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REND. SEM. MAT. UNIVERS. POLITECN. TORINO

Vol. 40°, 2 (1982)

Agostino Prastaro

GEOMETRODYNAMICS OF NON-RELATIVISTIC CONTINOUS MEDIA. I: SPACE-TIME STRUCTURES

Summary: In order to formulate the non-relativistic continuum mechanics as a unified field theory on Galilei space-time M, the geometrical structure of Galilean space -time is considered and the space and time resolution of bundles of geometric objects on M are analysed in detail. These results are then adopted to give a rigorous meaning to the empirical concept of observed physical entities.

1. Introduction

In these last fifteen years the continuum mechanics has been quickly developing into a well grounded mathematical formulation (see the papers of Coleman, Marsden, Noll, Truesdell, Wang, etc. [3,4,8,9,12,15,16]).

At present the standard approach used to describe a continuous body is to consider it as a 3-dimensional differentiate manifolds supported by a class $ of mappings k.&^i*, with values into a 3-dimensional Euclidean space £.

However, we think that more unitarity, from a structural point of view, can be obtained by adopting a 4-dimensional formulation. (Homogeneous formalism, based on geometrical structures, was already used to describe clas­sical mechanics [2,5,8,10,11,13,17,18]). By adopting Galilean space-time as the first structural element in a continuum mechanics theory, we obtain a conceptual economy and we clearly see which are the "frame independent elements" [19] or "intrinsic elements" [8], and which, instead, are connected with the introduction of a frame. (For example, a continuity equation is frame independent, whereas an energy equation will be frame dependent (see [19] p. 67)).

Classificazionepersoggetto: AMS (MOS) 1980 Suject Classifications: 83E05, 55R65, 57R25, 53C05, 58A20.

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Then, scope of this work is to give as rigorous an intrinsic formulation of continuum mechanics as possible by using a 4-dimensional approach and by considering a continuous body as a "continuum system" in the sense described in our previous work [14]. In this first paper we will analyse the kinematic aspect of the theory. Thus we will examine Galilean space-time, emphasizing its fiber bundle structure. Galilean group as the group of transformation which preserve the space-time structure will then be described. We will represent physical entities as fields of geometric objects in the sense of [15]. These geometric objects need not be tensor fields on the classical space-time. For example "apparent velocity", which is not a tensor field, will still be a field of geometric objects.

We will develop the space and time-resolution of geometric objects as­sociated with the presence of an observer. The resolution will be related to the notion of "framing" of space-time ([19] p. 26) and "slicing" of space-time ([9] pp. 1-4-21).

The concept of geometric object gives rigorous meaning to the concept of observed physical quantity. It clarifies the ambiguity of why "frame de­pendent" quantities are useful, even essential, in the kinematic description of continuum mechanical bodies. Moreover, it will clarify the paradosical nature of "frame indifferent statements about frame dependent quantities" ([20] p. 102), These will turn out to be simply statements about fields of geometric objects which are not tensor fields.

Notations — We shall always consider differentiable manifolds of finite di­mension and of class C°° and maps of class C°°

O(S) denotes the orthogonal group of Euclidean space S; SO(S) the special orthogonal group of S.

If {xl} is a coordinate system on the differential manifold V, we denote {bxi} and {dx1} the natural basis induced on C°°(TV) and C°°(T*V) respectively. Moreover, if n : W -> V is a locally trivial fiber bundle, {x\ yk} means a coordinate system on W such that xl = xl ° IT and {yk } are coordinates on the fibers. In par­ticular on the tangent bundle TV -* V, we recognize a canonical coordinate system induced fibers. In particular on the tangent bundle TV-* V} we recognize a canonical coordinate system induced from one on V: {x\ xk }, with xk(Xp) = <dxk(p), Xp >, for any Xp E TV, Vp £ V. Similarly, on the cotangent bundle we get the coordinate system {x\xk}, with Xk((Xp) = <cfy, dxk(p)>, for any otp€T*V.

bf:V-+TW, df(p) = Dlf(0,p),

where the bar above D denotes the composition of the derivative Df with the

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canonical projection R X TW -+ TW. Moreover, the index " 1 " means partial derivative with respect to the first space in R X V.

Locally one has bf= (3/*) byk ° / 0 , where {#a, y*} are local coordinates on W and f* =yk ° f If between W and V there exists a fiber bundle structure irw: W-* V and / is a section of irw, then one has: 3 / : V •+ f*vTW C TW, where vTW is the vertical tangent space of W. Thus if W is a vector bundle over Af and / is a section of nw, we can write bf: V-* W, since f*vTW is ca-nonically diffeomorphic to W.

2. Galilean space-time

The structure of the space-time in classical mechanics was studied from different points of view, by many authors, see e.g. refs. [2,5,8,9,10,13,17]). Here we shall give a sistematic presentation of this important structure em­phasizing its fiber bundle content.

We recall [8,9] that a space-time for Classical Mechanics is a 4-dimensi-onal fiber bundle n.M^ R with the following properties: (a) M admits a symmetric connection V with the property S/dn = 0 (b) Each fiber admits a metric gt compatible with the connection on the fiber S/t obtained by restricting V to vectors in T(Mt), S7tgt

= 0« A point pEAI is called an event and n(p)= t is its time.

Definition 2.1. (*) Let M b e a 4-dimensional affine space, the corresponding affine structure is (M, M, a) and let T be a 1-dimensional affine space, the corresponding affine structure is (T, T, j3), (T is supposed oriented).

The Galilean space-time structure is the couple (^ g), where: 1) ^ is the fibre bundle ^ = {r : M -* T), being T & surjective map of constant_rank ¥= 0. M is called Galilean spacejime and T time-space. We pose S= ker(Dr) C M, (space of space-vectors); (D is the symbol of the Frechet derivation). 2) g is a Euclidean structure on S.

We say that Galilean space-time is oriented if an orientation 7? on S is fixed. From the fibre bundle structure of M we have thatM= U Mt, where

the fibers Mt are 3-dimensional affine spaces, with the affine structure (Mt, S, at), where at=a\Mt. The fibers Mt can be seen as the maximal

(*) For a geometric definition of Galilean space-time see also ref. [10].

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connected integral sub-manifolds of M corresponding to the differential system S:p-+Spi where Sp is identified with S.

If g denotes the constant section over M of the trivial vector bundle vS°M = MX S*oS* given by g(p) = g, Vp GM, then Mt becomes a 3-dimensi-onal Riemannian manifold with metric field gt=g\Mt, and M can be considered a deformation of the Riemannian structure (Mt> gt), Vt e T. (For more details about deformations of Riemannian structures see e.g. ref. [7]).

If on each fiber Mt one fixes a point (origin) then one has the canonical identification of Mt with S so that M becomes a family of Euclidean vector spaces parametrized by the points of T. Thus the space-time defined in [19] p. 26 or [20] p. 81 can be considered Galilean space-time with the following additional structures: (a) Each fiber Mt is "a pointed set; (b) On T there is an Euclidean structure.

The following proposition gives the relation between Galilean space-time and classical space-time.

