Mechanics of affine bodies. Towards affine dynamical symmetry

J. Math. Anal. Appl. 446 (2017) 493–520

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Mechanics of affine bodies. Towards affine dynamical symmetry Jan Jerzy Sławianowski, Vasyl Kovalchuk ∗ , Barbara Gołubowska, Agnieszka Martens, Ewa Eliza Rożko Institute of Fundamental Technological Research, Polish Academy of Sciences, 5B , Pawińskiego str., 02-106 Warsaw, Poland

a r t i c l e

i n f o

Article history: Received 23 February 2016 Available online 25 August 2016 Submitted by W. Sarlet Keywords: Homogeneous deformation Structured media Affinely-invariant dynamics Elastic vibrations encoded in kinetic energy Calogero–Moser and Sutherland integrable lattices

a b s t r a c t In this paper we discuss certain dynamical models of affine bodies, including problems of partial separability and integrability. There are some reasons to expect that the suggested models are dynamically viable and that on the fundamental level of physical phenomena the “large” affine symmetry of dynamical laws is more justified and desirable than the restricted invariance under isometries. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Below we present ideas and models which are rather new and up to our knowledge quite rarely touched in the literature (the exceptions are the Marsden’s and Hughes’ theorem in their book [14] and Mariano’s theorem in [12]). In most papers we know (many of them quoted in the references [2,4,6,7,15,18,16,17,21, 26–28]) it is only kinematics that is based on affine geometry, but on the dynamical level affine symmetry is broken and restricted to the Euclidean group of motions or some of its subgroups. Unlike this we discuss models the dynamics of which is affinely-invariant. For example we show that instead of using some explicit potential energy expression one can encode the dynamics of elastic vibrations in an appropriate form of affinely-invariant kinetic energy. The resulting geodetic models in a sense resemble the procedure of Maupertuis principle, where the orbits of motion are geodetics of an appropriate metric tensor built as some conformal modification of the “true” geometric fundamental tensor [1]. There is also some similarity with the concept of effective mass known from solid state physics [9]. In a sense, our affinely-invariant * Corresponding author. E-mail addresses: [email protected] (J.J. Sławianowski), [email protected] (V. Kovalchuk), [email protected] (B. Gołubowska), [email protected] (A. Martens), [email protected] (E.E. Rożko). http://dx.doi.org/10.1016/j.jmaa.2016.08.042 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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geodetic models may be interpreted as a discretization of the Arnold description of ideal fluids in terms of geodetic Hamiltonian systems on the group of volume-preserving diffeomorphisms [1]. This is a very drastic discretization, reducing the continuum cardinality of degrees of freedom to the finite one, namely n(n + 1), where n is denoting the dimension of the physical space (n = 3 in realistic models). We mention about certain procedures based on modified finite elements methods [29] following the ideas of M. Rubin and his co-workers [19,20]. In its most traditional version the idea of finite elements was to cover the bulk of deformable body with a mesh of tetrahedrons or parallelepipeds (or sometimes other simple figures) of sufficiently small size; similarly, the surface was replaced by a mesh of triangles or parallelograms (or other simple two-dimensional figures). Those small parts were assumed to be (in a good approximation) homogeneously deformable. Basing on this assumption one replaced a system of partial differential equations by a discrete system of difference equations. And this approximation enabled one to perform directly the computer-aided numerical calculations. The time variable is then also discretized and its continuity is replaced by some mesh. But there is also another procedure, when instead of the spatio-temporal finite elements one uses also spatial ones. This hybrid method consists in an approximate representation of continuum as a system of mutually interacting affinely rigid (homogeneously deformable) bodies. But the time variable is continuous and for the mentioned system the usual methods of analytical mechanics and qualitative theory of dynamical systems are used. In various problems this hybrid discretized-analytical approach seems to be more appropriate. In any case the powerful methods of dynamical systems theory may be used. At the end of the Introduction section let us shortly summarize the results that will be presented in this article. Later on we will develop some models which are ruled by affine group (either in the physical space or in the material) not only on the kinematical but also on the dynamical level. We may expect some physical applications of those models, e.g., in the collective nuclear dynamics, for description of defects in solids, in certain special problems in mechanics of deformable objects, and so on. The presented concept of affine bodies is obtained in the framework of elaboration of collective degrees of freedom and moments methods in many-body problems, mechanics of continuum media, and field theory, which is connected with classical discretization procedures (e.g., Rietz, Galerkin, etc.) and finite-elements methods. The possibility to encode elastic interactions without the usage of the potential term in the very kinetic energy (metric tensor on the configuration space), i.e., like in the Maupertuis variational principle, is illustrated through the two-polar decomposition of internal configurations and splitting the motion into the isochoric (incompressible) and dilatational parts. The formal analogy to the lattice-like structures after the partial diagonalization of the internal part of the affinely-invariant kinetic energy has been also discussed. 2. Some basic concepts Although we admit discrete or even finite systems of material points, it is more convenient to use the standard terms of continua. Then configurations of continuous media are described by (even weak) diffeomorphisms Φ of the material space N (“Lagrange variables”) onto the physical space M (“Euler variables”) which are assumed to be affine spaces of the same dimension n (in applications n = 3, but sometimes also 2 or 1). Obviously, motion is described by the time dependence of Φ. Usually (not always) M and N are endowed with flat metric tensors g, η in M , N ; g ∈ V ∗ ⊗ V ∗ , η ∈ U ∗ ⊗ U ∗ where V , U are respectively linear spaces of translations (free vectors) in M , N , and V ∗ , U ∗ are their duals (spaces of linear functions respectively on V , U ). It is convenient to use rectangular coordinates aK , y i respectively in N and M . They are what is often referred to as Lagrange and Euler coordinates. The components of g, η are denoted by gij , ηAB ; obviously in rectilinear rectangular coordinates we have gij = δij , ηAB = δAB . The contravariant invariant metrics g −1 ∈ V ⊗ V , η −1 ∈ U ⊗ U have components traditionally denoted by g ij , η AB (the upper case indices), where g ik gkj = δ i j , η AC ηCB = δ A B .

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Analytically Φ is represented by the dependence of y i -variables on aK -variables, i.e., y i = y i (aK ). When dealing with motions, we have time-dependent Φ (t) and y i = y i (t, aK ). The deformation gradient is analytically represented by the matrix χ with components χi K = ∂y i /∂aK . Solving the above expressions with respect to the Lagrange variables, i.e., aK = aK (y i ) or aK = aK (t, y i ), we obtain the analytical description of the inverse mappings Φ−1 , Φ(t)−1 of M onto N .   The Lagrange velocity field is defined on N and takes values in V , i.e., V i (t, a) = ∂y i /∂t (t, a). The corresponding Euler velocity field is defined on M and takes values in V , i.e., v i (t, y) =

∂y i (t, a(t, y)) = V i (t, a(t, y)) , ∂t

(1)

where the partial differentiation ∂/∂t acts only on the first argument, not on the t-variable in a(t, y). One can easily show that v i (t, y) = −

∂y i ∂aK (t, y) ∂aK (t, y) i (t, a) = −χ (t, a(t, y)) . K ∂aK ∂t ∂t

(2)

In certain formulas we need locally co-moving, i.e., material expressions of those (and other) quantities. They are given respectively by K  K (t, a) := ∂a (t, a) V i (t, a) = χ−1 (t, a)K i V i (t, a), V ∂y i

vˆK (t, y) :=

∂aK K (t, a(t, y)) v i (t, y) = χ−1 (t, a(t, y)) i v i (t, y). ∂y i

(3) (4)

The Green and Cauchy deformation tensors G, C are meant as follows: ∂y i ∂y j = gij χi K χj L , ∂aK ∂aL ∂aK ∂aL Cij = ηKL = ηKL χ−1K i χ−1L j ; ∂y i ∂y j

GKL = gij

(5) (6)

to avoid the crowd of characters we do not write arguments. Using the modern notation of differential ∗ geometry, namely the pull-back symbols, we write G = Φ∗ g, C = Φ−1 η. We use also the contravariant KA ia tensors G−1 , C −1 , where, by definition, we have G−1 GAL = δ K L , C −1 Caj = δ i j . Obviously G−1

KA

=

∂aK ∂aL ij g , ∂y i ∂y j

C −1

ij

=

∂y i ∂y j KL η . ∂aK ∂aL

(7)

−1 In terms of the push-forward notation G−1 = Φ−1 , C −1 = Φ∗ η −1 . Additionally we have also metrically∗ g KL raised contravariant versions of G, C, i.e., G = η KA η LB GAB , C ij = g ia g jb Cab and mixed tensors  K L := η KA GAL , C  k l := g ka Cal . The tensorial “membership” of those objects (when taken at a fixed G  ∈ U ⊗ U ∗  L (U ), argument value) is G ∈ U ∗ ⊗ U ∗ , G−1 ∈ U ⊗ U , C ∈ V ∗ ⊗ V ∗ , C −1 ∈ V ⊗ V , G ∗  ∈ V ⊗ V  L (V ). Obviously the first four are symmetric and positively definite. The pairs of tensors C (G, η), (C, g) give invariants, which may be chosen in infinity of ways, e.g., we can take  rise todeformation   −a , a = 1, . . . , n. In the non-deformed situations we have G = η, C = g. a = T r C Ka [φ] = T r G Sometimes it is convenient to use the deformation measures vanishing in the non-deformed state, e.g., E := (G − η) /2, e := (g − C) /2 referred to as the Lagrange and Euler deformation tensors. Obviously, their tensorial “membership” is E ∈ U ∗ ⊗ U ∗ , e ∈ V ∗ ⊗ V ∗ and they are symmetric. We can also construct ∗ ∗ AC i ik  A the mixed tensors  E ∈ U ⊗ U , eˆ ∈ V ⊗ V , namely E B = η ECB , eˆ j = g ekj , and the corresponding  a , Tr (ˆ invariants Tr E ea ), which are obviously some functions of the previously introduced ones.

