ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS

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1 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS ADRIAN CONSTANTIN AND BORIS KOLEV Abstract. Accoring to the principle of least action, the spatially perioic motions of one-imensional mechanical systems with no external forces are escribe in the Lagrangian formalism by geoesics on a manifol-configuration space, the group D of smooth orientation-preserving iffeomorphisms of the circle. The perioic invisci Burgers equation is the geoesic equation on D with the L right-invariant metric. However, the exponential map for this right-invariant metric is not a C 1 local iffeomorphism an the geometric structure is therefore eficient. On the other han, the geoesic equation on D for the H 1 right-invariant metric is also a re-expression of a moel in mathematical physics. We show that in this case the exponential map is a C 1 local iffeomorphism an that if two iffeomorphisms are sufficiently close on D, they can be joine by a unique length-minimizing geoesic - a state of the system is transforme to another nearby state by going through a uniquely etermine flow that minimizes the energy. We also analyze for both metrics the breakown of the geoesic flow. 1. Introuction Motions of mechanical systems with no external forces are escribe in the Lagrangian formalism by paths on a configuration space G that is a Lie group. The velocity phase space is the tangent bunle T G of G. Let G be the Lie algebra of G - the tangent space at the neutral element of the group. For a nonegenerate inner prouct,, the quantity 1 v, v, v G, is calle the kinetic energy K. We can exten K by right- or left translation 1 to a right- or left-invariant Lagrangian L : T G R in orer to efine a natural Lagrangian system on G, cf. []. The action along a path g(t), a t b, in G is efine as ˆ b a L(g, g t ) t. The Action Principle cf. [41] states that the equation of motion is the equation satisfie by an extremal (a critical point) of the action in the space of curves on G, the paths g(t) over which we are extremizing satisfying the fixe en conitions g(a) = g an g(b) = g 1. In many cases, cf. [], the paths escribe by the motion of a mechanical system are not only extremals but also (local) minimal values of the action functional - the Principle of Least Action hols. Observe that if g(t), a t b, is a C 1 -regular path (i.e. g t on [a, b]) joining g(a) = g to g(b) = g 1, the action a(g) = 1 ˆ b other han, the length l(g) = a g t, g t t epens on the parametrization of the path. On the ˆ b a g t, g t 1 t oes not epen on the parametrization an Date: May. Mathematics Subject Classification. 35Q35, 58B5. Key wors an phrases. geoesic flow, iffeomorphism group of the circle. The authors are grateful to Professor Lars Hörmaner an to Professor Viktor Schroeer for useful iscussions. 1 In general, the Lagrangian is a scalar function L : T G R so that constancy uner particular transformation of its arguments is the only sort of symmetry to which it can be subject. This explains the preferre choice of right- or left invariance. 1

2 A. CONSTANTIN AND B. KOLEV l (g) (b a) a(g), with equality if an only if g t, g t is constant on [a, b]. From here we infer that the (local) minimum of the action is realize by the curve of minimal length joining g to g 1. In conclusion, for the Principle of Least Action to hol, it is necessary that the equation of motion is the geoesic equation on the configuration manifol. The configuration space of a rigi boy fixe at its centre of mass is the group SO(3) of rotations of R 3. An element g of the group correspons to a position of the boy obtaine by the motion g from some arbitrarily chosen initial state (corresponing to the ientity element of the group) an a rotation velocity g t of the boy is a vector in the tangent space T g G. The kinetic energy of a boy is etermine by the vector of angular velocity in the boy (obtaine by carrying the tangent vector to G, the tangent space at the ientity, by left translation) an oes not epen on the position of the boy in the space. Therefore, the kinetic energy gives a left-invariant Riemannian metric on the group. By the Principle of Least Action, cf. [], the motion of a rigi boy with no external forces is a geoesic in SO(3) with this left-invariant metric. The motion of a system in continuum mechanics is escribe by a path of iffeomorphisms ϕ(t, ) of the ambient space. The knowlege of ϕ(t, ) gives the configuration of the particles at time t. The material velocity fiel is efine by (t, x) ϕ t (t, x) while the spatial velocity fiel is given by u(t, y) = ϕ t (t, x) where y = ϕ(t, x), i.e. u(t, ) = ϕ t ϕ 1. In terms of u we have the spatial or Eulerian escription (from the viewpoint of a fixe observer) while in terms of (ϕ, ϕ t ) we have the material or Lagrangian escription (the motion as seen from one of the particles - the observer follows the particle). Note the following right-invariance property: if we replace the path t ϕ(t) by t ϕ(t) η for a fixe time-inepenent η D, then the spatial