Proposition 2.1. If an origin on T and a Euclidean structure on T are given, then Galilean space-time becomes a classical space-time.

Proof. See Appendix.

From now on we shall always assume that a Euclidean structure gQ on T is chosen.

Definition 2.2. AGalilean transformation (resp. special Galilean transformation) is a bijective map f:M-*M which preserves Galilean space-time structure (resp. oriented Galilean space-time structure). More precisely: (a) / is an affine bundle transformation of ^ , namely the following diagram

MXM a

M k

f

— T

a T MXM ** M —

h

P TX T

P TX T

T(fT) = <fT, fT)

is commutative; (b) / | S e O(S), / is called the associated linear Galilean transformation, (or if Galilean space time is oriented: / | SeSO(S) , then / is called spacial linear Galilean trans forma tion);

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(c) fT = idT, (namely fT is a rigid transformation of T). For the corresponding groups we use the following nomenclature:

— Galilean group: G; — Special Galilean group: SG; — Linear Galilean group: LG; — Special linear Galilean group: SLG.

Definition 2.3. An infinitesimal Galilean transformation is a Killing vector field X on M with respect to G.

Note that LG (resp. SLG) is the structure group of the principal bundle of the linear adapted orthogonal frames on M (resp. oriented linear adapted orthogonal frames on M) called principal bundle of Galilei frames (resp. prin­cipal bundle of oriented Galilei frames).

3. Frames, space and time resolution of bundles of geometric objects

In order to traduce into an axiomatic form the empiric concept of observer and observed physical entity, we introduce two new structural ele­ments: the bundle of geometric objects and the framing of space-time. By means of a frame we can project a bundle of geometric objects into two subbundles: the time-bundle and the space-bundle. In particular the space-bundle will allow us to give a precise meaning to the concept of observed fields of geometric objects and therefore of observed quantities like apparent velocity. In fact we assume that any physical entity can be represented by a field of geometric objects in the sense of [15]. This assumption is justified because in this way we satisfy the requirement of "covariance" for physical entities which can be more complicated then a tensor field.

On the other hand, we note that for the particular structure of Galilean space-time, many physical entities can not be represented by tensor fields on the space-time. For example the metric field is not a tensor field on M.

In fact g is a section of the vector bundle vS°M over M, but we have not a canonical embedding of vS°M into the tensor bundle S°M = MX M*oM* which is the manifold of the symmetric tensors of type (0,2) on M.

Definition 3.1. A frame is a map \p : TX M -• M, such that: 1) i//f: M -> Mt is a retracting map. (Therefore Mt is a retract of M) (*).

(*) Recall that a retracting map r: X -> A, where A is a subspace of a topological

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2) For each t, t' ST, idM is a fiber bundle isomorphism between the fiber bundles (M,Mt, ^ ( ) , (fittMtl\l/tt)t over the diffeomorphism ^(t,tf) = tyt'Wt', i.e. the following diagram

idM

M +M

**'

Mt -Af,'

is commutative, i.e. \pt'(^t(p))~ ^t'(p)>

Note. Definition 3.1. considers a frame as a "flow" on the space-time. A frame is related to a "framing" ([19] p. 26) and to "frame of reference" ([20] p. 82). A "framing" in fact, is obtained by relating Galilean space-time with the observed space by means of a frame (this will be clarified later). On the other hand a "frame of reference" can be considered a Cartesian co­ordinate system on Galilean space-time adapted to a frame (see section 5, definition of "frame-coordinate system").

Definition 3.2. The observed space by means of a frame \jj is the space of equivalence classes S^ =M/~, where ~ is the equivalence relation in M defined by

p ~p' <*> {3t' e T such that p'= i/v(p)}.

We shall write [p]^ to mean the equivalence class of p G M; this is identified with the world line \l/p:T-*M defined by \pp(t)= \jj(t, p).

Further, \jj/:M -+ S^ is the canonical projection. As a consequence of this definition the space-time can be represented

by a product by means of the bijective maps:

h:P.-* (r<P>> tPl*) * h1: ft \Ph)+ iK*.p)-

Moreover, S^ is a 3-dimensional differentiable manifold "modelled" on the affine spaces Mu more precisely we get the atlasJ/ = (S^J^t)tGr with

space X, is a map such that r°i = idj\f where i-.A-*X is the canonical inclusion [16]. Note, further, that a different, but equivalent geometric definition of frame is given in [10].

* .

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$t,t'= &t\Mt> as transition functions. We remark that S$ is precisely the Truesdell-Wang definition of absolute space relative \jj (see [20] p. 83) and the inverse map to i//, j^-.S^X T^-M is the Marsden slicing map ([9] pp. 1-4-21). See also refs. [10,13].

To a frame we can associate two important physical entities: a) velocity: \jj = (Dt i//) ° (r, idM)\ (^ will denote the free part of \p) b) acceleration: $ = V \j).

One has the following:

Proposition 3.1. \p is a "time-like" vector field, i.e.

(1) <oJ>=l,

and \jj is a "space-like" vector field, i.e.

(2) < a , i / !>=0 .

Proof. Eq. (1) follows directly from the following property of frame:

r° \pp-idTt for any peM.

Eq. (2) is a consequence of eq. (1) and the property of the canonical affine connection on M.

Remark 3.1. A particularly important category of frames is that of affine frames, i.e. framesj// such that \j/titt are affine maps for each t, t' G T. In such cases ^t'.t^^^t'j ls constant, so we have the identification of ypt>t

with an element of the general linear group GL(S) of S. Moreover Dip is a constant map on each fiber Mt, Site 7", so the in­

finitesimal strain e = (1/2) i^ g, the spin co = (1/2) rot^ and the angular ve­locity £2 = 'g(*co) are constant on Mt also. (Here 'g is the canonical fiber bundle isomorphism over M: vT*M — vTM, and *o; is the adjoint form of co defined by the contraction of co by the volume form 1?).

For a general frame, 5^ is not an affine space; however if \jj is an affine frame, then 5^ has the following affine structure (S^, S ,, a^), where: (a) $y is the space of equivalence classes induced by the equivalence relation in (TX S) given by:

(t, u) ~ {t\ u') <*u' = \l/t',t(u), (t, t' eTand u, « ' G S ) ;

(b) (ty ([p]^, [t, u]^) = [p + \pT{p)ttW]^ • The most important sub-class of affine frames is that of rigid-frames

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characterized by the condition that for any t, t' e T, ypt,t' ls a conformal

transformation between Mt and Mt>. In such cases \jjt'>t e SO(S) and e = JC g =

= 0. Hence, for rigid frames, \jj is an infinitesimal Galilean transformation, (jC^a= 0 is always verified! ).

Finally, for the inertia! frames, i.e. frames with constant i//, we get: (a) ^t',t = ids> (b) V ^ = 0, thus the world lines of an inertial frame are geodesies of the space-time.

In order to describe physical quantities in mathematical terms we define the bundles of geometric objects.