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Spatial gradient of the Euler velocity field (1), Ωi j = ∂v i /∂y j , is very important in hydrodynamics, nevertheless, it is also of some relevance for elasticity, and in any case it is generally well-defined in any  A B = ∂ˆ infinite continuum. One can also introduce its material representation Ω v A /∂aB . The tensorial  A B is Ω ∈ V ⊗ V ∗  L (V ), Ω  ∈ U ⊗ U ∗  L (U ). The g-skew-symmetric part “membership” of Ωi j and Ω of the first one describes the local field of angular velocity, ωi j =

 1 i  1 i Ω j − Ωj i = Ω j − gjk g il Ωk l , 2 2

(8)

and the g-symmetric part refers to deformation velocity, di j =

 1 i  1 i Ω j + Ωj i = Ω j + gjk g il Ωk l , 2 2

(9)

the quantity often used in mechanics of viscous fluids. Its material representation, DAB =    dij ∂y i /∂aA ∂y j /∂aB , i.e., strain rate, may be expressed as DAB = (1/2) dGAB /dt. If GAB is expressed as a function of Euler variables y i , then the usual time derivative is to be replaced by the substantial one. An important thing is the study of transformations and symmetries. Let A : M → M , B : N → N be, respectively, diffeomorphisms of the physical space onto itself and of the material space onto itself. Roughly speaking, they describe the spatial and material transformations. In particular, spatial and material symmetries are described in this way. Transformations A, B act on configurations Φ respectively through the following left and right superposition: φ → A ◦ Φ ◦ B = (LA ◦ RB ) (φ) = (RB ◦ LA ) (φ) ;

(10)

in the last two symbols LA and RB are to be meant as left and right regular translations by A and B. In particular, the homogeneity and isotropy (translational and rotational invariance) of space and material are described in terms of such mappings. 3. Symmetries and conservation laws The complete, systematic description of symmetries, conservation laws and their mutual relationships is based on variational principles and Hamiltonian formalism. Nevertheless, many partial results may be obtained within the more general Newton-d’Alembert framework, including the dynamics of non-conservative systems, in particular dissipative ones. Obviously, even if not used explicitly, the Lagrangian and symplectic concepts are always somehow hidden behind the treatment. Let us begin with translational invariance. In mechanics of affine bodies the problem of material translational invariance (material homogeneity) becomes diffused, and as a matter of fact, it disappears. There are two reasons for that. The first one is that on the level of dynamics one deals with global quantities obtained as mean values, integral averages performed over the material space. The second reason is that all formulas we use, in particular the one for kinetic energy, are expressed in Lagrangian coordinates vanishing at the center of mass. This fixed point and finite size of the body break translational symmetry in the material space. Nevertheless, translational symmetry in the physical space is still well defined. If the set of all a priori admissible configurations Φ is confined to the affine constraints Af I (N, M ), i.e., the n (n + 1)-dimensional manifold of affine isomorphisms of N onto M . Obviously, there are two open connected components in Af I (N, M ). They are mutually disjoint, nevertheless they infinitesimally approach each other along Afsing (N, M ) ⊂ Af (N, M ), i.e., along the closed subset of Af I (N, M ) consisting of dimension-degenerating affine mappings of N into M , i.e., such ones which “glue” different material points. Such transformations are non-admissible in continuum mechanics, nevertheless, at least some of them may

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be acceptable in mechanics of discrete affine bodies, e.g., nothing bad happens if all atoms of a four-atomic molecule in three-dimensional space are placed at some moment co-planarly or collinearly. If orientations of affine spaces N , M , i.e., those of linear spaces U , V are fixed, then the mentioned two connected components of Af I(N, M ) consist respectively of orientation-preserving and orientation-reversing affine isomorphisms. Only if we assume that N = M , then the two connected components are defined intrinsically, without fixing orientation. If we use rectilinear, in particular rectangular coordinates aK , y i respectively in N and M , then affine transformations, i.e., configurations of affine body, are analytically described in terms of linearnonhomogeneous transformations, y i = xi + ϕi K aK .

(11)

The quantities xi and ϕi K are generalized coordinates; motion is described by their time-dependence. Because Φ : N → M is an affine transformation, so it preserves all affine concepts, like straight-lines, parallelism etc. This means that there exists a linear mapping L [Φ] : U → V such that −−−−−−→ → − Φ(a)Φ(b) = L[Φ] ab

(12)

→ − for any a, b ∈ N . Obviously, the arrow symbol denotes the vector ab ∈ U originating at a ∈ N and terminating at b ∈ N . For brevity, the same symbol is used in (M, N ). It is clear that for any chain of affine spaces we have L[Φ1 ◦ Φ2 ◦ . . . ◦ ΦK ] = L[Φ1 ]L[Φ2 ] · · · L[ΦK ], assuming of course that superpositions are well defined. As a concrete example of affine bodies we can present the case of the body which can deform homogeneously in the two-dimensional central plane and simultaneously perform one-dimensional oscillations orthogonal to this central plane. This kind of situation is known in the theory of plates and shells as the Kirchhoff–Love constraints. Then the material space N is presented as the Cartesian product R+ × R2 and the group of material transformations has the form R+ × GL(2, R), where R+ is the dilatation group in the third dimension. The detailed analysis of the dynamics of such a two-dimensional affine body with “thickness” immersed into the three-dimensional physical space with imposed additional constraints of the Kirchhoff–Love type is presented in our previous works [11,10]. For an appropriate definition of model potentials (depending only on the deformation invariants) strongly nonlinear equations of motion in the general form as well as some special solutions, i.e., stationary ellipsoids, were obtained there. An interesting fact is that despite the stationary character of the obtained solutions the deformations are not constant in time, i.e., the expressions for both the Green and Cauchy deformation tensors in the general case depend explicitly on time. The obtained results can be useful in the theory of structured media for description of such bodies as one-dimensional structural elements of liquid crystals and flat molecules (H2 O, S3 , CO2 ) or supramolecular structures which consist of a large number of molecules but have an almost “flat” core. Moreover, some applications in nanophysics are possible, e.g., for description of the pure (the flat affinely rigid body almost with zero thickness) or chemical grafen with functional groups (the flat affinely rigid body with thickness that can change, e.g., depending on the environment in which the functional groups are located). In the general case we see that analytical representation (11) preassumes some choice of affine reference frames (O; E1 , . . . , E2 ), (o, e1 . . . en ) respectively in N and M . The points O ∈ N , o ∈ M are origins and EA ∈ U , ei ∈ V , where A, i = 1, . . . , n, are basic vectors. The elements of dual bases will be denoted as usual by E A ∈ U ∗ , ei ∈ V ∗ . The parameters xi refer to translational motion. They are just coordinates of the current position Φ (O) ∈ M of a fixed material point O ∈ N . The variables ϕi K are generalized coordinates of relative/internal motion. When O ∈ N is fixed once for all, our configuration space Af I(N, M ) may be identified with the Cartesian product Q = M ×LI(U, V ) = Qtr ×Qint where, obviously LI(U, V ) denotes the set of all linear isomorphisms

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of U onto V ; it is an open submanifold in L(U, V ), the linear space of linear mapping of U into V . Usually O is chosen as the center of co-moving, i.e., Lagrangian (therefore constant) mass distribution. If μ is a positive measure on N describing this distribution, then O ∈ N is uniquely defined by 

−−→ OA dμ(A) = 0.

(13)

Then oφ := Φ(O) ∈ M is the current position of the center of mass in M , thus it satisfies 

− y dμφ (y) = 0, oφ→

(14)

where μφ is the φ-transport of the measure μ to M . Equations (13), (14) are equivalent only for affine configurations Φ. If Φ is non-affine, then in general the current (Euler) center of mass oΦ ∈ M does not coincide with Φ(O). The set of affine isometries, i.e., Euclidean mappings of (N, η) onto (M, g), will be denoted by E(N, η; M, g). Analytically, in terms of (11), the membership Φ ∈ E(N, η; M, g) means that ηAB = gij ϕi A ϕj B , i.e., using the modern geometric notation, η = Φ∗ g. Obviously then G = η, C = g. This means that L[Φ] is an element of O(U, η; V, g), i.e., it is an orthogonal mapping of (U, η) onto (V, g). Just like orthogonal groups, the set O(N, η; M, g) is non-connected; it is a disjoint union of two connected manifolds. If some orientation standards are fixed in N , M (in U , V ), those are manifolds of orientationpreserving and orientation-inverting isometries. The same concerns E(U, η; V, g). The metrical rigid body configuration space is then identical with the manifold of orientation-preserving isometries. Affine constraints simplify the structure of many expressions used in mechanics of continua. The placement χi K = ∂y i /∂aK becomes constant both in the physical space and in the body; it equals the internal part of the configuration, i.e., χi K = ϕi K . Deformation tensors are also spatially and materially constant, for example the expressions (5), (6), (7) for the Green and Cauchy tensors become GKL = gij ϕi K ϕj L , Cij = ηKL ϕ−1 K i ϕ−1 L j , G−1 KL = ϕ−1 K i ϕ−1 L j g ij , C −1 ij = ϕi K ϕj L η KL . Then the expression (1) becomes the following affine (“linear-inhomogeneous”) function of the Euler variables y i : v i (t, y) =

   dϕi A −1 A  j dxi j + ϕ = V (tr)i + Ωi j y j − xj , j y −x dt dt

(15)

therefore we have dϕi A −1 A ∂v i i ϕ = Ω = j j. ∂y j dt

(16)

 AB = The co-moving (material) representation for general (not necessarily affine) bodies is given by Ω −1 A i −1 A i j ϕ i Ω j ϕ B. i dϕ B /dt = ϕ The co-moving (Lagrangian) mass distribution is described by the time-independent positive measure μ on the material space N . Any configuration Φ : N → M , not necessarily affine one, gives rise to the current (Euler) mass distribution in the physical space M . It is described by the positive measure μΦ on M , the Φ-transport of μ. Obviously, it is time-dependent because Φ is so. The total mass of the body is then   given by m = μ(N ) = N dμ(a) = μΦ (M ) = M dμΦ (x), roughly speaking, the monopole moment of μ or μΦ . Higher-order material (Lagrangian, co-moving) moments of inertia are given by the family of constant tensors in N , J(k) ∈ ⊗k U . Analytically we have  J(1)K =

 aK dμ(a),

J(2)KL =

aK aL dμ(a),

...

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 A1 ...Ak

J(k)

=

aA1 · · · aAk dμ(a),

....

(17)

All those tensors are symmetric. There are three important special cases: J(0) = m, J(1) ∈ U , J(2) ∈ U ⊗U . Usually the origin of material (Lagrange) coordinates is chosen in the reference center of mass, so by definition J(1) = 0, cf. also (13). In typical simple situations, the knowledge of the system of all J(k)-s is essentially equivalent to the knowledge of μ. Nevertheless, in the special case of affine motion it is only J(2) ∈ U ⊗ U that is relevant for dynamics. Then it is denoted simply as J without the label “2” and referred to as the “inertial tensor”. It is not the same what is called “tensor of inertia” in mechanics of rigid bodies, however, they are essentially equivalent concepts. More precisely, they are linear functions of each other. In certain considerations one uses also the Eulerian version of J. More precisely, there are two such versions: J[φ, o], J[ϕ], both being symmetric tensors, elements of V ⊗ V . Obviously, o ∈ M denotes here the fixed origin in M given by coordinates y i = 0. J[φ, o] denotes the Eulerian inertial tensor with respect to the fixed spatial origin, and J[ϕ] is the Eulerian inertial tensor related to the current position of the center of mass in the physical space, oφ ∈ M . It is easy to show that the following formulas hold: J[ϕ]ij = ϕi A ϕj B J AB ,

J[Φ, o]ij = m xi xj + J[ϕ]ij .

(18)

Unlike J, which is constant and depends only on geometry of the mass distribution in the body, J[ϕ] is configuration-dependent, therefore, also time-dependent. In dynamics of multiparticle systems, including continua, it is also convenient to use multipole moments for distributions of other physical quantities like linear momentum and forces. When dealing with affine motion we need only monopole and dipole moments of those distributions in both Euler (spatial) and Lagrange (co-moving) representations. In a general non-constrained motion the l-th order multipole moment of the distribution of linear momentum, calculated with respect to the fixed spatial origin o ∈ M is analytically given by  Ko (l)

i1 ...il j

=

 i1

il j

y . . . y v (y)dμφ (y) =

y(a)i1 . . . y(a)il V j (a)dμ(a).

(19)

As usually, μφ denotes the φ-transport of the measure μ from N to M , and v, V are, respectively, the Euler and Lagrange velocity fields; the coordinates y i are assumed to vanish at the origin o ∈ M . The tensor Ko (l) is symmetric in the first l-tuple of indices. It depends explicitly on the choice of o ∈ M . One uses also quantities Kint (l) for which the fixed reference point o is replaced by the current position of the center of mass oφ ,  Kint (l)i1 ...il j =



   y i1 − xi1 · · · y il − xil v i (y)dμφ (y).

(20)

Being tensor in V , Ko (l), Kint (l) are Euler-like quantities. In certain formulas it is more convenient to use the multipole moments with respect to Lagrange variables aB which we denote by K(l): A1 ...Al j = K(l)

 aA1 . . . aAl V j (a)dμ(a).