velocity u = ϕ t ϕ 1 is unchange. This suggests the choice of right-invariance rather than left-invariance. In the case of a perfect flui (nonviscous, homogeneous an incompressible) moving in a boune smooth omain M R k, k =, 3, the configuration space is the group of all volume-preserving ˆ iffeomorphisms of M. Arnol [1] observe that the kinetic energy of the flui, 1 u(t, x) x, M is invariant with respect to right translations. The invariance of the kinetic energy with respect to right translations is ue to incompressibility (the iffeomorphisms are volumepreserving), as one can see from a simple change of variables. The obtaine geoesic equation is the Euler equation of hyroynamics [1]. In this paper we consier the one-imensional compressible analogue of the escription of the Euler equation for a perfect flui in two an three imensions by means of geoesics on the group of volume-preserving iffeomorphisms, a escription establishe by Arnol [1] an place on a rigorous founation by Ebin-Marsen []. The group D of smooth orientation-preserving iffeomorphisms of the circle S (the real numbers moulo 1) represents the configuration space for the spatially perioic motion of inertial one-imensional mechanical systems. The choice of the L inner prouct on each tangent space oes not provie us with a right-invariant metric in the one-imensional compressible case - incompressibility in one imension woul force the iffeomorphisms to be linear. We are therefore le to efine an inner prouct on the tangent space at the ientity an prouce a right-invariant metric by transporting this inner prouct to all tangent spaces of D by means of right translations. For the L right-invariant metric one obtains the invisci Burgers equation as the geoesic equation on D, (1.1) u t + 3uu x =. A rigi boy is a system of point masses constraine by the fact that the istance between points is constant cf. [].

3 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 3 The geometric approach is meaningful if we are able to use some methos that have been evelope in finite imensional Riemannian geometry. Unfortunately, as we shall see in Section 3, the Riemannian exponential map is not a C 1 local iffeomorphism in the case of the L right-invariant metric. This raises the natural question whether another right-invariant metric may lea to meaningful results. In view of this, we stuy the geoesic flow on D enowe with the H 1 right-invariant metric 3. The choice of this metric is motivate by the fact that the corresponing geoesic equation is a re-expression of a moel arising both in shallow water theory [8] an in elasticity [18], (1.) u t + uu x + x (1 x) 1 u + 1 u x In any irection at a given point of D there exists a smooth geoesic on D. We show that the Riemannian exponential map of the H 1 right-invariant metric is a C 1 local iffeomorphism. We also prove that with the H 1 right-invariant metric D is not geoesically complete an we analyze the breakown of the geoesic flow. Finally, we show that if two iffeomorphisms are sufficiently close on D, they can be joine by a unique lengthminimizing geoesic of the H 1 right-invariant metric within D. This can be reformulate as a variational problem in the family of smooth iffeomorphisms of the circle an illustrates the power of the geometric approach. Intuitively, it says that a state of the system is transforme to another nearby state by going through a uniquely etermine flow of (1.) that minimizes the energy. =.. Right-invariant metrics on D In this section we present the manifol an Lie group structure of D, the group of orientation-preserving C -iffeomorphisms of the circle, an we iscuss the enowment of D with a Riemannian structure..1. The iffeomorphism group. D is a connecte manifol moele on the Fréchet space C (S) of smooth maps of the circle (the family of real smooth maps on R of perio one), cf. [6]. Recall that a Fréchet space is a complete metrizable topological vector space, its topology being efine by a countable collection of seminorms { n }: a sequence u j u if an only if for all n 1 we have u j u n as j. On C (S) we consier the seminorms to be the H k (S)-norms with k. If F 1, F are Fréchet spaces, U F 1 is open an f : U F 1 F is a continuous map, the erivative of f f(u + tv) f(u) at u U in the irection v F 1 is efine by Df(u) v = lim. We say t t that f is C 1 on U if the limit exists for all u U, v F 1, an if Df : U F 1 F is continuous 4. Higher erivatives are efine as erivatives of the lower ones. The composition an the inverse are both smooth maps from D D D, respectively D D, so that the group D is a Lie group cf. [6]. The Lie algebra G of D is the tangent space to D at the ientity, T I D C (S), with the bracket [u, v] = (u x v uv x ), u, v G. 3 H k (S), k N, stans for the Sobolev space of functions with istributional erivatives up to orer k having finite L (S) norm. 4 The efinition iffers from the case of Banach spaces ue to the fact that in general the space of linear maps of F 1 to F will not form a Fréchet space. See [6] for a review of the intricacies of the Fréchet ifferential calculus.