Definition 3.3. (Salvioli) A bundle of geometric objects L a 4-plet (W, V, 7T; IB), where n : W -> V is a fiber bundle and IB is a covariant functor IB : C(V) -+ •+ C(W), where the category C(V), (resp. C(JV)), has as objects the opens of V (resp. W), and as morphisms the local diffeomorphisms between these objects. Moreover, if U is an object of C(V) then JB(U)=-iTl(U). A section of 7r is called a field of geometric objects.

Examples. A trivial example of bundle of geometric objects is the tensor bundle: {VSV, V, ?; ®J), but there are important bundles of geometric objects which are not tensor bundles as for example the metric bundle (vS°M, M, 7r'; vS°)', the metric field g is a field of geometric objects of this bundle.

Further, an orientation structure 17 on the space-time identifies a constant field of geometric objects 77 of the following bundle of geometric objects: (vA°3M,M, ir";vA°3).

Another important example of bundle of geometric objects is the space of connections on a fiber bundle ir-.'W -* X over a manifold X. Such a space can be identified with the jet-derivative space J$(W) (see ref. [14]). A con­nection on W is a section : W -• J2{\V). J2{W) identifies the following bundle of geometric objects over W: (j©(W0, W, irit0\J$), where 7r1?0 is the canonical projection J@(W)-> W and / ^ i s the functor/^: C(W)-> C(J%W)), where the category C(W) (resp. C(J3)(W))) has as objects the open subbundles of W (resp. J@(W)) and as morphisms the local fiber bundle diffeomorphisms bet­ween these objects.

We now indicate how the choice of a frame allows us to decouple the bundle of geometric objects over space-time into bundles over the "space" and "time" manifolds respectively. To produce the decoupling, we introduce the notion of vertical functor.

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Definition 3.4. Let a covariant functor B .- v -+$Fv be given, where v is the category having manifolds as objects and differentiable maps as morphisms and &*v is the category having vector bundles as objects and vector bundle morphisms as morphisms. We also suppose that: (A) the functor JB associates to the "manifold" consisting of a single point the "vector space" consisting of the zero element.

On the subcategory^T) c v of bundles on T we associate to each IB its vertical i.e.

vB xSFiX) -*&v

vJB : W~*vB(W) = U IB(Wt), tET

where Wt is the fiber of W on t G T. We remark that similar considerations are held true if IB is controvariant

or a functor of two variables. Moreover with respect to the bundle idT :T^>T, we get

vB{T)= U JB(*)=Tx{0}=0. tET

Definition 3.5. Let JB be a covariant functor like in Definition 3.4. Let (W, M, irw; IB) be a fiber bundle of geometric objects, with W = B(M). We call W = vB(M) the space-bundle associated to W.

Let us be given a frame \p. It can be shown that the following sequence

(3) 0^wJt-+W*^>W°^0 IX*) n(i//)

is exact and splits, where: py°=r*2B(T); ~q(w) =B(iT(nw(W)) (w) is the canonical inclusion induced by it : Mt->'M; Jf = (irw, B{T)) is the canonical projection;

T(Tr\y(w))) (w) ~ w*\i> - space-component associated to w by of i//;

~l(i//)(p, XT(p)) =lB(i//p)(XT(p)). Moreover we can show that ~i(i//) gives an iso­morphism between T*B(T) and a sub-bundle of W, W®, that we call the time-bundle associated to W by means of \p by means of \jj.

Thus we have the projection:

J(l//) = -|(l//)°p: W-*W%

J(i//) (w) = w<ty = time-component associated to w by means of i//.

We call r(j//) (or L(i//) = ^ ° r ( i / / ) ) and J ( ^ ) the space and the £wz£-

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projections respectively.

Definition 3.6. If / : M •+ W is a field of geometric objects, we cal l /^ =F(\p) ° o f: M -> W" the space-component and fo^=J(\jj) o f: M -> Wj the ft'w^-component corresponding to / b y means of the frame i//.

(Similar considerations can be made even if W is not a vector bundle of geometric objects directly on M but on another fiber bundle E which is a bundle of geometric objects over the space-time M. See the next examples).

Note. Our space and time bundle projections generalize the procedure of lapse and shift operations of [8] p. 505. We note, in particular, that on Galilean space-time there is not a "4-metric", so the procedure of [8] fails to determine the space and time splitting of the metric field on Galilean space-time.

Examples

(1) Tangent bundle: (TM, M, nTM; T), (here T denotes the tangent functor). The space-bundle associated to TM is vTM — M X S. The space-projection as­sociated to a frame i// is the map

r(\p) : TM^vTM

F(\p) : (p, u) -> T(i//r(p)) (p, u) = (p, grad\p(p) (u)) = (p, u^(p)).

where grad\jj is defined by

gradxp = 02 *) ° (r, idM) : M -> M* ® S.

The projection ~p gives

p : (p, u) -> (p, T(T) (p, u)) = (p, r(p), < a, u » ; (a = Dr).

The time-bundle is 0 : (TAf)J = U < i//(p) > ^ M (1). The isomorphism bet-

ween T*TT and (TAOj is

~l(i//) : (p, £, X) -» X i//(p) and its inverse is

-|(i//)_1: u -> (0(«), r(0(a)), <"a, u » .

The time-projection associated to i// is

J(i//): TM^(TM)%

(1) < i//(p) > denotes the vector space generated by yp(p).

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such that

J(\p) : (p, u)-*<0~U>i>(p).

Thus we have the following exact sequence:

q. p 0->vTM*=^*TM*=r==±T*TT+ 0

rw -KiW

The splittings F(i//) and ~\(\p) induce the isomorphism

TM = vTM®(TM)\

that is

(p, u) = (p, grad\p(p) (u) + <o,u> J(p))

= (p, u^{p) + u0^(p)).

(2) Cotangent bundle (T*M, M, 7T; T*). AS T* is a controvariant functor we shall consider the following splitted exact sequence of vector bundles over M:

q p 0 -> T*T*T+-—± T*M « 1 (T*M)A -> 0

rw id//) where: 1) (T*MV = U T*Mt= U MtXS*=MXS* = space-cotangent bundle;

2) T*T*TC MX TX *;

3) q:(p, T(p),H*+ (p,HO); 4) r(\p):(p,a)^{p,T(p),<a,$(p)»;

5) p:(p,OL)»(p, (X|S);

6) ^)-ip,P)*+(p,P°grnd$(p)y, ~~l(t//) gives an isomorphism between (T*A1)A and a sub-bundle of T*M, (T*M)J that we also call space-cotangent bundle associated to T*M by means of \p. Further, r(i//) gives an isomorphism between T*T*T and a subbundle of T*M (T*M)° , that we call the time-cotangent bundle associated to T*M by means of \p. One has the projections:

7) time-projection: L( i//) = q ° r ( i//) : T*M -> (T*M)J = U < o >

(p, a) »-• (p, < a, ;//(p) > a) = time-component of (p, a);

8) space-projection: J(;//) = nd//) ° p :T*M->(T*M)J, (p, a)n> (p, a - <a , i//(p)> a) = space-component

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of (p, a). Thus we have the isomorphism: T*M=(T*M)l ©(r*Af')$.