(21)

This is a mixed tensorial quantity, namely, the l-th order symmetric contravariant tensor in U and the usual vector in V . Obviously, the monopole moments are identical with the total linear momentum in its kinematical version i = mdxi /dt = mv(tr)i . The dipole moments Ko (1), Kint (1), K(1) “inertia × velocity”: ki := Ko (0)i = K(0) have to do with velocities of rotational and homogeneously deformative motion, combined multiplicatively

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with appropriate internal objects. The doubled skew-symmetric parts of Ko(1), Kint (1) represent respectively the total angular momentum with respect to the fixed origin o ∈ M and the spin angular momentum (one with respect to the center of mass), both in kinematical versions (vector product of the radius vectors and kinematical linear momenta). In affine motion we obtain for l = 1 the following expression: Ko ij = m xi v(tr)j + ϕi A

dϕj B AB J . dt

(22)

This is just the additive splitting into translational (“orbital”) and internal (“affine spin”) parts, Ko tr ij = m xi v(tr)j ,

Kint ij = ϕi A

dϕj B AB J . dt

(23)

The internal part does not depend on the choice of o. To avoid the crowd of symbols we denote it simply by Kij without the label “int”. The corresponding translational and internal angular momenta, both meant in the kinetic sense, are denoted respectively by Lo ij = Ko tr ij − Ko tr ji ,

S ij = K ij − K ji .

(24)

The total angular momentum with respect to o ∈ M is denoted by Jo ij = Lo ij + S ij .

(25)

In the physical dimension n = 3, the skew-symmetric tensors (24), (25) are identified with the corresponding axial vectors with components Lo i , S i , Jo i . The nice and intuitive translational-internal splitting like (22) holds also for non-affine bodies, but only for the dipole moments, i.e., for Ko (1), no longer for Ko (l) with l > 1. This fact has some deep geometric split in this way. reasons. Unlike this, all Lagrange moments K(l) To obtain the equations of motion in the form of balance laws, we need also to express the distribution of forces in terms of its multipole moments which play the role of generalized forces responsible for the dynamics of some collective modes of motion. Then the balance form of the equations of motion is obtained as a relation between corresponding moments of the distribution of linear momentum and the distribution of forces within the body. In the case of continuous media, this system of moments equations provides some kind of infinite discretizations of the problem (for details of the general theory of simple force and stress multipoles see, e.g., [8]). As expected, the monopole and dipole moments of forces are sufficient for describing affine motion. Multi (l) of the distribution of forces are given by formulas obtained from (19)–(21) pole moments No (l), Nint (l), N by substituting instead velocity field the density of forces per unit mass. So, in affine motion, when configurations are given by (11) and placements by (12), the equations of motion are given by the effective system of ordinary second-order differential equations imposed on the time dependence of generalized coordinates xi , ϕi A :

i

ϕ

K

  mx ¨i = F i t; xj , x˙ j ; ϕj B , ϕ˙ j B ,   ϕ¨j L J KL = N ij t; xj , x˙ j ; ϕj B , ϕ˙ j B .

(26) (27)

Equations (26), (27) are invariant under spatial translations when the total force (monopole moment of forces) F i and affine torque (dipole moment of forces; cf. the tensor balance of torques in theories of granular media [3,5]) N ij acting on the system do not depend on the spatial position x ∈ M . Let us observe that this does not imply the conservation of kinematical linear momentum even if F i do not depend on internal

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generalized coordinates ϕi k , on their generalized velocities and on the time variable t. Indeed, if they depend only on translational velocities v i = x˙ i , equations of translational motion are translationally-invariant in M , but as seen from the balance form of the above equations of motion, i.e., k˙ i = F i ,

K˙ ij = N ij + ϕ˙ i A ϕ˙ j B J AB = Ωi k Ωj m J[ϕ]km + N ij ,

(28)

ki is not a constant of motion. It becomes a conserved quantity only if the total force vanishes, i.e., F i = 0. If F i is constant but non-vanishing (homogeneous field of forces), then equations of translational motion are translationally-invariant, however K i is not a constant of motion, either. This is obvious, because the center of mass motion is then uniformly accelerated. Nevertheless, there is some explicitly time-dependent constant of motion somehow corresponding to translational symmetry. This is κ i := ki − F i t. Obviously, κ i /m is the initial velocity at the time instant t = 0. Galilean boosts do not preserve equations of motion, but they preserve their general solution, i.e., the set of uniformly accelerated motions (and separately preserve the subset of uniform motions). There is some prescription which associates with this symmetry some time-dependent constants of motion, i.e., ξ i = xi −

Fi 2 ki t+ t . m 2m

(29)

It is seen that those are initial coordinates at the initial time instant t = 0. In this article we are interested in problems concerning dynamical symmetries of internal degrees of freedom, so to simplify discussion we neglect translational degrees of freedom and considerate internal dynamics as autonomous one. Let us now begin with the spatial internal transformations, i.e., ϕ → a ϕ b,

(30)

where we put b = IdU . Equations of internal motion (27) and their byproducts are invariant under such transformations, i.e., their general solution is transformed onto itself if and only if the affine torque satisfies the following transformation rule: N ij (aϕ, aϕ, ˙ t) = ai k aj l N kl (ϕ, ϕ, ˙ t) ,

(31)

i.e., the prescription for N as a function of state variables is “transparent” under the left action of any a ∈ GL(V ). In terms of the co-moving description this simply means that  AB (aϕ, aϕ,  AB (ϕ, ϕ, N ˙ t) = N ˙ t) .

(32)

 must be built exclusively of quantities If such a rule is to be satisfied for all a ∈ GL(V ), then, obviously, N with the capital (material) indices. It is clear that without additional geometric objects, such a prescription does not exist. The only variable defined exclusively in the material space which we have then at disposal  and its tensorial byproducts Ω  m , m being non-negative integer; there are also invariantly defined is Ω p  are mixed ones in U , whereas N   must be twice scalars T r(Ω ). But the only second-order tensors built of Ω A  B may be obviously raised with the help of η, the material metric, contravariant. The second index of N AB A CB   = N C η ; but one must be aware that the occurrence of η restricts the material GL(U )-symmetry N to isometries O(U, η). But well, we are fighting now for GL(V )-symmetries. Therefore, any material tensors  K L , e.g., in the simplest case we might use  AB from Ω might be formally fixed and used for producing N  p )Ω  K L , etc. Obviously, such strange models of N  AB = T AB K L (T rΩ  are completely useless the scheme N for describing the elastic-like behavior. For such purposes we need the Green deformation tensor GKL (5). Incidentally, let us remind that G may be also used for shifting the material indices, like η above, although this is not always physically motivated.

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Summarizing: realistic dynamical models compatible with the idea of invariance under spatial transformations have the following form:   CD .  AB GKL , Ω  AB = N N

(33)

Incidentally, let us remind that the isotropic constitutive laws for unconstrained continua, when the second Piola–Kirchhoff stress tensor T AB is expressed as a function of the Green deformation tensor GKL  C D . Their analogy to (33) is obvious and certainly and the material representation of the velocity gradient Ω non-accidental, namely, in virtue of  N ji = −

σ ij ,

Ki = − N

 T Ki ,

 AB = − N

 T AB ,

(34)

where σ ij is the Cauchy stress field, T Ki denotes the first Piola–Kirchhoff stress tensor, and integration is performed with respect to the corresponding spatial and material rectangular coordinates, we have that (33) is obtained as the material average of the unconstrained constitutive law. The spatial invariance is then automatically reduced to g-isometries E(M, g) because G is algebraically built of the spatial metric tensor g. In the hyperelastic case, when the following formula holds:

  A B , V = −ϕi B ∂V ,  A B [V ] = Σ N ∂ϕi A

(35)

the potential energy of V of the internal O(V, g)-invariant dynamics is given by some function of G, V =  is the derivative of W with respect to G, W(G). Then, roughly speaking, N  (ϕ) = −2DG W, N

 AB = − ∂W , N ∂GAB

(36)

or, more precisely (GAB are not independent because G is symmetric), d 1  AB W(G + xε) = N εAB , dx 2 t=0

(37)

 AB obtained in this way is symmetric, and, obviously, so for any symmetric ε. By the very construction, N is N ij . Therefore, (28) implies that spin (internal angular momentum) is a conserved quantity,  d  ij dS = K − K ji = 0. dt dt

(38)

 is derived from the potential V , this conservation law is a consequence of the Noether theorem. Indeed, If N equations of motion are then derived from Lint = Tint − Vint =

1 gij ϕ˙ i A ϕ˙ j B J AB − W(G). 2

(39)

And this Lagrangian is invariant under all transformations ϕ → aϕ, a ∈ O(V, g); the resulting Euler– Lagrange equations are so as well, however the invariance of Lagrangian itself is something more. And it is just the invariance of Lagrangian that implies the conservation of angular momentum. The invariance  AB is of equations of motion alone, i.e., condition (33) does not imply spin conservation. Nevertheless, if N  symmetric, this conservation law is satisfied, even if N is structurally non-variational and dissipative forces occur. It is seen that the relationship between symmetries and constants of motion is a rather delicate matter when beyond the variational framework.

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It is seen that Noether theorem excludes higher spatial symmetries than the isometry group. This is because the metric tensor gij is explicitly present in Lagrangian, both in the kinetic energy and in the potential term, where it enters via the Green deformation tensor GAB .  in (33), one must  on G and Ω Obviously, to construct any explicit prescription for the dependence of N use some constitutive tensors in the material space U . As a rule, this restricts the a priori material symmetry group GL(U ) to some proper subgroup. A typical example is the anisotropic nonlinear Hooke law (just the sort of neo-Hookian material with tangential stiffness): N AB = C ABKL EKL =

1 ABKL C (GKL − ηKL ) . 2

(40)

The above expression is linear in E, if C is configuration-independent, but it is explicitly nonlinear in generalized coordinates ϕi K . There is an obvious analogy with the constitutive laws for anisotropic continua subject to large elastic deformations, e.g., polymer media. However, usually in such realistic nonlinear models one prefers rather the situation when nonlinearity appears already on the level of the very relationship between E and N . The relationship (40) between E and N is invariant under such material mappings which preserve the tensors C, η. They form a subgroup of the material orthogonal group O(U, η). The simplest possibility is, however to rely only on the material metric η. Then (40) becomes the isotropic “nonlinear Hooke law”, and C is given by C ABKL = λη AK η BL + μη AB η KL . Such a model is invariant under O(U, η) and it is the simplest model with the symmetry group O(V, g) ×O(U, η) acting through (30). It was mentioned above that in certain formulas the material indices are shifted with the help of the Green deformation tensor G[ϕ]. One can modify (40) along such lines, namely replacing the constant (configuration-independent) constitutive ABKL = λG−1AK G−1BL + μG−1AB G−1KL . We do not tensor C by the configuration-dependent tensor C quote the corresponding formula for N . In spite of the formal similarity of the above two formulas for C, the second model is completely different from the previous one, in particular, its nonlinearity is much stronger.  K L := η KA GAL . Being linear mappings in U , It is convenient to use the mixed Green deformation tensor G  may be multiplied by each other and give rise to monomials like G b G  d , etc., where  Ω a Ω c Ω the tensors G,  a, b, c, d are integers, non-negative ones when used with Ω. The traces of those monomials are scalars in U , some O(U, η)-invariants. Combining the monomials with coefficients depending on the mentioned scalars,  Then  Ω. one obtains some mixed tensors, elements of U ⊗ U ∗ ∼ = L(U ), correctly defined as functions of G, AB AB raising their second indices with the help of η or G , one obtains twice contravariant material tensors, AB   elements of U ⊗ U , just N (G, Ω) in (33). Seemingly one might think about infinite series defining (33), however, the number of essential monomials will be finite in virtue of the Cayley–Hamilton theorem. The class of models described above and based on using merely the material metric η for producing  in (33) is distinguished among other ones by the very geometry of the material space.  N from G and Ω Nevertheless, other constitutive material tensors are also admissible. The peculiarity of the situation when  into N  Ω  is that one deals the prescription (33) uses only the material metric η as a tool for “gluing” G, then simultaneously with the spatial g- and material η-isotropy. Let us now just consider the problem of material affine symmetry and corresponding conservation laws. As mentioned, the only thing to be discussed is the action of center-affine material symmetry group isomorphic with GL(U ). Again, if we start from the invariance of equations of motion, without any reference to Lagrangian (existing or not), we conclude that the general solution is transformed onto itself if and only if the following two conditions hold for any b ∈ GL(U ): bK M bL N J M N = J KL ,

N ij (ϕb, ϕ˙ b, t) = N ij (ϕ, ϕ, ˙ t) .