4 4 A. CONSTANTIN AND B. KOLEV Each vector fiel v on S (equivalently, each v T I D) gives rise to a one-parameter group of iffeomorphisms {η(t, )} obtaine as solutions of the ifferential equation (.1) η t = v(η) in C (S) with initial conition η() = I D. On the other han, each one-parameter subgroup t η(t) D is uniquely etermine by its infinitesimal generator v = t η(t) t= T I D, the limit being consiere in the C (S) topology. Evaluating the flow t η(t, ) etermine by (.1) at t = 1 we obtain a iffeomorphism exp L (v). The iffeomorphism η(t, ) is given explicitly by η(t, x ) = x(t, x ), x S, where x(t, x ) is the unique global solution of the orinary ifferential equation x = v(x) with ata x() = x t, cf. [35]. The map v exp L (v), calle the Lie-group exponential map, is a smooth map of the Lie algebra to the Lie group. Although the erivative of exp L at the zero vector fiel is the ientity, exp L is not locally surjective cf. [35] so that the Lie-group exponential map cannot be use as a local chart on D. This failure is possible since the inverse function theorem oes not necessarily hol in Fréchet spaces cf. [6]. Note the contrast with the case of finite-imensional Lie groups where the map exp L is always a local iffeomorphism from the Lie algebra to the Lie group [37]. Let F(D) be the ring of smooth real-value functions efine on D an X (D) be the F(D)-moule of smooth vector fiels on D. For X X (D) an f F(D), we efine in a local chart the Lie erivative L X f as f(ϕ + h X(ϕ)) f(ϕ) L X f(ϕ) = lim, ϕ D. h h To efine the Lie bracket of X, Y X (D) we also procee in local charts, cf. [35]. If U C (S) is open an X, Y : U C (S) are smooth, we enote Y (ϕ + h X(ϕ)) Y (ϕ) D X Y (ϕ) = lim, ϕ D. h h We are le to efine the vector fiel [X, Y ] = D X Y D Y X. This efinition is covariant an efines globally L X Y = [X, Y ]. Let X R (D) be the space of all right-invariant smooth vector fiels on D. Note that X X R (D) is etermine by its value u at I, X(η) = R η u for η D, where R η stans for the right translation. The bracket [X, Y ] of X, Y X R (D) is a right-invariant vector fiel 5 an [X, Y ](I) = [u, v], where u = X(I), v = Y (I), cf. [35]... Right-invariant metrics. T I D C (S) is not a Hilbert space. We efine a weak right-invariant Riemannian metric on D as follows. We consier on T I D C (S) a nonegenerate continuous inner prouct,,. That is, u u, u is a continuous (hence smooth) map on C (S) an the relation u, v = for all v C (S) forces u = ; a typical example woul be the H s (S)-inner prouct with s. To efine on D a smooth right-invariant Riemannian metric, we exten this inner prouct to each tangent space T η D by right-translation, i.e. (.) V, W (η) := V 1 η 1, W η for V, W T η D. Each open set of the topology inuce by this inner prouct is open in the Fréchet space C (S) but the converse is not true - we efine a weak topology on C (S). 5 In view of the above iscussion of exp L, every right-invariant vector fiel has a smooth flow on D. The proof that the bracket preserves right invariance is therefore stanar.