(3) Deformation gradient space F = T*M ® vTM ^ M X M X M * ® S . In this case we have a functor of two variables: IF: v X v -> #i>, contro-

variant in the first and covariant in the second. Let us put

# = U T*Mt, ® TMt=MX MX S*®S; (t',t)GTXT

F ° = ( r X r ) * F ( r , T)=MXMX{0} .

Let \jj be a frame, then we have the following splitted exact sequence of vector bundles over the manifold M X M:

A qW p 0 -> F <• > F 5 = ± F° -» 0

r i where: 1) r : (p, p ' , a ® z>) >-* (p, p ' , (a|S) ® w); 2) #(i//): (p, p', 7 ® w) H> (p, p' , (T o grad\p(p)) ®v);

3) p:(p,p',(X®v)» (p,p ' ,0);

4) H:(p,p' ,0)H>(p,p' ,0).

Hence F is isomorphic to a subbundle (F)J of F called the space-deformation gradient space. Of course the time-deformation gradient space (F)° is the zero bundle over MX M. Further, we have the following projection {space-projection): L(i//) = q(\jj) ° r : F -> (F)J given by (p, p ' , a ® z?) (-> »-• (p, p ' , (a o gradtyip) ® z>).

(4) Derivative space @)(W) of a bundle of geometric objects (W, M, IT; IB) over M. Let IB be a covariant functor which satisfies the properties specified in

Definition 3.4. Recall, (see ref. [14]), that the derivative space of the fiber bundle IT : W-> M is defined by Ql(W) = ir*T*M ® TW. It is a vector bundle over W with projection 7r1)0. @(W) is also a fiber bundle over M with pro­jection 7T!. The derivative Df of a section f of n over Af is a section of TTi. The space @)(W) identifies the functor of two variables

£>(--) = T*(~) ® T o JB(-) : */ X P ->^v

which is controvariant in the first and covariant in the second. The space and time analysis induced by a frame \p is conduced by

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considering the following splitted sequence of vector bundles over W:

where: 1) J(W) = U D(M„Af t) = U 7r*T*Mt®(ToiB)(M,);

tGr *Gr 2) ®(W0° =2B(r)* D(T, T )cWX it*T*T • T o iB(r), being * the canonical

projection 2B(T) -> T; 3) q(\p) :(w, a® v)*+ (w,(a° gradi//(7r(zo))) ® (To JB)(jt)(#)), where zt is the

canonical inclusion Mf **• M; 4) r (* ) : (w,P-®«)^(w, (a |S )®(roJB)(^ ) (« ) ; 5) p(^) : (w, /5«»)H.( w , <p^(p)>9(ToB)(T)(u)h 6) n ( * ) : (w, X ® M ) ^ ( ^ , Xa®(To JB) (T ) (« ) ) ;

Thus we can identify two vector subbundles of ^(W) over W: 7) @(W)^ = Im q(\p) = space-derivative bundle associated to^(W0 by means

of ^; 8) £^(W)° = Im ~l(i//) = space-derivative bundle associated to @(W) by means

of \p. We have also the following natural projections:

9) space-projection: \_(\p)=q(\lj)°r(\lj):gi(W)-*@(W)y, 10) time-projection: J(\p) = 1(\p) o p(i//) :^(W) -^(W)J.

Then, if / is a section of 7r, by composition of the above projections we obtain the space and time component of the derivative Df. 11) space-component: \Df)^ =L(\jj) ° Df: M-*/*0(W)J ; 12) time-component: (Df)0^ = J(\jj) o Df :M-+ f* @(W)%.

In the last section we shall give the coordinate expression of these fields.

(5) Derivative space of order k@k(W) for a bundle of geometric objects (W, M, ir: B) over M. Recall, (see ref. [14]) that @k(W) is defined by iteration on k :@k(W)=.

= (@k~l(yV)). 9k{W) is a vector bundle over0*-1(W). Then the space and time analysis by means of a frame \jj is conduced by considering the following splitted sequence of vector bundles over^*_1(W):

0 -> ®k(WY ~ — -±@k(W) «*- — ±@k(W)° -> 0

where the bundles @k(Wy and @k(W)° as well the morphisms involved can be built in a similar way to the ones in the above example. Thus we have two

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subbundles of ®k(W): 0*(W% = Im q(\p) = space-derivative bundle of order k; and @k(W)§ = Im ~l(i//) = time-derivative bundle of order k. Further, we have the corresponding projections: l_(i//) : @k(W) -+ ^*(W% ; J(i//) : @k(W) -»

(6) Jet-derivative space of order kJ@k(W) for the bundle of geometric objects (W,M, IT, IB) over M. Recall, (see ref. [14]), that J@k(W) is a subbundle of 9k(W) defined by

j9k{W) s {ue@k(W)\3fe Ck(W)\Dkf(7rk(u)) = «},

being 7T* the canonical projection @k(W) •+M. J@k(W) can be identified with the jet space JkW by means of the canonical embedding jkf(p)*-> Dkf(p). J@k(W) has a natural structure of affine fiber bundle over J@k~l(W) with associated vector bundle S°Af ® j@k-uW)v^- Then in order to identify the space and time resolution of j3>k(W) by means of a frame \jj we must use the canonical embedding jk: jffi(W) -• i^*(W0.

In fact, by restriction of the projections L(i//) and J(t//) given in the above example we get the projections: (a) L(i//) | : J@k(W) -» J^k(W)^ s Im L(i//)l C 0*(JV) (space-projection) (b) J(i//) I : J®*(W0 -> J@k(W)l = Im J(i//)| c 9k{W) (time-projection).

(7) Space of connections over a bundle of geometric objects (W, M, TT-,B)

over M. Recall that such a space can be identified with the bundle J@>(W) -» W,

and a connection over W is a section 1 : W -> J@(W). Thus the considerations made in the above example can be applied to this case taking k = l.

(8) Space of velocity of motions over M. A motion in the Galilean space-time is a section m of the fiber bundle

over T. A velocity m of a motion m is just the derivative of m\ m- Dm. The space of velocity of motions can be identified with the-jet derivative space over <S : J ^ ( ^ ) C & (eg) s TM. Thus, by means of a frame \p we get the pro­jections: (a) (space-projection) L(i//): J®<&) -> Jtyig) J = vTM, vp»vp- \jj(p); (in this case L(i//) is also injective, i.e. L(i//) is a fiber bundle isomorphism over M); (b) (time-projection) J(i//) : / 0 ( # ) -> J9{<3)\ C (TAflJ, z>p »-» tf(p).

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4. Observed bundles of geometric objects

We can now give a precise meaning to the empirical notion of a physical entity as seen by an observer. We accomplish this by introducing the observed bundle of geometric objects.

Definition 4.1. Let (W = B(M),M, irw;B) be a bundle of geometric objects, with B covariant functor. Let \jj be a frame. We call W^ = B(S^) the observed bundle, corresponding to W, by means of \JJ.

The observed bundle of geometric objects B(S^) and the space bundle vB(M) can be related. In Pv = vB(M) let us define the following equivalence relation:

Up ~ ay <» a y = B(\pT{p>)Mp))(cjp) .

We write [cop]^ to mean the equivalence class of ay and $/~ to mean the space of equivalence classes. Then we have the following.

