, we use the canonical identification T G ≅ G × g and write elements of g in the form (Z, z) ∈ so(n) × R n . The shape operator of N j will be written S (j) . We omit the superscript when it is clear from the context to which body the operator is associated. Then the tangent space of M e at a point q = (b 1 , σ 1 , b 2 , σ 2 , g, s) is given by . Tangent spaces to ∂M e consist of those vectors for which = 0.
Let G q and H q represent the orbits through q ∈ ∂M e of the (right and left, respectively) actions of G and H on M e . The tangent spaces at q of the respective orbits will be written g q and h q . Then At any q ∈ ∂M e the differential of Ψ is where, denoting Ad σ (W ) ∶= σW σ −1 , The next proposition contains as a special case Proposition 2.1. It uses the notation for any q = (b 1 , σ 1 , b 2 , σ 2 , g, s). We allow s to be non-zero, in which case S (j) is the shape operator of the level hypersurface of M e H corresponding to value s. Assuming for simplicity that U of Proposition 5.1 is all of M e (we only need what follows on some neighborhood of the boundary of M e ), it is useful to know whether the principal bundle M e → M admits a G-invariant connection since the associated horizontal subspace H q can then serve as a proxy for the tangent space of TqM , without having to go to the quotient. A principal H-connection on M e is given by a one-form ω taking values in h and satisfying the properties: where Y q is the vector induced by the infinitesimal action of h; Let Π be the orthogonal projection from R n to R n−1 = e ⊥ n .
Proposition 5.2. For any real constants c 1 , c 2 , c 3 such that c 1 + c 2 + c 3 = 1 the h-valued one-form ω on M e given by Proof. This is a simple check.
Let us choose ω q (ξ) ∶= ΠV 1 Π and denote by H q the horizontal subspace defined by this choice of connection form. Recall the maps S j on R n−1 for q = (b 1 , σ 1 , b 2 , σ 2 , g, 0) ∈ ∂M e given by S j = σ −1 j S bj σ j . The b j are determined uniquely from Ψ(q) and the σ j are determined uniquely up to an overall common element of H acting on the right.
Proof. The proof is elementary, but we show the main point. Letξ = (Z 1 , z 1 , Z 2 , z 2 ) ∈ g×g. We wish to show the existence of a unique ξ = (v 1 , V 1 , v 2 , V 2 , Z, z, ) ∈ H q that is sent toξ under d q Ψ. The components of ξ satisfy: and Z j , z j are related to theses quantities by Writing g = (A, a) and Ψ(q) = (g 1 , g 2 ), we have g j = (Aσ −1 j , a − Aσ −1 j b j ). Note that from which we obtain and σ −1 1 v 1 − σ −1 2 v 2 ∈ R n−1 in terms of the Z j and z j . Also from which we obtain σ −1 1 v 1 in terms of the Z j and z j under the assumption that S 1 + S 2 is invertible. From Proposition 3.1, item (8), we deduce so that V 1 is also uniquely determined by the Z j and z j . From 1 Z 1 we obtain V 2 uniquely and from the above 5.4 we obtain v 2 uniquely. From these we easily obtain Z and z as well, proving the first part of the proposition. The second part follows from the observation that a vector is tangent to ∂M if and only if = 0.

The non-slipping, rolling, and diagonal subbundles
Let γ(t) be a smooth curve in ∂M e such that q = γ(0) = (b 1 , σ 1 , b 2 , σ 2 , g, 0). We omit the variable s, which is set to 0 for a boundary point. Let (γ 1 (t), γ 2 (t)) be the image of γ under Ψ and write γ j (0) = g j ,q ∶= (g 1 , g 2 ), where g j = (A j , a j ) and g = (A, a). Denote the components of the infinitesimal motion in M e by ξ ∶= γ ′ (0) = (v 1 , V 1 , v 2 , V 2 , Z, z), omitting = 0. The two bodies in configuration q are in contact at g 1 (b 1 ) = g 2 (b 2 ). Definition 5.1 (Non-slipping and non-twisting conditions). The infinitesimal motion ξ ∈ T q M e is said to satisfy the non-slipping condition if the velocities of the material points b j at the contact configuration are equal. It is said to satisfy the non-twisting condition if the tangent planes to N j at b j do not rotate relative to each other under ξ.