5 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 5.3. Covariant erivative. In orer to efine parallel translation along a curve on D an to erive the geoesic equation of the metric efine by (.), it is necessary to show the existence of a covariant erivative which preserves the inner prouct (.). Let us point out that, given a smooth Riemannian metric on D, the existence of a metric covariant erivative is not ensure on general grouns as we eal with a Fréchet manifol. For the group of volume-preserving iffeomorphisms, the existence of the metric covariant erivative has been establishe in []. We shall see that a evelopment relate to the ieas consiere in [1] an [] yiels an existence result for the covariant erivative in the case of a right-invariant metric on D. As the existence of such a covariant erivative is assume in the literature [3], it is of interest to provie a rigorous proof for it. Recall that a covariant erivative is efine as a R-bilinear operator : X (D) X (D) X (D) with the following properties: (i) X(η) = implies X Y (η) = (punctual epenence in X), (ii) X Y Y X = [X, Y ] for X, Y X (D) (torsion free), (iii) X (fy ) = (L X f) Y + f X Y for f F(D) an X, Y X (D), (iv) an for all X, Y, Z X (D), L X Y, Z = X Y, Z + Y, X Z (compatibility with the metric). Observe that (i) an the R-linearity in X force X Y to be F(D)-linear in X. In finite imensions, punctual epenence on X an F(D)-linearity in X are equivalent but this cannot be ensure in infinite imensions cf. [3] pp. -3. Since D is a Fréchet manifol with a weak Riemannian metric, in general the existence of a covariant erivative is not ensure, cf. [3], [3]. A sufficient conition for the existence of a covariant erivative is given by Theorem.1. Assume that there exists a bilinear operator B : C (S) C (S) C (S) such that 6 (.3) B(u, v), w = u, [v, w], u, v, w C (S). Then there exists a unique Riemannian connection on D associate to the rightinvariant metric,, given by ( X Y ) η = [X, Y Yη R ] η + 1 [Xη R, Yη R ] η B(Xη R, Yη R ) η B(Yη R, Xη R ) η, where for X X (D), we enote by Xη R the right-invariant vector fiel whose value at η is X η an we exten B to a bilinear map on the family X R (D) of right-invariant vector fiels, B : X R (D) X R (D) X R (D) by B(Z, W ) η = R η B(Z I, W I ) for η D an Z, W X R (D). In the proof of Theorem.1 we will use Lemma.. Consier on D a smooth right-invariant metric inuce by an inner prouct,. If X, Y, Z X (D) an Y η = at some η D, then L X Y, Z η = [X, Y ], Z η. Proof. Write the relation to be prove as L X R h 1Y h, R h 1Z h (η) = R η 1[X, Y ] η, R η 1Z η, e 6 The operator B was introuce by Arnol [1] in the Lagrangian formulation for Euler s equation of motion of a perfect flui in a boune omain Ω R 3. For Hilbert manifols the existence of B is guarantee by the Riesz representation theorem, cf. [3]. e

6 6 A. CONSTANTIN AND B. KOLEV where R ϕ stans for right translation an e = I. Being in the Lie algebra of D, we may specify L X R h 1Y h, R h 1Z h (η) = D X R h 1Y (h) (η), R η 1Z η so that is suffices to show that D X R h 1Y (h) e (η) = R η 1[X, Y ] η. This last relation is true in a Hilbert space H as we can erive R η 1 which belongs to the Hilbert space L(H, H) of continuous linear operators from H to H (note that L(C (S), C (S)) is not a Fréchet space, cf. [6]). Therefore the last equality hols in each H k (S), k, an we infer the result from here if we take into account the efinition of convergence on C (S). Proof of Theorem.1. As the proof is rather involve, we procee in several steps. We first show that uniqueness is ensure. Assuming the existence of, we erive its expression on right-invariant vector fiels an show that this etermines completely. Our last task will be to show that the obtaine explicit formula