Proposition 4.1. ZB(S ) is canonically diffeomorphic to W/~.

Proof. In fact by means of the following bijections

a) by : TX 07~ -» W, £„, : (*, [Wp]^)^ B(^tMp)){wp);

b) ib^ : f r /~->#(M,) ;

c) B(j-jt):B(SJ->B(Mt). A

We can define a canonical diffeomorphism ly.Wy = W/~, induced by the following commutative diagram:

l

*• J5(Mt) 4

*</;v> A

So we can identify W^ with W7~ : W^ = W/~ and therefore a point ? € W , with the equivalence classe [oy]^,.

Corollary 4.1. We get the canonical projection 7ty = 1$ ° P/\p ° L(i//): W-* W^ i.e. the following diagram

104

w w ** , r

P/+ W+* - W/~

is commutative. Obviously 7ty = ZB(«///).

Proposition 4.2. Let us make the hypotheses of Definition 4.1. If f:M-> W is a field of geometric objects, then the observed field of geometric objects corresponding to / , by means of a frame \jj, is the map

f^ is a time-dependent field of geometric objects of (W^, S^, P^ : 2B), where P^ is the canonical projection P^ : W^-> S^, i.e. / ^ t : S^-t-W^ is a section of P^.

The relation between fy and / ^ is the following:

U = *v* ° /^ ° ;V Examples

(1) The observed tangent bundle TS^= (TM)^. Let us build explicitly the diffeomorphism TiS^) = (TAf)7~. We have

the following bisections:

a) b^ : T X vTM/~ -> z>™ fy, :(*, Ip, U]^ Mt,p), \l>t,T<p)<p)(u))i

b) T ^ ) = (7X«), 7X*/)) : ™ - 7T X T50

jT(/V) : <P> U^ Wp)> <°,U>; [p, U^(p)]^)

and its inverse

W ) •' (', MA W]*) = Mt,p),ff1 \p(t,p)(\) + B2 Mt,p)(u)). Thus in particular

7V#t) (lp, «]*) = W*,p), ^2 *(*,/>) («))• and taking into account that ir^ :TM -> TS^, 7ty : (p, u) *-* [p, «A^(p)]^, we conclude that for any £GT one has the bijection (77W)J -* T(5^) given by jjp, »]*»-• [ty(t,p),D2\lj(t,p)(u)]^. For another t'e r we get \p,u]++ [\Jj(t',p), T)2ty{t',p){u)])p. On the_other hand \p(t',p) = (\pt')t ° \p^(t,p)-, and one has [p, «]„,*• [\pt'lt(ip(t,p)), D2(i>t',t ° *) (p) («)]* = [W*,P)> D2 Ht,p) («)]*. Thus

105

the diffeomorphism does not depend on the particular value of t chosen. The observed vector field X^ corresponding to a given vector field Xon M is the following time-dependent vector field on S\p:

X^ = 7^ o X o j -1: T X 5^ -» (TM) J,

Xt(t,\p]}„)=Mt,p)t(x-<o,x>ifr)(}Kt,p))]}ll.

(2) The observed cotangent bundle (T^M)^ = T*(5^,) As T* is a controvariant functor we can obtain results similar to the ones

in the above examples by substituting \pt}t' by i//~y. More precisely, we in­troduce in vT*M the following equivalence relation:

S->ZR (p, a) ~ (p1, a') *p'~p,a' = *•"» r(p), Tip) WW**"*"1 r(p'), rip)**1

We have the following bijective map h^ :TX vT*M/~ -• vT*M given by £ty :(*, [p, a ^ W Wt, p), ^*T(p)(p)(a)). Thus, we get the diffeomorphism ib^t: vT*M/~ = T*Mt X S* for any t e T . Then, by a similar proceeding to the one in the above example we get the identification (T*M)^= T*(S^) = = vT*M/~. Moreover, we have the following canonical projection 7ty = /,^° ° P/i// o J(i//) : T*M -» (T*Af)^, i.e. the following diagram

T*M ——*~ (T*M)^ = vT*M

is commutative. If a is a differential /-form on M the corresponding observed object by means of a frame is the following time-dependent /-form on S^: a ^ ^ o a o y-i: T X 5^ -> TS*, cfyfo [p]^) = [i//(f,p), a(i//(£,p)) |S]^.

(J) Observed deformation gradient space F^ . F^ is given by F^ = r*5^®T5^. Moreover, in the space-deformation

gradient bundle (F)J we introduce the following equivalence relation

wp,q~ wPW * Up>,q' = F ( ^ ) T ( p ) , trW)Mq))(o>p>q).

By arguments similar to the above ones, we get the canonical diffeomorphism lyp : F^ = (F)J/~ and the canonical projection 7ty = l^1 ° P/i// o |_(i//): F -* F^.

f-f) Observed derivative space of a bundle of geometric objects (W, M, ir,IB) over M. The observed derivative space Q)(W)^ is given by @(W^). So if 7ty is the

106

canonical projection @(W) -*@(W)y one has that the observed derivative (Df)^ of the derivative Dfoi a section/of 7r is given by the following time dependent section of ^(W)^ over S^:

@(W) *@(W)+ 4 •

Df ; - i (Dfo^TT+oDfojj

M ** T X S^

Similar considerations can be made for jet-derivative spaces and the space of connections on a bundle of geometric objects (W, M, ir,lB) over M.

Let now 0 be a second frame. Then 0A^ is the space-component of the 4-velocity observed with respect to i//. We define observed frame with respect to ^ the following map given by composition: 0^ = i///°0 : T X M -> S^.

Then 0^ = (Dl((>^) ° (r, id^) ° j ^ = (0)^ is the apparent velocity relative to i//, which is the "relative velocity" one would suspect. In fact one has

M * , IP]*)=[*(*,p),*(p)- W l * -

Let us now calculate the equation of transformation for the apparent velocity under a change of frame. Let 1//1 and \p2 be two other frames.

We define a change of frame the map i//12 : TX S^-* S ,2 given by com­position: i//12= 4>2^°j\p • (Note that i//12 corresponds to the change of framing corresponding to two Newtonian slicings of [9] p. 1-4-25) . Thus we have the following equation relating (f>^1 and 0^2 : 0^2 = \pl2 ° (id^, <t>^x)-

By taking the derivative of the above equation one has:

(4) 0^ 2 =r ( i / / 1 2 )o ( l , 0^ )o ; 1 2

where j12 is the canonical diffeomorphism TX S^2=TX S^v From (4) we obtain

(5) 0,/,2(*, [p]+2)= [^2(t,p),0^1(^a,p))+ (JMlM*,p))~ MMt,))]^

for any (t, \p]^2)<T X S^.

Remark 4.1. If \p is a rigid frame there is on S^ the following metric tensor field:

7* : S+ -> S 2 % s S„ X £j © $} , T^([pU) = ([/>]*, y^lph))

y^(lPh)(U>uhy It', u%) = \ptf,M •"«',

107

i.e. y^ : Sy ->• $$ ® ^J is a constant map which can be identified with an element of $$ ®^J. Further, the following diagram

ir

T X S^ »- S2 S^

1*

g* TXS^

,2

Sty

is commutative. In fact if (t, u) ~ (t, u) and (?, «') ~ (t\ u') then

[?, a]^[?, w']^= fo',,(w) •*? =(\lrt'tt(\pit(u)) -(fa'j'iu'))

= ^t',*(») •« ' = [£, u]} [t\ u']^ .