(41)

This is the invariance under (30) with a = IdV . Therefore, b must be J −1 -orthogonal, i.e., b ∈ O(U, J −1 ), and N must be a function of J[ϕ], Ω, where J[ϕ]ij = ϕi A ϕj B J AB and represents the internal inertia with

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  respect to the space-fixed reference frame, i.e., N ij = N ij J[ϕ]ab , Ωk l . In the hyperelastic case this means that the internal potential energy V depends on ϕ through J[ϕ], i.e., V = V (J[ϕ]). Obviously, the only possibility to construct scalars from the twice contravariant tensor J[ϕ]ab is to use some other tensor in the physical space, e.g., twice covariant one (the simplest possibility). Similarly, in the elastic case one must use some additional tensor objects in V to construct N ij from J[ϕ]ab . Obviously, the most natural possibility is just g itself, i.e., the metric tensor of the physical space. Usually we are interested in problems of spatial and material isotropy, i.e., invariance of equations of motion under (30) with a, b running over the symmetry groups O(V, g), O(U, η) respectively. The material isotropy of equations of motion is possible only when the inertial tensor is isotropic, i.e., J AB = Iη AB . Then J[ϕ] above has to be replaced by the inverse Cauchy deformation tensor C[ϕ]−1 (6), thus, we can write   N ij = N ij C[ϕ]ab , Ωk l . In the case of hyperelastic body we have, in analogy to (36), N (ϕ) = −2DC W,

N ij = −∂W/∂Cij ,

(42)

where the potential energy V depends on ϕ through C, i.e., V (ϕ) = W(C). Let us summarize the above invariance analysis of affine dynamics:  is an 1. Equations of motion (27) are invariant under internal spatial rotations when the affine torque N  algebraic function of G, Ω, i.e.,    .  =N  G, Ω N

(43)

More precisely, one should explicitly insert into this expression some constitutive material tensors CU in the material space U ,    CU .  =N  G,  Ω; N

(44)

 And  from the quantities G,  Ω. Those tensors are algebraically necessary for producing the quantity N physically they give an account of the structure of internal interactions. 2. Equations of internal motion (27) are invariant under material rotations in U when the inertial tensor J is isotropic, i.e., J AB = Iη AB , and the Eulerian torque N is algebraically built of C, Ω, N = N (C, Ω).

(45)

And again some spatial constitutive tensors CV in V are algebraically necessary for prescribing N as a function of C, Ω, so more precisely, we have N = N (C, Ω; CV ).

(46)

3. Equations of motion are simultaneously spatially and materially isotropic when both (45), (46) hold and are equivalent. But this means that in (44) CU is built algebraically of η and in (46) CV is built algebraically of g, thus we have two equivalent representations:  η),  =N  (G, Ω; N

N = N (C, Ω; g)

(47)

 η),  =N  (G,  Ω; N

 Ω; g), N = N (C,

(48)

or

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 C  denote the mixed tensors obtained from G, C respectively by the η- and g-shift of where, as usual, G, indices. Unfortunately, there is some disorder in notation, because characters are missing. For example, the “roof” symbol is used to denote the co-moving representation, but at the same time, it is also used for the metric-based shift of indices. It would be perhaps better and certainly non-ambiguous to use the symbols like η G, g C for the η- and g-raising of the first index, A

(η G)

B

:= η AC GCB ,

i

(g C)

j

:= g ik Ckj ,

(49)

 η , Σg , Ω  η , Ωg , N η , Ng , etc. for the metrical raising/lowering of the second index: and similarly Σ 

η Σ

AB

 A C η CB , := Σ



ij

(Ωg ) := Ωi k g kj ,



η Ω

ij

(Σg ) := Σi k g kj , η N

A B

 AC ηCB , := N

AB i

(Ng )

j

 A C η CB , := Ω

:= N ik gkj .

(50)

Let us repeat more carefully what we said above about the structure of affine torque for the doubly invariant models, using conventions (49), (50) and also the following ones concerning the metrical transposition of mixed tensors X ∈ U ⊗ U ∗  L(U ), Y ∈ V ⊗ V ∗  L(V ), i.e., 

XT

A B

= ηBC X C D η DA = XB A ,

 T i k mi Y = Yj i . j = gjk Y m g

(51)

Obviously, those are metric-dependent operations, to be quite pedantic, we should have written X T (η) , Y T (g) , however, we avoid the crowd of symbols and always keep metric tensors η, g implicitly assumed.  For velocity-independent internal forces, e.g., elastic ones, the doubly-isotropic affine torques N (g-isotropic in physical space, η-isotropic in the material space) have the following algebraic structure: η = N

n−1

a

la (K1 , . . . , Kn ) (η G) ,

(52)

a=0

where, obviously, the matrix exponents are meant, and Ki are basic deformation invariants, e.g., in the form   η A B = n−1 la (Ki ) η GA C η GC D . . . η GL B Ki =T r η Gi , and la are scalar functions of Ki . Analytically, it is N a=0    a factors  n−1 AC DE M N LB AB  or, equivalently, N = a=0 la (Ki ) η GCD η ...η GN L η . This resembles some known constitu   a factors of G

tive rules used in continuum mechanics, as expected in view of (34). Nevertheless, the above formulas may be derived without averaging continuum mechanics expressions. They apply also to discrete affine bodies like molecules (e.g., fullerens) in an appropriate approximation. Obviously, from the naive point of view one might have expected the infinite series in (52), but of course, this series compresses to the finite sum in virtue of the Cayley–Hamilton theorem. Because of the same reason the summation may be extended over any range of integers from some m ∈ Z to (m + n − 1). Obviously, this changes the functions la . As usual in matrix calculus, for any square matrix X, the zero-th exponent is taken to be identity matrix X o = I. When written in terms of M -spatial geometric objects, (52) becomes respectively Ng =

n−1

 a la (K1 , . . . , Kn ) Cg−1 ,

 −1 i  −1 im Cg gmj , j = C

(53)

a=0

 −1 i  −1 k  −1 r Cg or, in other words, N ij k Cg m . . . Cg j    factors  −1 ik  −1 mn  a −1 zj  −1 rs n−1 C C C l (K ) g . . . C g . a i km sz a=0   

equivalently, Ng i j

=

n−1

a=0 la (Ki )

a factors of C

=

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If the internal forces are derivable from some potential depending only on deformation invariants, i.e., V = U (K1 , . . . , Kn ), then the above formulas are characterized by the special form of controlling functions la , namely, la (Ki ) = −2a∂U/∂Ka . From the formal point of view, (53) is much more general than the above special form, because nothing like the vanishing of the expression b (∂la /∂Kb )−a (∂lb /∂Ka ) is assumed (which takes place when we define this controlling function as la (Ki ) = −2a∂U/∂Ka ). But, one must mention, quite often some doubts are raised if non-potential velocity-independent forces are physically realistic (although, of course, mathematically well defined). Let us now go to velocity-dependent affine torques, i.e., we mean their dependence on internal velocities. Everything expressed in the formulas (43)–(48) is still true, but the analogues of (52), (53) are more complicated. Namely, (52) has to be replaced by a sum of monomials like     ρ  σ  T (η) . . . (η G)γ Ω  T (η) . κ Ω λ Ω  Ω  T (η) (η G)α Ω la Inv η G, Ω,

(54)

 ρ α  κ  T (η) Ω In other words, we take some products of linear operators (η G) Ω , multiply those monomials by η T (η)  Ω  coefficients depending on scalar invariants built of G, Ω, , and take the sum of resulting expressions. Obviously, the mentioned scalars are traces of operator monomials. The exponents at η G are integers and  Ω  T (η) are non-negative integers. As usual, it is sufficient to take exponents from the range those at Ω, (0, . . . , (n − 1)); this is a consequence of the Cayley–Hamilton theorem. Similarly, (53) is replaced by a sum of the corresponding operator monomials in L(V ),     ρ  σ α γ la Inv g C, Ω, ΩT (g) (g C) Ωκ ΩT (g) . . . (g C) Ωλ ΩT (g) .

(55)

Literally meant expressions like (54), (55) are, so-to-speak, “neurotically” too general. They represent what algebra does answer to the inquiry concerning the most general doubly isotropic prescriptions for the affine torque. But this is pure algebra, in physics only some special simple models are realistic.  , very often is given as the In applications the affine torque N , just as its co-moving representation N sum of two terms: one depending only on the configuration ϕ and the other one describing generalized forces which depends in an essential way on velocities. The first term describes in particular the elastic and hyperelastic behavior. If one deals only with purely internal interactions, then it is g-isotropic, whereas the constitution of the object does not distinguish any “material” direction, then it is η-isotropic. These  two situations are described respectively by (43), (44) but without the Ω-dependence and by (45), (46), so    =N  (G; η), respectively N = N (G; CU ), N = N (C; CV ). For the doubly-invariant models this reduces to N N = N (C; g) and this may be alternatively written as (52), (53). If the system is potential, e.g., hyperelastic, then la (K) = −2a∂U/∂Ka holds. What concerns velocity-dependent forces, the simplest model in the special case of discretized continua is given by the following expression: 

ij Nint.diss

    det[gij ] det ϕi A η Ωij + Ωji det[ηAB ]      i  2η det[gij ] k ij det ϕ A Ω kg , − Volo ζ− det[ηAB ] n

= −Volo

(56)

which is obtained on the basis of linear isotropic viscoelasticity: 

ij σvis

2η = 2η d + ζ − n ij

 gab dab g ij ,

dij =

 1  ij Ω + Ωji , 2

(57)

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507

and equations (34). However, this does not seem adequate when one deals with initially discrete systems (when we have no predefined continuous notions of viscosity coefficients), and then it is rather more natural to postulate the internal friction term in affine torques in more general form:   Nint.diss = −α Ωg + ΩgT − β T r(Ω) g −1 ,

(58)

or using the notation (9) and (50), we can rewrite it as follows: Nint.diss

g

  = −α Ω + ΩT (g) − β T r(Ω)IdV = −2αd − β T r(d)IdV .