for satisfies properties (i)-(iv). STEP I. We show the uniqueness of an, assuming existence, we erive its expression on right-invariant vector fiels. Let us write own (iv) for a cyclic permutation of X, Y, Z X (D), L X Y, Z = X Y, Z + X Z, Y, L Y X, Z = Y Z, X + Y X, Z, L Z X, Y = Z X, Y + Z Y, X. Aing the first two relations an substracting the thir, the following ientity can be erive e (.4) X Y, Z = [X, Y ], Z [Y, Z], X + [Z, X], Y, L X Y, Z L Y X, Z + L Z X, Y, if we take into account (ii). Since the inner prouct, is non-egenerate, the previous formula shows the uniqueness of. Let X R (D) be the space of all right-invariant smooth vector fiels on D. Due to the right-invariance of the metric, Y, Z is constant for Y, Z X R (D) so that L X Y, Z = for all X X (D). Therefore, (.4) reuces to X Y, Z = [X, Y ], Z [Y, Z], X + [Z, X], Y, X, Y, Z X R (D). We evaluate this relation at e = I to obtain by means of (.3) that R η 1( X Y ) η, Z e = [X, Y ] e, Z e e [Y, Z] e, X e e + [Z, X] e, Y e e, e = [X, Y ] e, Z e e B(X e, Y e ), Z e e B(Y e, X e ), Z e e, as R η 1X η = X e by right-invariance an since the Lie bracket of two right-invariant vector fiels is a right-invariant vector fiel. We get (.5) ( X Y ) η = 1 [X, Y ] η R η B(X e, Y e ) R η B(Y e, X e ), η D, which is the expression of on right-invariant vector fiels.

7 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 7 STEP II. Assuming the existence of we erive an explicit formula for it. If X X (D), we enote by Xη R the right-invariant vector fiel on D whose value at η D is X η. If exists, then (.6) ( X Y ) η = [X, Y Y R η ] η + ( X R η Y R η ) η, η D. Inee, by (ii) must be torsion free so that an (i) yiels X (Y Yη R ) (η) (Y Y R η )X (η) = [X, Y Yη R ] η X (Y Yη R ) (η) = [X, Y Yη R ] η. Combining this with (i) we obtain (.7) ( X Y ) η = [X, Y Y R η ] η + ( X Y R η ) η = [X, Y Y R η ] η + ( X R η Y R η ) η, which is the only possible formula for. STEP III. We efine by (.7) an check that it satisfies all require properties (i) (iv). It is useful to write own (.7) in the more etaile form ( X Y ) η = [X, Y Yη R ] η + 1 [Xη R, Yη R ] η B(Xη R, Yη R ) η B(Yη R, Xη R ) η, where we extene B to a bilinear map B : X R (D) X R (D) X R (D) by B(Z, Z ) η = R η B(Z e, Z e) for η D an Z, Z X R (D). Since the vector fiel (Y Yη R ) is zero at η, we have [X, Y Yη R ] η = [Xη R, Y Yη R ] η as one can see by going to local charts. We have therefore a secon equivalent explicit form of (.7), ( X Y ) η = [Xη R, Y Yη R ] η + 1 [Xη R, Yη R ] η B(Xη R, Yη R ) η B(Yη R, Xη R ) η. Clearly is R-bilinear. The above explicit form of (.7) shows that ( X Y ) η epens only on the value X η of X at η. Property (iii) can be easily checke as the expression [Xη R, Yη R ] η B(Xη R, Yη R ) η B(Yη R, Xη R ) η is tensorial. To verify that is torsion free, note that the above two explicit forms of (.7) yiel ( X Y ) η = ( X R η Y ) η so that, by these formulas ( X Y Y X) η = ( X R η Y Y X) η = [X R η, Y Y R η ] η [Y, X X R η ] η + [X R η, Y R η ] η which cancels to [X, Y ] η. To complete the proof, we have to check that efine by (.7) is compatible with the metric. To prove (iv) at a given η D amounts to show that L X Y, Z η = [X R η, Y Y R η ], Z η + [X R η, Z Z R η ], Y η as the remaining parts cancel. Due to bilinearity, it will be enough to verify the above equality for the triples (X, Y Yη R, Z), (X, Yη R, Z Zη R ) an (X, Yη R, Zη R ). The first two triples satisfy the equality in view of Lemma 1 while for the thir triple the verification is obvious as both sies are zero. We prove therefore that there exists a unique Riemannian connection on D associate to the right-invariant metric,. From its explicit form we see that maps right-invariant vector fiels into right-invariant vector fiels.