We now derive an explicit expression for these conditions. The infinitesimal motion of B j is given by ξ j ∶= (Z j , z j ) ∈ g, which is obtained from dΨ q ξ. We know that The non-slipping condition, V ξ1 (b 1 ) = V ξ2 (b 2 ), then reduces to Turning now to the non-twisting condition, let u j be a tangent vector to N j at b j such that, in the contact configuration given by q, is sent to a common vector, for j = 1, 2, in the plane of contact. Thus A 1 u 1 = A 2 u 2 . The infinitesimal rotation of A j u j at the point of contact is The orthogonal projection to the plane of contact is AΠA −1 , recalling that Π is the orthogonal projection to R n−1 . (It may be helpful to keep in mind Figure 13.) Because A = A j σ j , the non-twisting condition takes the form ΠV 1 Π = ΠV 2 Π and since ΠV 1 Π = 0 holds for horizontal vectors, ΠV j Π = 0 for j = 1, 2.
Now let Ψ(q) = (g 1 , g 2 ), g j = (A j , a j ) andξ = (Z 1 , z 1 , Z 2 , z 2 ) = dΨ q ξ. The non-slipping condition expressed in terms ofξ becomes and the non-twisting condition becomes for a W ∈ so(n) independent of h and w j ∈ T bj N j .
Definition 5.2 (Non-slipping, rolling, and diagonal subbundle). The non-slipping subbundle of T (∂M ) consists of all tangent vectors satisfying the non-slipping condition. The rolling subbundle of T (∂M ) consists of all tangent vectors satisfying both the non-slipping and non-twisting conditions. The diagonal subbundle of T (∂M ) is the tangent bundle to the orbits of the action of G on ∂M defined by g(g 1 , g 2 ) = (gg 1 , gg 2 ). We denote these three subbundles, respectively, S, R, D. We refer to these collectively as the kinematic subbundles of T (∂M ). Notice that D Ψ(q) = g q , using previous notation.
Starting from this definition rather than Definition 2.1, the content of the latter becomes a statement, which is proved by the above remarks.

Collision maps
Let now M ⊂ G × G be the configuration manifold of two rigid bodies in R n , where G = SE(n). By condition 2 of Proposition 5.1 M has smooth boundary and boundary points represent configurations in which the bodies are in contact at a single point. Let the state of the bodies before and after collision be given by the element of T q M , q ∈ ∂M , represented by (Z ± 1 , z ± 1 , Z ± 2 , z ± 2 ) ∈ g × g. Here the sign '+' indicates post-collision velocities and '−' pre-collision velocities. We obtained in 4.6 a condition on the pre-and post-collision velocities due to impulsive forces that act at a single point of the body. We restate it here. Let the common point of contact be Q = A j b j + a j , where b j is the material point in standard body configuration which corresponds to Q in configuration g j = (A j , a j ), where a j = x cj is the center of mass of body j in the given configuration. Then we obtain from expression 4.6: where u j = A −1 j I cj m j . We should add to these equations I c1 + I c2 = 0 for conservation of (linear) momentum.
Proof of Theorem 2.1. A simple dimension count gives dim C q = n and dim S q = 2 dim g−n so that the sum of the two dimensions equals dim T q M . Therefore, it suffices to show that these subspaces are orthogonal. The Riemannian metric on M is the restriction of the product metric on G × G (each factor having a possibly different metric as the bodies may have different mass distributions.) Explicitly, let u, v ∈ T q M and write v = ((Y 1 , y 1 ), (Y 2 , y 2 )), w = ((Z 1 , z 1 ), (Z 2 , z 2 )).