(59)

Obviously, IdV denotes here the identity operator in V , and α > 0, β > 0. Let us notice that in the physical case of weakly compressible (almost isochoric) objects, the formula (56) contains higher-order terms, nonlinear in state variables. It is seen that (59) is nothing else but the very special case of (55). Using the technique of invariant tensor expressions we can write the general expression for the isotropic fluid-type internal dynamics in the following form:

Nint.diss

g

=

n−1

fa (L1 , . . . , Ln ) da ,

(60)

a=0

  where La are scalar invariants built of d according to the standard trace prescription Lb = T r db . The above formula may also contain the non-dissipative pressure term Npr g = −p In , i.e., Npr = −p g −1 . Obviously, if the internal friction is anisotropic, the prescription for N as a tensorial function of d, must ij contain some constitutive tensors, e.g., in the linear case Nint.diss = −C ijab dab , the shift of indices meant in the metrical g-sense. It is easy to reformulate the above expressions into language of co-moving geometric objects (tensors in U ). Finally, let us mention about other very important dissipative problems, namely, the external surface friction. Applications are obvious: imagine a homogeneously deformable small suspension or inclusion moving in fluid. It is not only translational motion but also the internal motion in ϕ-degrees of freedom that is faced with frictional obstacles, we mean the friction between the surface of “suspension/inclusion” and the surrounding medium. As usual, the simplest and most natural assumption is that of generalized friction forces linear in generalized velocities, e.g., in the isotropic case Next.diss g = −ν Ω. This simple expression, however, looks rather not very adequate, because it is physically natural to expect that the internal motion is obstacled different way in the special cases of rotational, shear-like, and dilatational motion. So, it is reasonable to suppose something like ij ij ij ij Next.diss = Next.diss.rot + Next.diss.shear + Next.diss.dil   1 = −α ω ij − β dij − gab dab g ij − γ gab dab g ij n   β gab dab g ij , = −αω ij − βdij − γ − n

(61)

where α, β, γ are positive constants and the meaning of symbols ω, d is like in (8), (9) respectively. Obviously, one can also discuss anisotropic models when the above constants α, β, γ are replaced by some fourth-order tensors. Another a priori possible modification is the external friction nonlinear in velocities. In the isotropic a case this will be again obtained from the combination of operator monomials (dg ) , a = 0, . . . , n − 1, with  b coefficients depending on the scalar invariants of d, i.e., quantities T r d , b = 0, . . . , n − 1.

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4. Affine dynamical symmetry Two very important and at the same time very delicate points have to be stressed here. Let us discuss them to be able to finish this step of investigation with some conclusions opening the perspective on further developments. 1. There are actually nine ways of deriving equations of motion of complex and constrained systems [13], but two most popular ones are as follows: (a) procedure based on Newton equations and d’Alembert principle, (b) procedure based on the variational Hamiltonian principle, Hamiltonian formalism and Poisson brackets. 2. Kinematics of our system is based on affine geometry, however, its dynamics is not invariant under the action of the affine group. The highest dynamical invariance we were dealing with above, was that under isometry groups, both in the spatial and material sense. Therefore, there is only partial analogy between our equations of motion and gyroscopic Euler equations. This is at least aesthetically non-satisfactory and disappointing, and brings about some questions concerning the status of dynamical affine symmetry in mechanics and fundamental physics. The procedure 1a in the item 1 is rather more popular among specialists in continuum mechanics. It is directly applicable both to conservative and dissipative systems. The balance form of equations of motion appears there in a rather natural way, nevertheless, there is no direct and systematic relationship between symmetries and conservation laws, only some intuitive hints do exist. Unlike this, the procedure 1b offers a systematic theory of that relationship; it is based on Noether theorems. On the other hand, dissipative terms of equations of motion are then introduced “by hand”, and they are always more or less external, exotic corrections, a “foreign body” in Hamiltonian framework. But the balance form of equations of motion appears there simply in a canonical way. The item 2 is strongly related to that problem. Namely, the search of affinely-invariant dynamical models is much more easy within the Hamiltonian approach with its direct relationship between symmetries and conservation or balance laws. Namely, it is seen that the first and main obstacle against dynamical affine symmetry is due to the position of the spatial and material (reference) metric tensors g, η respectively in the “usual” expression for the kinetic energy. In other words, it is due to the very particular Euclidean structure of the configuration space, namely, one implemented by Euclidean structures in the physical and material spaces M , N according to T = Ttr + Tint =

m 1 gij x˙ i x˙ j + gij ϕ˙ i A ϕ˙ j B J AB , 2 2

(62)

this metric Γ is given by Γ = mg ⊕ (g ⊗ J) ,

Γ−1 =

 1 −1  −1 g ⊕ g ⊗ J −1 . m

(63)

However, from the purely geometric point of view other metrics on the configuration space are much more natural, ones partially or completely independent of metrics in M , N . The alternative geometrically-motivated version of expressions (62), i.e., Ttr =

m GAB vA vB , 2

Tint =

1  AK Ω  B L J KL = 1 gij Ωi k Ωj l J[ϕ]kl , GAB Ω 2 2

(64)

is a good starting point. Whereas in (62) we were dealing with a quadratic form of generalized velocities with constant coefficients built of g and J, in (64) the kinetic energy is expressed as a quadratic form of geo-

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509

metrically nicely-interpretable non-holonomic velocities, however with configuration-dependent coefficients built of G [ϕ] and J or g and J [ϕ]. Summarizing: either we have constant coefficients but representation of velocities non-adapted to geometry of the problem, or geometrically-nice affine velocities but variable coefficients. Therefore why not to take the “good” features from both schemes and just to unify, join together their advantages?  A B with constant coefficients. There are two possibilities of expressions quadratic in v i and Ωi j or vA and Ω The first one consists in replacing GAB in (64) by η, i.e., Ttr =

m ηAB vA vB , 2

Tint =

1  AK Ω  B L J KL . ηAB Ω 2

(65)

The second possibility fixes the spatial metric g and some additional spatial tensor h ∈ V ∗ ⊗ V ∗ substituted instead of the configuration-dependent J [ϕ]: Ttr =

m gij v i v j , 2

Tint =

1 gij Ωi k Ωj l hkl . 2

(66)

Let us observe that (65) may be alternatively written as follows: Ttr =

m C [ϕ]ij x˙ i x˙ j , 2

Tint =

1 C [ϕ]ij ϕ˙ i A ϕ˙ j B J AB . 2

(67)

This is like (62) but the difference is that the usual metric tensor g is replaced by the Cauchy deformation tensor C[ϕ]. Because of this there is something like the mentioned similarity to the concept of effective mass in solid state physics. Similarly, (66) may be written down as follows: Ttr =

m gij x˙ i x˙ j , 2

Tint =

1 AB gij ϕ˙ i A ϕ˙ j B h [ϕ] , 2

(68)

where h[ϕ]AB = (ϕ−1 )A i (ϕ−1 )B j hij is the co-moving configuration-dependent representation of the spatial tensor h. For the special case of maximal symmetry under isometries in U , V , i.e., J KL = Iη KL , hij = Ig ij respectively, we obtain for (65), (66) the following expressions: Tint =

I  AK Ω  B L η KL , ηAB Ω 2

Tint =

I gij Ωi k Ωj l g kl . 2

(69)

An important property of the model (65) is that it is invariant under the total affine group GAf (M ) in the physical space. What concerns material invariance, the first term in (65) is invariant under orthogonal group O(U, η) and the second one is invariant under O(U, η) ∩ O(U, J −1 ). The latter group becomes simply O(U, η), when then internal tensor J is isotropic. As seen from (67), the corresponding metric tensor Γ of the configuration space is given as follows: Γ = m C[ϕ] ⊕ (C [ϕ] ⊗ J) ,

Γ−1 =

  1 −1 C[ϕ]−1 ⊕ C [ϕ] ⊗ J −1 . m

(70)

Unlike (63), it is curved, i.e., the corresponding geometry in the configuration space Q is essentially Riemannian. It has a large isometry which contains GAf (M ) acting through (10) with B = IdN and O(U, η) ∩ O(U, J −1 ) (in particular just O(U, η) when J = Iη −1 ) acting through (30) on Qint and trivially on M . Quite symmetrically, the model (66) is invariant under the total GL(U ) acting through (30) on the configuration space and under E(M, g) ∩ E(M, h) (in particular, under E(M, g) when h = Ig −1 ) acting through (10) with B = IdN . Then the corresponding metric tensor Γ on Q is given as follows: Γ = m g ⊕ (g ⊗ h [ϕ]) ,

Γ−1 =

 1 −1  −1 −1 g ⊕ g ⊗ h [ϕ] . m

(71)

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The internal parts of metric tensors (70), (71), i.e., the corresponding metric tensors on Qint , do factorize into tensor products of V - and U -terms, just like (63). But if we once dare to give up the d’Alembert form (62) there is no reason any longer stick to factorization. The most general kinetic energy (Riemannian metric) on Q splitting into translational and internal parts and affinely invariant in M (invariant under GAf (M ) acting through (10) with B = IdN ) has the following form: T = Ttr + Tint =

m 1  AB Ω C D, ηAB vA vB + LB A D C Ω 2 2

(72)

where the coefficients L are constant (obviously they are components of some fourth-order tensor in U ).  they are symmetric in bi-indices, i.e., LB A D C = Moreover, being coefficients of a quadratic from of Ω LD C B A . It is clear that the metric tensor Γ underlying (72) has the following form:  A  C Γ = mC [ϕ]ij dxi ⊗ dxj + LB A D C ϕ−1 i ϕ−1 j dϕi B ⊗ dϕj D .

(73)

It is curved and there are no generalized coordinates in which its components may become constant. Let us observe that the dϕ ⊗ dϕ-part is autonomous, unlike this, the dx ⊗ dx-part is ϕ-dependent. Similarly, for kinetic energies affinely-invariant in N we have T = Ttr + Tint =

m 1 gij v i v j + Rj i l k Ωi j Ωk l , 2 2

thus, the corresponding metric tensor Γ on Q has the following form:  A  B Γ = m gij dxi ⊗ dxj + ϕ−1 j ϕ−1 l Rj i l k dϕi A ⊗ dϕk B ,

(74)

(75)

where the internal and translational parts are mutually independent and R is a constant fourth-order tensor in V , symmetric in biindices, just like L. There is no model of kinetic energy, i.e., no Riemannian structure on Q which would be affinely-invariant simultaneously in N and M . The reason is that the affine group is not semisimple and its translations subgroup is a normal divisor, e.g., the model (72), (73) is affinely-invariant in M but its maximal group of N -symmetries consists of isometries in U . And conversely, the model (74), (75) is affinely-invariant in N but in M it is invariant at most under isometries. The mentioned situations of maximal two-side symmetry occur when the tensors L, R are algebraically built respectively of (η, IdU ) or (g, IdV ). For kinetic energies (metric tensors on Q) affinely-invariant in M and only isometrically invariant in N we have LB A D C =

I A B ηAC η BD + δ B C δ D A + δ B A δ D C , 2 2 2

(76)

where I, A, B denote constants which are generalized scalar moments of inertia in affine motion. Similarity, for models affinely-invariant in N and only isometrically invariant in M we have Rj i l k =

I A B gik g jl + δ j k δ l i + δ j i δ l k 2 2 2

with the same meaning of constants. Therefore, explicitly we have     B  2 I T Ω 2 + T r Ω  ,  + A Tr Ω Tint = T r Ω 2 2 2    B  A I 2 Tint = T r ΩT Ω + T r Ω2 + T r (Ω) , 2 2 2 respectively for (76) and (77).