8 8 A. CONSTANTIN AND B. KOLEV.4. Derivative along a curve an parallelism. Let us now construct a erivation along curves. Let J R be an open interval an consier a C 1 -curve α : J D. By a lift γ of α we mean a C 1 -curve γ : J T D lying above α. If Lift(α) is the set of lifts of α, we efine the erivation D αt : Lift(α) Lift(α) along α in local coorinates by (.8) D αt γ = γ t Q(α t α 1, γ α 1 ) α, γ Lift(α), where the bilinear operator Q : C (S) C (S) C (S) is efine by Q(u, v) = 1 u x v + uv x + B(u, v) + B(v, u), u, v C (S). If α is inuce by a vector fiel, we recover the expression of the covariant erivative. Inee, let X, Y X (D) be such that γ(t) = Y (α(t)) on J an α t (t ) = X(α ), α = α(t ) for some t J. If X R, Y R, are the right-invariant vector fiels on D whose values at α D are X(α ), respectively Y (α ), we have, cf. STEP III in the proof of Theorem.1, that ( X Y ) (α ) = [X R, Y ] (α ) 1 [X R, Y R ] (α ) Accoring to Section.1, in local coorinates, [X R, Y R ] (α ) = γ(t ) +B(X R, Y R ) (α ) + B(Y R, X R ) (α ) [X(α ) α 1 ] x α X(α ) On the other han, writing out explicitly the efinition, we see that [X R, Y ] (α ) = γ t (t ) γ(t ) [X(α ) α 1 ] x α, thus D αt γ(t ) = ( X Y )(α(t )). Let us now prove. [γ(t ) α 1 ] x α. Lemma.3. Let J R be an open interval an consier a C 1 -curve α : J D. If γ 1, γ Lift(α), then (.9) t γ 1, γ = D αt γ 1, γ + γ 1, D αt γ, t J. Proof. The metho is quite similar to the one we use in the case of vector fiels. Let us fix t J. First we establish, the same way as in Lemma., that (.1) t γ 1, γ = ( t=t t γ 1)(t ), γ (t ) if γ 1 (t ) =. Then we prove that (.9) is satisfie at t = t by the three couples (γ 1 γ R 1, γ ), (γ R 1, γ γ R ), an (γ R 1, γ R ), where γi R (t) = R α(t) R α(t ) 1 γ i(t ), t J, i = 1,. Inee, efining the right-invariant vector fiels Yi R whose values at α(t ) are γ i (t ), observe that γi R (t) = Yi R (α(t)), t J, i = 1,. That (.9) is true for the first two couples is a irect consequence of (.1). On the other han, since γi R erive from the vector fiels, by the compatibility of the covariant erivative with the metric we have Y R i L X R Y1 R, Y R α(t) = X RY1 R, Y R α(t) + Y1 R, X RY R α(t) = D αt γ1 R (t), γ R (t) + γ1 R (t), D αt γ R (t), where X R is the right-invariant vector fiel on D whose value at α(t ) is α t (t ). But L X R Y1 R, Y R α(t ) = t γr 1, γ R, t=t

9 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 9 as one can check using the fact that X R has a flow cf. the iscussion of the Lie group exponential map on D. Therefore the thir couple satisfies (.9) at t = t too. Aing up these three relations, we obtain (.9) at t = t. Due to the arbitrariness of t D, the proof is complete. If ϕ : J D is a C -curve, we say that a lift γ : J T D is ϕ-parallel if D ϕt γ on J. In local coorinates, enoting ϕ t ϕ 1 = u, γ ϕ 1 = v, this is equivalent to requiring that v C 1 (J; C (S)) is a solution of the equation (.11) v t = 1 vu x v x u + B(u, v) + B(v, u), A C -curve ϕ : J D is calle a geoesic if D ϕt ϕ t on J. With u = ϕ t ϕ 1 T I D C (S) we can write the geoesic equation as (.1) u t = B(u, u), t J. Both (.11) an (.1) are ifferential equations in the Fréchet space C (S). The classical local existence theorem for ifferential equations with smooth right-han sie oes not hol in C (S), cf. [6]. We aopt the following approach. We complete C (S) uner the H k (S)-norm (k ), eal with the resulting Hilbert manifol D k, an then show that the solutions of the equation uner stuy actually are C if the ata is