From a physical point of view it is interesting to consider linear con­nections defined on a bundle of geometric objects. Thus we are led to give a coherent meaning to the concept of observed connection.

Definition 4.2. Let us consider the bundle TM with the given connection 2 : TTM -• vTTM for a classical space-time (not necessarily Galilean).

Then the observed connection, by means of a frame \Jj, is the map

2* = ttfXity) o 2 o 7T(;V) o ?: TX 7T(S„) -> *7T(S„),

where t :TX TT(S^) -* TT(TX S^) is the canonical inclusion. For each tGT the map 2 ^ t : 7T(5^) 7* vTT(S^) can be considered a good linear connection, thus we define the absolute differential associated to 2^, of a field of geometric objects s :S -> 75^ as the map

V 5 : T X 5 ^ T % ® z ; T r ( 5 ^ )

such that

(&)*= 2^,,o Ds = (idT*Si//® 2^,,) o Ds.

Remark 4.2. It is illuminating to observe that for X, Y vector fields on M,

(a) The observed intrinsic covariant derivative of Y by X is the pullback ('VxY)\ij)t= j*(VxY) °f t n e intrinsic covariant derivative by the slicing map

jt = /<M a t t i m e *> (b) the apparent covariant derivative, computated relative to the observed connection, is the pull-back of the intrinsic covariant derivative of the space

108

component of the vector field: VA>,*V, = / ? ( V Y ^ 1 % ) .

In particular, the observed intrinsic acceleration of the frame ((V^),//)* = = j*(Vih and the apparent acceleration ( v t ) ^ ) / ^ , ) ^jfflj,^ ^ ) = 0 are not equal. This example leads to the characterization of an inertial frame on a classical space-time (M,V), not necessarily Galilean, as a frame \jj for which the observed intrinsic covariant derivative equals the apparent covariant de­rivative.

Let us now specialize the case of the tangent bundle over the Galilean space-time endowed with the canonical connection G.

Proposition 4.2. Let (TM, M, 7T;T) be the tangent bundle over Galilean space-time and let 2 be the canonical affine connection G; then we get:

1. observed connection: S^ = vT{T(ty/))oGoTl(j-J)o<i : T X T2S^ -> vTTS

G^(t, [p,u,v,w]^)=[\lj(t,p), ^,T(p)(p)(«), 0, ^t,T<p)(p)(«0+ $lT<p)(p)(u, *>)]*•

3- \Vtgip,t)t - 0, i.e., for each £ G T, 5^ has the structure of a Riemannian manifold with Riemannian connection G^,tt.

4. If \jj is an affine frame, then we have a canonical connection of TS^, the observed affine connection, i.e. the map

given by

1* : ([p]+, [t, «]^, [*', *]„, [*", w]^) »-> ([p]^, [*, «]^, [*", w ]^) .

Moreover, the relation with G^ is given by the following commutative diagram:

% vTTS^,

Proof. See Appendix. •

Corollary 4.1. The equation of transformation for the apparent acceleration of a frame 0 under a change of affine frame in Galilean space-time is the following:

109

5. Coordinate representation

The results of the previous sections have been stated in a coordinate free formalism. Explicit computations in terms of a coordinate system have been delayed to this section, in order to avoid interruptions of the previous con­ceptual development.

The space-time M can be considered as a trivial 4-dimensional manifold. In fact for any o G M and for any {ea}, basis of M, we can determine a

global chart 0 : M -> IR4 such that the coordinate functions xa = if- ° 0 : M -• IR are given by xa(p) = < ea, (p - o) >, being {ea} the dual basis of {ea}.

Of course the corresponding coordinate lines [14], xa>p: IR -» Af, are straight lines passing for pGM. The corresponding natural basis of C°°(TM) is {3 #o; : M -+ TM | 3 xa(p) = (p, ^a) = (/?, d xa(p))} Vp G Aff, i.e. 3 ^ a are con­stant vector fields on M. Moreover, <dxa, dXp>= d® :M -> JR.

A very important family of coordinate systems onM is that of the adapted coordinate systems. More precisely an adapted coordinate system on M is a fiber bundle diffeomorphism, x : M -+ IR4, over a coordinate X on T, i.e. the following diagram

x M-

X

is commutative (1). In such cases we get dx°= o, <o, dXk>= 0 and < o,dx£> = 1, namely

one has dx^ip) G vTM and dx0(p) G J ^ ( ^ ) or, equivalently, dxk(p) G S and.

(1) Adapted coordinate systems to Galilean space-time (or to a classical space-time) are the so-called "distinguished coordinate systems" [22] associated to the canonical fo­liation induced by means of the surjective map T. M has just a structure of 3-dimensional foliated manifold embedded in the 4-dimensional structure. In this structure the leaves, namely Mt = T (t), t G T, are the connected components. Then T results identifiable with the quotient manifolds M/T.

For a physical picture of erecting an adapted coordinate system see [8] p. 299, exercise 12.8.

— ^ R 4

110

bx0(p) £ I, where I = { V 6 M | < D , O > = 1}. Hence with adapted coordinates, the coordinate lines Xk,p have values in MT(P) only and the coordinate lines '%fp are not generally straight lines, in fact dxa are not constant vector fields. We call {xk} the space-coordinates and x° the time coordinate of {xa}. (Latin indices run from 1 to 3. Greek indices run from 0 to 3).

If the adapted coordinate system x : M -» R4 is an affine fiber bundle diffeomorphism then we get an adapted Cartesian coordinate system, i.e. {xa} is individuated by a basis {ea} of M such that {e1} is a basis of S.

Let JB-.1F-+V be a covariant functor between the category J* having fiber bundles as objects, and fiber bundle diffeomorphisms as morhisms, and the category v having differentiable manifolds as objects and diffeomorphisms as morphisms. Then an adapted coordinate system on M, x :M-> R4, induces an adapted coordinate system on 1B(M) given by B(x) : B(M) -* 1B(R4). For example considering the functor vT we get vT(x) = {xa

f x1} : vTM -* ZR4X R3.

Proposition 5.1. Let {x®} be an adapted coordinate system on M. {xa} will be another adapted coordinate system on M if and only if

dxa • x^- A&

where \& :M-> R is such that (A^(p)) is a matrix like I --j™ I with b e R3

and BGGL(3,R).

2. {xa} and {#a} are Cartesian also if and only if A^ = cost, i.e.

x^A^xa + b^

with (tf) 6 R4.

3. {£a} and 0ca} are Cartesian orthogonal also if and only if B G 0(3), i.e. ((A&), (&<*)) e L G (4) X/R4, where LG(4) is the group of 4X4 matrices like

l-rr- with beR3 and 5 e 0(3).

The following proposition allows us to relate the Galilean group with the group of change of Cartesian orthogonal adapted coordinate systems.