(77)

(78) (79)

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511

 ∈ L(U ), Ω ∈ L(V ) is always meant in the metrical sense, Let us stress that the transposition of Ω DC = Ω  T )A B = η AC ηBD Ω  B A , (ΩT )i j = g ik gjl Ωl k = Ωj i . respectively of the metric tensors η, g, i.e., (Ω The two last terms in (78), (79) are pairwise identical, but nevertheless it is convenient to distinguish  and Ω under the corresponding trace expressions. This is not only consequently between the symbols Ω more “aesthetic”, but also prevents from some unintentional mistakes in calculations. There exist models of Tint simultaneously invariant under GL(V ) and GL(U ). They correspond to the vanishing value of I:  2   A   2 B  = A T r Ω2 + B T r (Ω)2 . Tint = T r Ω (80) + Tr Ω 2 2 2 2 Except the singular case n = 1 such a “kinetic energy” (metric tensor on Qint ) is never positively definite. Nevertheless it may be physically useful and the negative configurations may be interpreted as an alternative description of elastic forces, without any use of potential energy term, just within the framework of purely geodetic models. The first term in (80) has the signature (n(n + 1)/2, n(n − 1)/2). Obviously, this is the main term and the second one is a mere correction because for A = 0 the corresponding “metric” would be strongly degenerate. For the special case A = 2n, B = −2 (or, more “generally”, for the ratio A : B = n : (−1)) one obtains the Killing “metric” on the linear group (or, more “generally”, something proportional to it). Obviously, this “metric” is degenerate and has a one-dimensional singularity corresponding to the dilatational normal divisor of the linear group. For the generic choice of (I, A, B) the metrics/kinetic energies (78), (79) are non-degenerate. There exists an open subset of triples (I, A, B) ∈ R for which these metrics are positively definite (Riemannian). With exception of the singular dimension n = 1, for all such triples I must be non-vanishing. It is instructive to notice that the total kinetic energy corresponding to (65), (78), i.e., for the M -affine and N -metrical models, may be written down in the alternative forms: m I KM Ω  LN + A Ω J I + B Ω J J IJ Ω II Ω ηAB vA vB + ηKL η M N Ω 2 2 2 2 m I A B Cij v i v j + Ckl C mn Ωk m Ωl n + Ωi j Ωj i + Ωi i Ωj j . = 2 2 2 2

T =

(81)

Similarity, for the kinetic energy corresponding to (66), (79), i.e., one affinely invariant in N and metrical in M , we have the following forms: m I KM Ω  LN + A Ω IJ Ω J I + B Ω II Ω J J GAB vA vB + GKL GM N Ω 2 2 2 2 m I A B gij v i v j + gkl g mn Ωk m Ωl n + Ωi j Ωj i + Ωi i Ωj j . = 2 2 2 2

T =

(82)

A complete description of the scheme of breaking the affine symmetry and reducing it to the metrical one is achieved when some additional metric-dependent terms are admitted. The corresponding expression for the kinetic energy, i.e., for the metric tensor on Q, is given as follows: T =

1 1  (m1 GAB + m2 ηAB ) vA vB + I1 GKL GM N + I2 ηKL η M N 2 2  K  LN + A Ω IJ Ω J I + B Ω II Ω J J  MΩ + I3 GKL η M N + I4 ηKL GM N Ω 2 2

(83)

or, alternatively, T =

1 1 (m1 gij + m2 Cij ) v i v j + (I1 gkl g mn + I2 Ckl C mn 2 2 A B + I3 gkl C mn + I4 Ckl g mn ) Ωk m Ωl n + Ωi j Ωj i + Ωi i Ωj j . 2 2

(84)

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512

If the translational degrees of freedom are active, then the two-side affine invariance is not possible. It is possible only when we formally put m1 = 0, m2 = 0 (translational degrees of freedom neglected), and in addition I1 = 0, I2 = 0, I3 = 0, I4 = 0. Then the metric tensor on Qint is affinely-invariant both on the left (in space) and on the right (in the body). The total affine invariance in space is obtained when m1 = 0, I1 = 0, I3 = 0, I4 = 0. The total affine invariance in the material sense corresponds to the choice: m2 = 0, I2 = 0, I3 = 0, I4 = 0. For any choice of constants in (83), (84) the corresponding kinetic energy (metric tensor on Q) is invariant under spatial and material isometries. All those metrics are curved (essentially Riemannian), except the special case m2 = 0, I1 = 0, I2 = 0, I4 = 0, A = 0, B = 0. In this special case the metric (83), (84) becomes flat (Euclidean) and reduces to (62), i.e., (64), or, more precisely, to its particular case J AB = Iη AB , i.e., the spherically symmetric top subject to homogeneous deformations. Therefore, m1 = m, I3 = I, m denoting the usual mass of the body and I the scalar inertial moment of the isotropic top. It is both easy and instructive to write down explicitly the Riemannian metrics Γ on the configuration space Q, underlying the above kinetic energies. Namely, for (83), (84) they are given by Γ=

 1 AB (m1 gij + m2 Cij ) dxi ⊗ dxj + I1 gij G−1 + [I2 Cij + I3 gij ] η AB 2  AB A B A B + I4 Cij G−1 + A ϕ−1 j ϕ−1 i + B ϕ−1 i ϕ−1 j dϕi A ⊗ dϕj B .

(85)

This is the family of metric tensors on Q, ordered in a hierarchic way on the basis of their isometry groups. As mentioned, certain choices of constants correspond to isometry groups containing GL(V ) acting on the left, GL(U ) acting on the right, and sometimes both of them if translational degrees of freedom are neglected. More general choices of controlling parameters correspond to situations when the isometry groups in Q are smaller and based only on isometries in M, N . If they contain transformations induced by affine isomorphisms of M or/and N , then certainly the metrics Γ are curved (essentially Riemannian). Their useful feature is that such affine models may describe bonded elastic vibrations without any use of potential energy. The elastic dynamics may be encoded then in the very form of appropriately chosen kinetic energy, as a purely geodetic motion in Q. Without affine invariance this would be impossible. In particular, if m2 = 0, I1 = 0, I2 = 0, I4 = 0, A = 0, B = 0, the corresponding metrics on Q are flat (Euclidean) and the general solution consists of straight-line in Q, evidently non-bounded, non-physical behavior. When dealing with models admitting hypothetic affine symmetry, at least partial one, we cannot rely upon the d’Alembert principle in its traditional formulation. The only natural procedure one has at disposal then, is based on variational principle and Hamiltonian formalism. Dissipative forces are then postulated as some correction terms motivated by some phenomenological and intuitive guiding hints. This procedure, based on Poisson brackets and Legendre transformation, which was described earlier, however it was specialized there to non-affine models. Let us now write some explicit formulas for metrics affinely invariant in M or in N . In the case of affine symmetry in space, (72), (73), Legendre transformation has the form: pA = m ηAB vB ,  A B = LA B C D Ω  D C . Inverting it we obtain the following formula for the kinetic Hamiltonian, i.e., expression Σ of energy through canonical variables: T = Ttr + Tint =

1 AB 1 BA Σ DC , η pA pB + L A B C D Σ 2m 2

(86)

where L A B K L LL K C D = δ A D δ C B . The corresponding contravariant inverse of the metric tensor Γ (73) is given by Γ−1 =

1 ∂ ∂ ∂ ij ∂ C[ϕ]−1 ⊗ + L A B C D ϕi A ϕj C ⊗ . i j i m ∂x ∂x ∂ϕ B ∂ϕj D

(87)

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513

Similarity, for the affine models (74), (75) Legendre transformation is given by pi = m gij v j , Σi j = Ri j k l Ωl k , and the phase-space expression for kinetic energy becomes T = Ttr + Tint =

1 ij 1 a c b d g pi pj + R b d Σ aΣ c, 2m 2

(88)

a b k l Rl k c d = δ a d δ c b . The corresponding inverse metric has the following form: where R Γ−1 =

1 ij ∂ ∂ i a j b ϕa B ϕb D ∂ ⊗ ∂ . g ⊗ +R m ∂xi ∂xj ∂ϕi B ∂ϕj D

(89)

For the general models (72), (74) it is rather difficult to find the explicit expressions for (86), (88), i.e., for R. It is also difficult for more specified models (83), (84) controlled by eight scalar the inverse coefficients L, coefficients. However, it may be easily done explicitly for the very special models (76), (78) and for (77), (79), and the more so for the simplest models based on (80), i.e., corresponding to vanishing I. And it is just these particular models which seem to be most useful in dynamical applications and theoretical analysis. In the sector of internal variables (relative motion) Legendre transformations for the L-models (76), (78) and R-models (77), (79) have respectively the following forms:  A B = LA B C D Ω DC , Σ i

Σ

j

=

Ri j l k

k

Ω l,

M M ,  K L = I η KM ηLN ΩN M + A Ω  K L + B δK L Ω Σ i

Σ

j

=Ig

im

n

gjn Ω

m

i

+AΩ j +Bδ

i

j

m

Ω

m.

(90) (91)

In the sector of translational variables, we have respectively pA = mηAB vB ,

pi = mgij v j .

(92)

Obviously, (90) and the first formula in (92) may be as well expressed through the spatial V -representation, whereas (91) and the second formula in (92) may be expressed through the material U -representation. This is, however, non-natural (although sometimes useful in a sense). The constant tensors on the right-hand sides are then replaced by ϕ dependent ones. For example, in (90)–(92) the tensors η, g are replaced respectively by the Green and Cauchy deformation tensors G [ϕ], C [ϕ]. In general it is rather difficult to inverse effectively the first formulas in (90), (91), however this may be easily done for their special cases, i.e., the second formulas in (90), (91), and it is just these special cases that are particularly useful both from the point of view of geometry and applications. One can easily obtain then the explicit form of Hamiltonian formalism. The above-mentioned inverses may be respectively expressed as follows:  K L = 1 η KM ηLN Σ N M + 1 Σ M M ,  K L + 1 δK LΣ Ω I A B 1 1 1 Ωi j = g im gjn Σn m + Σi j − δ i j Σm m , I A B

(93) (94)

    A, B, are given by I = I 2 − A2 /I, A = A2 − I 2 /A, B = where the inverse inertial constants I, − (I + A) (I + A + nB) /B. The simplest situation is when the internal kinetic energy is affinely-invariant A = A, simultaneously on the left and on the right, i.e., when I = 2. Then, obviously, 1/I = 0 (infinite I), B = −A (A + nB) /B. If there is no B-correction term, B = 0, then, similarly, 1/B = 0 (infinite B). In virtue of (93), (94), the corresponding kinetic Hamiltonians are 1 KM Σ  L N ηN M + 1 Σ  LK + 1 Σ  LL,  K LΣ KKΣ ηKL Σ 2I 2A 2B 1 1 i j 1 i j gik Σi j Σk l g jl + Σ Σ + Σ Σ . = j i 2B i j 2I 2A

Tint =

(95)

Tint

(96)