Now consider the vectors
where Z j = Z − V j and z * = z − z ′ . Observe that Then ⟨v, w⟩ q = j m j 1 2 But m 1 A 1 u 1 + m 2 A 2 u 2 = 0 by the definition of C q so the two vectors are orthogonal.
Proof of Corollary 2.3. Let C be a linear involution in O(n−1). Then C is diagonalizable over R with eigenspace decomposition R n−1 = (C + I)R n−1 ⊕ (C − I)R n−1 and eigenvalues 1, −1 having multiplicities n − k − 1 and k, respectively, where k ∈ {0, 1, . . . , n − 1}. Thus for each such C there is k and A ∈ GL(n − 1, R) such that C = A −1 J k A where J k is the diagonal matrix diag(I n−k−1 , −I k ) and I l indicating the l × l identity matrix. We can take A to be orthogonal. In fact, let A = SU be the polar decomposition of A into a positive symmetric part S = √ A † A and orthogonal part U . The condition C † C = I implies that S 2 and J k commute, from which it follows that S 2 , hence S, is also a block matrix with 0 on the off-diagonal blocks of size k × (n − k − 1) and (n − k − 1) × k. Therefore, S commutes with J k whence the claim. Thus the set of all orthogonal involutions in dimension n − 1 is the disjoint union of the sets J k = {U † J k U ∶ U ∈ O(n − 1)}. It is clear from this description that J k is the homogeneous space O(n−1) L, where L is the isotropy group of J k . Equivalently, L is the subgroup of all U that commute with J k , which is easily seen to be the product O(n − k − 1) × O(k).
The following proposition gives a concrete expression for the unit normal vector field. space at (q, v) of the base-point projection map π ∶ T M → M . We briefly recall the definition of the vertical and horizontal subbundles E V and E H of T (T M ). For simplicity of notation we denote points in T M by v rather than (q, v). Then the fiber E V v above v is the kernel of dπ v and E H v is the kernel of the connection map K v ∶ T v (T M ) → T q M , defined as follows: if ξ = w ′ (0) where w(t) is a curve through v representing ξ, then K v ξ = ∇ dt t=0 w(t). If now X and Y are vector fields on T M , then (7.1) dθ See [5] for more details. Now let S denote the boundary of M , N the pull-back to S of the tangent bundle T M under the inclusion map and for each value E > 0 define So N E is a level set of the kinetic energy function. It is shown in [5], that the pull-back of dθ to N E under the inclusion map is non-degenerate on N E ∖ T S, and so it defines there a symplectic form. If the ambient space of the system is R n then N E has dimension 2n − 2. The canonical billiard measure is now the measure associated to the (2n − 2)-form Ω = (dθ) n−1 pulled-back to N E ∖ T S. The smooth field q ↦ C q of collision maps defines a smooth map (away from singularities) C ∶ N E → N E . The pull-back of θ under this map is easily shown to be (C * θ) v (ξ) = ⟨v, C q dπ v ξ⟩ q .
Note that dπ v ξ ∈ T q S whenever ξ ∈ T v N . Define the projections Π ± q from T q S to the eigenspaces of C q associated to eigenvalues ±1. The assumption that C is parallel is equivalent to one of these projections (equivalently, both) being parallel. Now define θ ± v (ξ) = ⟨v, Π ± q dπ v ξ⟩, so that θ = θ + + θ − and C * θ = θ + − θ − . Consequently, The projections Π ± can also be defined on T N E by requiring Using these maps we define 2-forms ω ± by ω ± v (ξ, η) ∶= dθ v (Π ± v ξ, Π ± v η). We now wish to relate ω ± and dθ ± .
First define a tensor field ϑ ± on S such that for u, v ∈ T q S and any vector fields X, Y on S such that X q = u and Y q = v, we have It is not difficult to verify that this is indeed a tensor field and the definition does not depend on the extensions X, Y of u, v. Furthermore, ϑ ± vanishes under the conditions of Theorem 2.2. A straightforward calculation now shows that dθ ± v (ξ, η) = ω ± v (ξ, η) + ⟨v, ϑ ± (dπ v ξ, dπ v η)⟩.