Lemma 5.1. If /GG then we get:

a) < o, v > = < o, f(v) >, v e M;

b) f\ I G GL(ir) = group of automorphisms of I C M.

I l l

Proposition 5.2. For any adapted coordinate system {xa} we realize the following isomorphisms:

a) LG^LG(4);

b) G s LG(4) X R4 A(4) = LG(4) X R4 = 4-dimensional affine group.

More precisely f = xa of=A<x x? + ba, VfeG and with (ba)eR4, G4<*)GLG(4).

Further i f / e S G one has the alternative isomorphisms: / l ! 0

a') SLG = 5G(4), where SG(4) is the group of 4 X 4 matrices like I - - j —

with b e R3 and B e 50(3).

b') SG^SLG(4)XR4.

Let us now calculate the space and the time-components of the natural basis {dxa}, associated to an adapted coordinate system {xa}.

Since the frame velocity, in adapted coordinates, is given by

i// = dx0+ \jjl dx( (i.e. i//°= 1),

we get

a) time-component: i (bxQ)0^ - \p (d*a)oi{, = <o,dxa> \jj= S ;

( (9^)o^ = 0 b) space-component: L (dxo)*^ = dx0 - \p

( O ^ ( ^ ) A ^ = OX(x~ \OX(x)o\jj ~ S \ yoXj)*^ — oXj

So if X = X® dxa is an arbitrary vector field on M, we get X0rp = X° \p and X^ = (X1 - X° \jj) 3*,-.

Let us now calculate the coordinate expression of the space and time-components of the derivative of a section f-.M-* W of a bundle of geometric objects:

a) time-component: (Df)0^ = \pOL5oCdx°®dxQ+\ljl\jja&adx0® dxi + (3xa • f) . \padx°® (dyp °./)0,// where {xa, y®} is a coordinate system on W and/** =y& of

b) space-component: {Df)^ = (5* - ty 5?) (dxf ~ V dx°) ® dxj + (3x,o /**) (<&'- frdx^mbypofl.

Now we are able to calculate the corresponding observed quantities.

I 0*oM*, lpU)=[^p),(3^o-*)(*(^p))U

{(9*,%, J is a basis for C°°{TS^), VteT. Thus we call {(8*,-) ,} a time-de-

112

pendent observed natural basis. For a vector field X-.M^- TM, such that in adapted coordinates it is represented by X=X0Lbxa we have:

X^t, [p]^ = W(t,p), dxi(Wt,p))]^ (X1 - X° ft) (i//(t,p)),

i.e. X^ = (X1 - X° fr) o y-1 (bx{)^ thus (X1 - X° \jj() o y"1: T X 5^ -> 1R are the

observed components of X

b) (D/)^ = 5{(<fc% ® 0*y)„ + (3*,. • f) o # (<&% ® (typ ° /)•

With respect to the affine connection G on Galilean space-time we have.

Proposition 5.3. In adapted coordinate the connection symbols G®y of G have the following properties:

a) G{k = [ik, s]^ where [ki, s] are the Christoffel symbols given by [ki, s] = = (1/2) [(3xfe -gis) + (3a?f- -g5*) - (bxj • g*,-)]; We call G%jk the sp<zc£ connection symbols. D ) ^/3a = ^a/3 = 0-

Proof, a) follows by a direct computation from equation Vg = 0 and b) from equation Va = 0. •

Attached to any frame \jj we recognize privileged coordinate systems: the frame-co ordinate systems, i.e. adapted coordinate systems {xa} such that \jj = dx0. For such systems the events belonging to a same world line 4>p: T -» ->• M, Vp EM, have the same space-coordinates -{V}. In fact we get: £>! (#' o i//) = 0 -> i//p is identified with the coordinate line x0p passing for p.

With respect to a frame-coordinate system we have further information about the connection symbols of G.

Proposition 5.4. Let ^ be a frame and {xa} an associated frame-coordinate system. Then we get:

1. Gj, = CJ. = 0.

2. {Gl00} represent the components of the acceleration of the frame, i.e.

V^ = G%QQbxi.

3. Further if \jj is an inertial frame Gloa- G^0 = 0.

An adapted coordinate system {xa} on M induces a time-dependent coordinate system on Sy, i.e. a map £' =xl° j~^\ T.XS^-+ IR such that {£J}, VtET, represents a coordinate system on S$. In fact, for each t£T, £,= = x°° $t : 5^ ->JR is a constant map. If {#a} is a frame-coordinate system,

113

then £Li :T-»ZR is a constant map, thus we obtain a (time-independent) coordinate system on S^ :y* :5^-> ZR, i.e. the following diagram

_ — • — ' . . i

is commutative. Then the observed connection coefficents G* : T X S^ -» 1R defined by the equation

are given by

where

[ft;, /]„,, = (1/2) [(3^ t(^), ,y/) + 0«y.« • (g*)M*> - 0&, t • fe#)*y)]

are the observed Christoffel symbols. If i// is a rigid frame we can express 5£- by means of 7^; more precisely

one has

* . *

with

r?i = (7*)li[kj,(\*s^iR

being

\kj, l \ = (1/2) [(dyk • (7„)y/) + (dyy • (7*)/*) - (8y, . (7*)*/)].

Therefore, when i// is a rigid frame 5^ is a Riemannian manifold with [&;, /] as Christoffel symbols.

Concluding remarks

The considerations developed in this paper allow us to emphasize the role of the bundle of geometric objects as the natural environment for physi­cal entities. On the other hand it is our opinion that a satisfactory theory of continuum mechanics must be a unitary field theory. (Here, of course, the unitarity must be understood in the framework of Newtonian physics). In

114

other words, a continuous body must be described by a suitable fiber bundle C on Galilei space-time, (configuration bundle), and by a differential equation on C (dynamical equation) (see ref. [14, HI]). Thus the configu­ration bundle must be a bundle of geometric objects in the sense of [15] in order to represent physical entities. In the next article we will develop this unified geometric picture considering the dynamic and constitutive aspects.

Thanks. I would like to thank Professor J. Marsden for his encouragement during the development of this research. I am also grateful to Dr. J.F. Pierce and the referees for their criticism and the detailed revision of this paper.

BIBLIOGRAPHY

[I] Bourbaki N., Topolope Generate, Hermann, Paris 1971.

[2] Cartan E., Ann. Scient. Ec. Norm. Sup., 40 (1923) 325, 41 (1924) 1.

[3] Coleman B.D., Thermodynamics of Materials with Memory, Arch. Rational Mech. Anal, 17 (1964) 1, 46.

[4] Coleman B.D. and Owen D.R., A Mathematical Foundation for Thermodynamics, Arch. Rational Mech. Anal, 54 (1974) 1, 104.

[5] Kunzle H.P., Galilei and Lorentz Structures on Space-time, Ann. Inst. Poincare, A, 17, 4 (1972) 337, 362.

[6] Kobayashi S., Theory of connections, Ann. Mat. Pura Appl. 43 (1975) 119,194.

[7] Lafontaine, J., Proceedings of Symposia in Pure Mathematics, 27 (1975) 3, 32.