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When I = 0 (1/I = 0) then the general solution of the geodetic problem is given by exponentials: ϕ(t) =  where E is an arbitrary element of L(V ) or equivalently, exp(Et)ϕ0 = ϕ0 exp(ϕ0 −1 Eϕ0 t) = ϕ0 exp(Et),  is an arbitrary element of L(U ), and ϕ0 is an arbitrary element of LI(U, V ). Roughly speaking, they are E  constants of motion, or initial conditions ϕ(0) = ϕ0 and ϕ˙ (0) = Eϕ0 = ϕ0 E. If I = 0, the above expression is no longer a general solution of the geodetic problem. Nevertheless, even then there exist special solutions of the same type, so-called stationary solutions. They are special in that  initial conditions are subject to certain restrictions; namely, if we use the representation ϕ(t) = ϕ0 exp(Et) ηT ηT  E  is η-normal in the sense that [E,  ] = 0, where E  for stationary solutions of (95); then E is the ηT A AC D ηT η     η-transpose of E, i.e., (E ) B := η ηBD E C . Roughly speaking, E does commute with E . This holds  η . Similarly, for stationary solutions  is η-symmetric or η-antisymmetric: E  ηT = ±E in particular when E of (96) we have ϕ(t) = exp (Et) ϕ0 , where ϕ0 ∈ LI(U, V ) is arbitrary, just like above, but E ∈ L(V ) is   g-normal, i.e., E, E gT = 0, where (E gT )i j := g ik glj E l k . This type of “stationary solutions” is just some counterpart of stationary rotations in mechanics of anisotropic rigid body. There are doubly affinely-invariant geodetic models (95), (96) with I = 0 (incidentally, they are identical in both versions of the formula). Then the general solution is given by the above-mentioned matrix exponents. Of course, one can admit in addition to T some potentials V and consider other models, then no longer ones admitting exponential solutions. Nevertheless even within purely geodetic framework one can describe strongly nonlinear elastic vibrations. Dynamics is not then encoded in anything like V , but just in the kinetic energy, i.e., in the metric tensor of the configuration space. This resembles some properties of the Maupertuis principle. More precisely, this is true for the isochoric (incompressible) part of motion, when we consider only degrees of freedom ruled by the special linear groups SL(V ), SL(U ). General solution contains then an open subset of bounded motions and an open subset of non-bounded, escaping and collapsing solutions. Roughly speaking we are dealing with some dissociation threshold and bounded, non-linearly vibrating processes. And all this without potential, and analytically based on the properties of exponents! The purely dilatational part of geodetic motion is non-bounded or collapsing, except, of course the constant solution. But this purely dilatational part may be stabilized by introducing some auxiliary dilatational potential in one dimension, some oscillator, potential well, etc. All this is a very important argument for investigating affinely-invariant dynamical models, although from some point of view they might seem “exotic”. But they are not more exotic than the concept of effective mass in solid state physics. Analytical tools of the analysis are based on the properties of the matrix exponential map and the polar and two-polar decomposition of the matrix ϕ representing the internal configuration. First of all, let us notice that the expressions (95), (96) may be written respectively in the following forms:    2   1  2 + 1 T rΣ  − 1 Tr V 2 , Tr Σ 2α 2β 4μ     1 1 1 2 T r Σ2 + (T rΣ) − T r S2 , = 2α 2β 4μ

Tint =

(97)

Tint

(98)

where the tensors S and V denote the canonical spin S i j = Σi j − Σj i = Σi j − gjk Σk l g li and vorticity  C D g DA . The constants α, β, μ are expressed by A, B, C as follows:  AB − Σ BA = Σ  A B − ηBC Σ V AB = Σ α = I + A, β = −(I + A)(I + A + nB)/B, μ = (I 2 − A2 )/I. The formulas (97), (98) may be also expressed as follows: Tint =

1 1 1 2 C(2) + C(1)2 +

V , 2α 2β 2μ

(99)

Tint =

1 1 1 2 C(2) + C(1)2 +

S , 2α 2β 2μ

(100)

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     k , and V , S are where C(k) denotes the k-th degree Casimir quantity, C(k) = T r Σk = T r Σ     2 2 the magnitudes of vorticity and spin: V = −(1/2)T r V 2 , S = −(1/2)T r S 2 . It is seen that the difference in the invariance properties between (95) and (96), i.e., between (97) and (98) is reflected only by the last, i.e., third, terms in (97), (98). For more general models, like (83), (84), situation is more complicated and the relationship between velocity-based and canonical models is more complicated, although it still does exist. Let us stress that (99), (100) and the doubly (left and right) affinely invariant geodetic models (80) 2 2 of internal dynamics differ by constants of motion proportional to V and S . They are, so-to-speak, half-affine, i.e., affine in the space and metrical in the body (99) or conversely, metrical in the space and affine in the body (100). They are special cases of (83), (90). We consider also their combination given by Tint =

1 1 1 1 2 2 C(2) + C(1)2 +

V +

S , 2α 2β 2μ 2ν

(101)

again with α, β, μ, ν being some inertial constants. However, for us the most important expression would be one containing only the Casimir α, β-terms, Tint = (1/2α)C(2) + (1/2β)C(1)2 , or merely, the main second-order Casimir Tint =

1 i j 1 1 A B C(2) = Σ jΣ i = Σ B Σ A. 2α 2α 2α

(102)

It is just this simplest model where one can observe the mentioned encoding of nonlinear elastic vibrations in negatively-determined part of the kinetic energy expression. To see this explicit, one should use so-called two-polar splitting for the mapping ϕ ∈ L(U, V ) describing the internal configuration. Any linear isomorphism ϕ of U onto V induces in linear spaces U , V two metric-like tensors, just the Green and Cauchy deformation tensors G[ϕ] ∈ U ∗ ⊗ U ∗ , C[ϕ] ∈ V ∗ ⊗ V ∗ . But the material and physical translation spaces U , V are endowed also with some fixed, thus also constant in time, metric tensors η ∈ U ∗ ⊗ U ∗ , g ∈ V ∗ ⊗ V ∗ . And, as we saw, these metrics enable one to construct the mixed   tensors G[ϕ] ∈ U ⊗ U ∗ , C[ϕ] ∈ V ⊗ V ∗ . They have eigenvalues λa , λa −1 , a = 1, . . . , n. In certain formulas √ it is also convenient to use the quantities denoted by Qa , q a , a = 1, . . . , n, where Qa = exp(q a ) = λa . Obviously, λa -s are positive, because η, G[ϕ] and g, C[ϕ] are positively-definite tensors. Using the quantities Qa as diagonal entries, we can construct the diagonal matrix D = diag(Q1 , . . . , Qn ). It may be identified with some linear isomorphism of Rn onto Rn , D : Rn → Rn . The quantities Qa = Daa (no summation convention!) are the basic stretchings of the system. They contain the information about how much the object is elongated/shortened in principal directions. But they do not tell us anything about the orientation  of those stretchings in U and V . This information is encoded in orthonormal systems of eigenvectors of G a a  The eigenvectors satisfy equations GR  a = exp(2q )Ra , CL  a = exp(−2q )La . The bases Ra , La are and C. K L assumed to be orthonormal, i.e., η (Ra , Rb ) = ηKL R a R b = δab = g (La , Lb ) = gij Li a Lj b . To be more precise, they are assumed to be unit vectors; their orthogonality is a generic situation for the case of simple  C.  The dual bases in U ∗ and V ∗ will be denoted by Ra , La , thus, Ra , Rb  = δ a b , spectra of operators G, La , Lb  = δ a b . Therefore, the deformation tensors may be expressed as follows: G[ϕ] =

exp (2q a [ϕ]) Ra [ϕ] ⊗ Ra [ϕ],

(103)

exp (−2q a [ϕ]) La [ϕ] ⊗ La [ϕ].

(104)

a

C[ϕ] =

a

Every mapping of internal configuration ϕ is represented by three kinds of objects, namely, by a pair of metrically rigid bodies with configurations (. . . , Ra [ϕ], . . .) and (. . . , La [ϕ], . . .) in U and V , and the n-tuple of fictitious material points with positions (. . . , q a [ϕ], . . .) in the real axis R. Those subsystems have respectively n(n − 1)/2, n(n − 1)/2, n degrees of freedom. It is important to note that this representation

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  is not unique. It is so even in the special case of ϕ with non-degenerate spectra of G[ϕ], C[ϕ], because   the permutation of indices a in (103), (104) does not affect the sum. When the spectrum of G[ϕ], C[ϕ] is degenerate, the non-uniqueness of (103), (104) becomes continuous. This singularity of coordinate system resembles that of spherical or polar coordinates at r = 0 (ϕ, ϑ completely non-determined), although in concrete details it is much more complicated. Obviously, the linear frames L = (. . . , La , . . .), R = (. . . , Ra , . . .) may be canonically identified with = (. . . , La , . . .), some isomorphisms L : Rn → V and R : Rn → U . And, obviously, their dual co-frames L a −1 n = (. . . , R , . . .) correspond canonically to some linear isomorphisms L : V → R and R−1 : U → Rn . R If the diagonal matrix with deformation invariants Qa on diagonal is identified with a linear isomorphism D of Rn onto Rn , then, making use of all the above identifications, we can simply write ϕ = LDR−1 ; in matrix terms this means: orthogonal times diagonal times orthogonal. It is clear that the spatial and material isometries preserving orientation in V and U , A ∈ SO(V, g), B ∈ SO(U, η) act on the left on the L- and R-gyroscopic part of ϕ, because the mappings L → AL, R → BR imply that ϕ transforms under them as ϕ → AϕB −1 . Their Hamiltonian generators are respectively the spin (for A) and the negative vorticity (for B). Let us now consider in a more detail the structure of L- and R-rigid bodies. Their “physical spaces” are respectively V and U . And their material spaces are simply identified with Rn . The group SO(n, R) acts on them on the right, L → LC, R → RD, where C, D ∈ SO(n, R). These mapping are not expressible by an action on ϕ as a whole. Nevertheless, locally they are well-defined and so are their Hamiltonian generators. One can define their “spatial” (in the V , U -sense) and co-moving (in the Rn -sense) angular velocities and canonical spins. The “co-moving” and “spatial” components for the angular velocity of the L-top are defined as follows:   χ a b := La , L˙ b = La i L˙ i b ,   a b La ⊗ Lb . χi j := L˙ i a La j , or χ = χ

(105) (106)

Obviously, χ ∈ SO(V, g) , χ  ∈ SO(n, R) . The first latin characters a, b are simply labels, i.e., indices in Rn , whereas the middle ones i, j are tensor indices in V . Similarly, the “co-moving” and “U -spatial” components of the angular velocity of the R-top are given by analogous formulas:   ϑa b := Ra , R˙ b = Ra M R˙ M b ,   ϑL M := R˙ L a Ra M , or ϑ = ϑa b Ra ⊗ Rb .