[8] Misner C, Thome K., Wheeler J.A., Gravitation, W.H. Freeman and Co., San Francisco, 1973.

[9] Marsden J. and Hughes T., The Mathematical Foundations of Elasticity, (in pre­paration).

[10] M. Modugno, On the structure of classical kinematics, Riv. Mat. Univ. Parma, (4) 5 (1979), 249-269; (4) 6 (1980), 135-149.

[II] M. Modugno, On the structure of classical dynamics, Riv. Mat. Univ. Parma (4) 7 (1981).

[12] Noll W., A New Mathematical Theory of Simple Materials, Arch. Rational Mech. Anal, 48 (1972) 1, 50.

[13] A. Prastaro, Geometrodynamics of some non-re lativistic incompressible fluids,

115

Stocbastica 3 (2) (1979), 15-31.

[14] Prastaro A., On the General Structure of Continuum Physics. I: Derivative Spaces, Boll. Un. Mat. Ital. (5) 17-B (1980) 704, 726. II: Differential Operators, Boll. Un. Mat. Ital. Suppl. 1 (1981) 69-106. Ill: The Physical Picture, Boll. Un. Mat. Ital. Suppl. 1 (1981) 107-129.

[15] Salvioli S.P., On the Theory of Geometric Objects, J. Diff. Geom. 7 (1972) 257, 278.

[16] Switzer R.M., Algebraic Topology, Springer-Verlag, Berlin 1975.

[17] Trautman A., C.R. Acad. Sc, Paris, 257 (1963) 617; Fiber Bundles Associated with Space-time, Rep. Math. Phys. 1 (1970) 29, 62.

[18] Truesdell C. and Toupin R.A., The Classical Field Theories, in Flugge (ed.), Handbuch der Physik, Bd. 11 I/I, Springer, Berlin 1960.

[19] Truesdell C, A first Course in Continuum Mechanics, Academic Press, 1977.

[20] Truesdell C. and Wang C.-C, An Introduction to Rational Elasticity, Nordhoff International Publishing Co., Leydon, 1973.

[21] Vilms J., Connections on Tangent Bundles, J. Diff. Geom., 11 (1967) 235, 243.

[22] For an introduction an survey of the subject of foliations see e.g. the paper. Lawson H.B., Foliations, Bull. Am. Math. Soc, 80 (1974) 369, 418, and the book Kamber F.W. and Tondeur P., Foliated Bundles and Characteristic Classes, Springer Lecture Notes, Springer 1975.

APPENDIX

Al. Proof of Proposition 2.1. We need to prove that T can be identified with IR. Further, with respect to the canonical connection on M we shall prove that S/o ~Vg~0, where o — dr, and that this connection is symmetric. We shall proceed by steps. (1) The choice of an origin on T induces the identification T' = T. (2) The choice of an Euclidean structure on T_ induces the identification T = IR. (3) Equation Vc = V g = 0 can be seen as a particular case of a more gene­ral result. More precisely, let N be an affine space and let TTW-WNXW-+N be a trivial vector bundle on N. Let s = (idN,s~) be a constant section of TTW, (i.e. s-.N -> W is a constant map), then with respect to the affine connection we get Vs = 0.

In fact by considering the following commutative diagram

116

T*N ® TW = 0 (N, W)-~v&(N, W) = T*iV ® vTW DS \ y/lS

v . — where 0 = idT*N® 6 is the derivative operator canonically associated to the canonical connection 6 on W (see [14, II]), we get

V$(tf) (e, u) = (6°T(s)) (e, u) = 8(e, s~(e), u, o)~ (e, s(e), o, o), Ve £M, uG N = = space of free vectors of N.

Now a = dr and g are constant sections on M of the trivial vector bundles r * M = M X M * and vS°2M = M X S* ® S* respectibely. So we get Va = Vg = o.

(4) Finally, let us prove that the canonical connection associated with the affine structure on M is symmetric. We must prove that the torsion map T is zero. Recall [21] that

T=(l/2)(ir'0G-ir'0GoS): T2M-+TM,

where: (a) G is the connection on M seen as the splitting on the left of the following exact sequence:

0 -* vTTMj* TTM -> ir*TM -+0; (n is the canonical projection G TM-+M);

(b) 7r' is the canonical map vTTM^-M; (c) 5 is the isomorphism

s : T2M = T2M given by s(p, u, v, w) = (p, v, u, w).

Then one has:

ir' O G(/7, &, if, Zi;) = 7r'(/7, &, O, W) = (p , Itf),

7r' O G O 5(/7, w, v, zu) = TT' ° G(p, v, w, w) = ir\p, v, o, w) = (p, w). D

^42. Proof of Proposition 4.3. Let us prove the points 2, 3 and 4.

2. X ^ . W K / " - , 1 ,

(VX)* =ir*°GoDX°$ =f[*o(idT.M9G)oDXoj$ = [?*®(r2(#/)oG)]°DXo/;j},

= [5F* ®(T2(^/)oGo 7 V * K ° 7*(¥/)]o£>Xo#

117

where 7Ty:T*M -> T*Sy is the projection induced by ty. On the other hand we get (T2(^/)oG) (p, «, v, w) = [p, i^(p) , 0, ^* (p ) ]* ,

GyitoT2(V/)(p, u, v, w) = G%t([p, w.*(p), *.*'(p), w.*(p) 4- gradtf2(p)(ii, *)]*)

= TH^/^GoT2^)^, O, O, O, [p, u^(p), v.*(p), w^(p) + ^rld^2(p)(w, v)U

= T2(^/)oG(^a, p), Z32 *(*,p)(«.*(p)), D2*(t, p)(v^(p)),

522V(t,p)(u^(p), v^(p)+D2^(typ)(w^(p)+B2^(t,p)(^M2(p)(u, v))

= T2(W)(V(t,p),D2V(t,p)(u^(p)\ o,B22V(t,p)(u^(p\ v^(p)) +

S2:¥(t, p)(w.*(p))+52tt(f, p)(grld#2(p)(™, »)))

= [tf(*,p), Vt,t(p)(p)(u-*(p)), o, %MP)(P)(W~*(P» +

+ ^lr(p)(p)(u^(p), V„v(p)+%Mp)(p)(glC2i<M2(p)(u, V))]y .

3. Dg%t = r*(;^t)®T(52(^/))oDg o /"£„

Vg =G^ t oDg^ t ,

* _ (V g )([pk)([p, uU)^G%toDg^t([ph)([p, uh)

t-v,t

= G%toT(S20W))(Dgmt,p)))Wt, p), *ttT(p)<p)(u)))

= (G^,oT(52(^/))) <¥<t,p),£ %Mp)(p)(u\ o)

= G%t([*(t,p),g, %Mp)(p)(u), oU)

4. G*{t, [p, u, », w]*) = [i//(*,p), ^>T(p)(«), ^ (T(p)(w)k = [p, «, w]*;

AGOSTINO PRASTARO, Dipartimento di Matematica, Universita della Calabria, Roges (Cosenza), Italia.

Lavoro pervenuto in redazione il 20- V-1981