(107)

Again for this rigid body U plays the role of the “physical” space in spite of its being originally “material”. The “material” space for the R-top is again Rn . The choice of non-holonomic velocities as (q˙a , χ a b , ϑa b ) or a i L (q˙ , χ j , ϑ M ) depends on particular purposes. Just as it was the case with the usual, physical rigid bodies, it is convenient to introduce non-holonomic, just Poisson-non-commuting canonical momenta (pa, ρa b , τa b ) or (pa , ρi j , τ A B ). Here pa are canonical momenta conjugate to deformation invariants q a . The quantities ρa b , τa b and ρi j , τ A B are canonically conjugate to χ a b , ϑa b , χi j , ϑL M . Obviously, ρ, τ ∈ SO(n, R) , ρ ∈ SO(V, g) ,  τ ∈ SO(U, η) . The meaning of the mentioned “conjugacy” is encoded in the following duality formulas: 1 T r(ρχ) + 2 1 ρχ ) + = pa q˙a + T r( 2

(ρ, τ, p), (χ, ϑ, q) ˙ = pa q˙a +

1 T r(τ ϑ) 2   1  = (  q) T r( τ ϑ) ρ, τ, p), ( χ, ϑ, ˙ . 2

(108)

The Hamiltonian interpretation of ρ and −τ as generators of the left- and right-orthogonal translations of ϕ (V -spatial and U -material ones) implies that they are simply identical with spin and vorticity. They are

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 Unlike ρ and τ , their Rn -material representatives ρ, respectively g- and η-skew-symmetric parts of Σ and Σ. τ fail to be constants of motion for geodetic systems and for Lagrangian models with potentials depending only on deformation invariants q a . It is also clear that the constants of motion ρ, τ are related to ρ, τ as follows: ρ = ρa b La ⊗ Lb , τ = τa b Ra ⊗ Rb . Let us stress that when dealing with the traditional, orthogonally invariant kinetic energy (62), (63), the traditional form of deformation invariants Qa is more convenient than q a which are particularly suited to models with affine symmetry. It was mentioned that the two-polar expansion is not unique. At the same time, it is well-known that the usual polar decomposition is unique. It is convenient to mention here about the polar splitting in this description. Namely, La [ϕ], Ra [ϕ] are two orthogonal bases, respectively in the (V, g)-sense and (U, η)-sense. Therefore, there exists the unique isometry U [ϕ] which establishes a one-to-one relationship between them: La [ϕ] = U [ϕ]Ra [ϕ]; U [ϕ] ∈ O(U, η; V, g). One can show that ϕ = U [ϕ]A[ϕ] = B[ϕ]U [ϕ], where the linear mapping A[ϕ] ∈ GL(U ), B[ϕ] ∈ GL(V ) are respectively η-symmetric, g-symmetric and positive. Therefore, η(x, A[ϕ]y) = η(A[ϕ]x, y), g(w, B[ϕ]z) = g(B[ϕ]w, z), η(x, A[ϕ]x) > 0, g(w, B[ϕ]w) > 0, for any vectors x, y, w, z. In the last two inequalities the vectors x, w are non-vanishing. The symmetric elements of the left and right polar decomposition are related to each other as follows: B[ϕ] = U [ϕ]A[ϕ]U [ϕ]−1 , i.e., by the U [ϕ]-similarity transformation. In the usual matrix representation we have ϕ = U A = BU . The symmetric terms, e.g. A, may be diagonalized by the orthogonal similarity: A = RDR−1 , R-orthogonal. Then we have ϕ = U RDR−1 = LDR−1 , U = LR−1 . Although L, R separately taken are not unique, U is so, and therefore, A, B are unique. Let us define some objects instead of ρ, τ, which enable one to perform a partial diagonalization of the affinely-invariant kinetic energy of internal motion, ρ − τ, N = ρ − τ. After this substitution  i.e., M := −  2 , becomes the second-order Casimir invariant, C(2) = T r Σ

C(2) =

a

1 (M a b ) 1 (N a b ) − a b a b . 16 16 sinh2 q −q cosh2 q −q 2

pa 2 +

a,b

2

2

a,b

(109)

2

The first term in (109) admits a nice representation as a sum of the relative and over-all contributions,    2  . p2 /n + (1/2n) a,b (pa − pb ) , where p is the first-degree Casimir invariant, p = C(1) = T r (Σ) = T r Σ It is a quantity canonically conjugate to the center of mass of logarithmic deformation invariants: q =  (1/n) a q a , {q, p} = 1. The corresponding kinetic energy based on the second-order Casimir invariant as the main term, T =

1 C(2), 2α

(110)

has, as seen from (109), a very peculiar structure. It consists of the first term which formally has a structure of the kinetic energy of the n-particle q a -objects on R, and of the next two terms. The first of these terms, i.e., sinh−2 (q a − q b )/2-term describes centrifugal repulsion between invariants, but the second one,   i.e., cosh−2 q a − q b /2, due to its minus-sign, represents a kind of “centrifugal attraction”. The strengths coefficients (M a b )2 , (N a b )2 are positive, so really one has to do with repulsion and attraction of deformation invariants. Repulsion is singular at the coincidence q a = q b , whereas attraction is then finite. But at large |q a − q b |-distances, attraction prevails if |Nab | > |Mab |. This is the typical shape of “intermolecular” forces 2 between deformation invariants interpreted as indistinguishable material points. Admitting the C(1), S , 2

V -terms, we do not change this interpretation. Before commenting it more, we mention only about some related models. It is seen that (109), (110)-models resemble the hyperbolic version of the Sutherland lattice. There are both similarities and differences. The main difference is just the occurrence of attractive terms, the negative contributions to (109). Their interpretation is slightly obscured because the coupling amplitudes M a b , N a b

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are not constant, they satisfy together with q a -s closed system of equations of motion. These equations may be easily expressed in terms of Poisson brackets between basic state variables (q a , pb , M a b , N a b ); obviously {q a , pb } = δ a b , {q a , M c d } = {q a , N c d } = 0, and {pa , M c d } = {pa , N c d } = 0. The brackets for M a b , N c d may be easily obtained from their definitions and from the standard Poisson rules for ρab , τc d as Hamiltonian generators of the action of SO(n, R) × SO(n, R). The family of functions (q a , pb , M a b , N a b ) forms a Poisson algebra. In the case of non-geodetic models with doubly isotropic potentials, depending only on deformation invariants q a , equations of motion may be obtained in terms of the Poisson brackets as F˙ = {F, H} ;

(111)

for F one has to substitute q a , pb , M a b , N a b . Obviously, this gives us only a partial description of motion, without the time dependence of gyroscopic variables La , Ra . But in practical applications like, e.g., vibrations of molecules, this time dependence is less important. In any case, having solved (111), we know the time dependence of ρ, τ. Then, inverting the Legendre transformation, one can express the time dependence of gyroscopic angular velocities ρ, τ. And then, finally, for the time dependence of La , Ra one obtains first-order differential equations for the n-legs La , Ra : L˙ i a = Li b χ b a (t), R˙ i a = Ri b ϑb a (t). So, after solving the system (111) for (q a , pb , M a b , N a b ), i.e., after obtaining the main deformation description, we can in principle solve the next, less important step and determine the time dependence of the main axes of inertia. It is so for any doubly-isotropic model. The very idea is that in (109), (110) the “plus” and “minus” terms act in opposite directions, representing, respectively, the repulsion and attraction of deformation invariants even in the completely geodetic, potential-free models. The question is, however, if the idea is correct, because the repulsion and attraction strengths are not constants. There is, however, some special case where they are constants of motion. It is in two-dimensional models, when n = 2. In fact, in a consequence of the fact that the orthogonal group SO(2, R) is Abelian (because it is one-dimensional), in this special case we have ρa b = ρa b , τa b = τ a b , and because of this, M a b , N a b are constants of motion (for the doubly-isotropic models), just a 2 as ρa b = S a b , τ a b = −V a b are. Because of this, the effective strengths (M a b )2 , (N   b ) in (109), (110) are  cos α − sin α cos β − sin β constant in time. The matrices L, R, D become respectively L = ,R= , sin α cos α sin β cos β     1 0 0 exp q 1 Q . It is convenient to represent GL+ (2, R) as R+ SL(2, R) by introducing D= 2 = 0 Q 0 exp q 2  1  the “center of mass”, q = q + q 2 /2, and the displacement between q 1 and q 2 , i.e., x = q 2 − q 1 , and their conjugate momenta p = p1 + p2 , px = (p2 − p1 ) /2.     0 −1  0 −1  and ϑ have the following obvious form: χ  = χ = α˙ , ϑ = ϑ = β˙ , The angular velocities χ 1 0 1 0     0 1 0 1 ,N =n , where, obviously, m = pβ − pα , whereas the quantities M , N become M = m −1 0 −1 0 n = pβ + pα . Therefore, for B = 0, the doubly affine-invariant kinetic energy of internal degrees of freedom becomes T =

 m2 n2 1  2 1 1 p 1 + p1 2 + − . 2 1 2A 16A sinh2 q −q 16A cosh2 q2 −q1 2 2

(112)

Using the above-defined new variables one obtains T =

px 2 m2 p2 + + 4A A 16A sinh2

x 2

−

n2 16A cosh2

x 2

.

(113)

For a bit more general models (78), (79), (97), (98) one obtains essentially the same, up to certain modifi2 2 cation of constants and correction terms proportional to V , S . The essence of the problem remains 2 2 the same, because V , S are constants of motion.

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It is seen that in the geodetic problems like (113), the shear-part, i.e., SL(2, R)-component of dynamics admits a continuous family of bounded elastic vibrations. This is the part characterized by |n| > |m|, where at large |x|-distances the “centrifugal attraction” prevails the centrifugal repulsion. This is in spite of the non-compactness of the x-variable and of SL(2, R) at all. The family of bounded motions has the dimension six, i.e., twice the dimension of SL(2, R). Above the threshold, i.e., for |m| > |n|, one deals with the six-dimensional family of unbounded motion. This is the typical threshold behavior, which does exist on the purely geodetic level due to the curved geometry of SL(2, R). In this way, the isochoric (incompressible) elastic motions may be described in purely geodetic terms, without any use of the potential. The only failure of this threshold behaviors on the level of the total GL(2, R) are just dilatations. According to (113), the variable p occurs in a quadratic way with constant coefficient, and its conjugate configuration variable q does not occur at all. Therefore, except the solutions q = const, p = 0, all solutions are singular with respect to q. They are either infinitely expanding, or contracting to the singularity q = 0. This is a consequence of the fact that GL+ (2, R) is not semisimple, being isomorphic to R+ SL(2, R) = exp(R)SL(2, R). Of course, this is not a dangerous failure for our philosophy. The q-motion may be stabilized by introducing any bounding potential, e.g., oscillator, thin potential well, etc. Exactly the same holds for any n > 2. The difference is only that for n > 2, the spin quantities M a b , N a b fail to be constants of motion, they also vibrate. But the SL(2, R)-threshold behavior is a consequence of the commutation rules SL(2, R), GL(2, R). More or less, the same holds also for other models with the partial affine invariance of kinetic energy, like (78), (79), i.e., (99), (100). Let us mention also about some other models more or less related to the above ones. First of all, let us remind the traditional d’Alembert model (62), (63) with the materially isotropic co-moving inertial tensor J AB = Iη AB . Then the kinetic energy is given in the two-polar representation by T =

2 2 1 2 1 (M a b ) 1 (N a b ) Pa + + . a b 2 a b 2 2I a 8I 8I a,b (Q − Q ) a,b (Q + Q )

(114)

Let us notice that now it is not logarithmic invariant q a , but just Qa that is convenient. Obviously, this expression contains only centrifugal repulsion of invariants. Therefore, (114) is completely useless as a model of nonlinear elastic vibrations. To obtain a viable model, one has to modify it by adding some potential   energy. In the doubly isotropic models, it will be a function V Q1 , . . . , Qn of deformation invariants Qa alone. Obviously, the first two terms of (114) resemble the Calogero–Moser integrable lattice and indeed there is some relationship here. It was mentioned also that (109), (110), (112) resemble the hyperbolic version of the Sutherland lattice. It turns out that some relationship with the usual, i.e., trigonometric Sutherland lattice does exist as well. To obtain it, one should compactify the deformation invariants by the formal substitution of Qa = exp (iq a ) to the positively definite kinetic energy Tint = −(A/2)T r(Ω2 ) = (A/2)T r(Ω+ Ω), A > 0. Then GL(2, R) is  = ϕ−1 dϕ/dt, ϕ ∈ U (n). formally replaced by U (n), i.e., the compact real form of GL(n, C). Obviously, Ω After this formal substitution we obtain 1 2 1 (M a b ) 1 (N a b ) Pa + a −q b + a b . q 2 2 q −q 2A a 32A 32A a,b sin a,b cos 2 2 2

Tint =

2

(115)

The relationship with the usual Sutherland lattice is obvious. Let us observe that now the deformation invariants run over the circle. Because of this there is no problem with the positive definiteness of the terms in (115). Because in the compact space with the circular topology it is practically impossible to distinguish between repulsion and attraction. Summary: There are also other important problems concerning affinely rigid body and the problem of dynamical affine invariance. They concern mainly two fields: the theory of systems of affine bodies and

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