smooth. More precisely, for k, let D k = η H k (S), η is bijective, orientation preserving an η 1 H k (S). D k, k, is only a topological group an is not a Lie group as the composition map an the inverse map D k D k D k, (f, g) f g, D k D k, f f 1, are merely continuous, not C. For ϕ D k, right composition is a C map but left composition R ϕ : D k D k, R ϕ (η) = η ϕ, η D k, L ϕ : D k D k, L ϕ (η) = ϕ η, η D k, is continuous without being locally Lipschitz. However, the composition regare as a map D k+n D k D k an the group inverse regare as a map D k+n D k are both of class C n. D k, k, is a Hilbert manifol moelle on T I D k H k (S); see [3] for a etaile treatment of these matters. In our approach, the stuy of the structure of all the D k, k, with respect to a given right-invariant metric will enable us to obtain results for D as the geoesic flow on D k preserves D. 3. The L right-invariant metric Since T I D is a smooth function space, the most natural inner prouct to start with woul be the L inner prouct ˆ u, v L = u(x)v(x) x on T I D C (S). S

10 1 A. CONSTANTIN AND B. KOLEV In the case of the smooth right-invariant metric obtaine by right-translation by means of (.), it is easy to check that B(u, v) = u v uv for u, v C (S). The geoesic equation for the L right-invariant metric is (3.1) u t + 3uu x =, where t ϕ(t) is the geoesic curve starting at time t = at the ientity I in the irection u T I D, u = ϕ t T ϕ(t) D. Note that (3.1) is a ifferential equation in C (S). Equation (3.1) is part of the system (3.) with initial ata ϕ() = I, u C (S). ϕ t = u(t, ϕ) u t + 3uu x = 3.1. Burgers equation. Equation (3.1) is the well-known invisci Burgers equation [7]. Though rather simple, it is a successful mathematical moel of gas ynamics [4]. This partial ifferential equation was investigate in great etail. If u H k (S) with k, then equation (3.1) with initial ata u(, ) = u has a unique solution u C([, T ); H k (S)) C 1 ([, T ); H k 1 (S)) for some maximal time T >, cf. [31]. Moreover, on [, T ), u(t) epens continuously on the initial ata in the H k (S)-norm, while Höler continuity with any prescribe exponent generally oes not hol - see [3]. Equation (3.1) can be analyze by the metho of characteristics. If u H k (S), k, then the solution u C([, T ); H k (S)) satisfies (3.3) u(t, x + 3t u (x)) = u (x), t [, T ), x S. Using this, one can see 7 that the maximal existence time is precisely (3.4) T = min {x S: u (x)<} 1 3u (x) Since u is perioic, we euce that all solutions but the constant functions have a finite life-span. The evelopment of singularities is also well-unerstoo: if u H k (S), k, is not constant, then while max x S ª >. u(t, x) = max u (x), t [, T ), x S min x S u x(t, x) as t T <. Note that on [, T ) we have u(t, ) H (S) C 1 (S). Relation (3.3) is useful in etermining the blow-up rate lim (T t) min {u t T x(t, x)} = 1 x S Existence of geoesics. It is quite natural to view (3.1) as the geoesic equation for the right-invariant L -metric on D k with k. However, this nees further justification since, in contrast to the case of D, we can not start from the notion of covariant erivative to efine the geoesics. Note that the allege covariant erivative given by Theorem 1 is not well-efine on D k ue to loss of smoothness. We woul also like to point out that if X X (D k ), k, then the map η X(η) η 1 is only continuous on D k so that the L right-invariant metric on D k is not smooth whereas the L right-invariant metric on D is smooth. To fully justify why we are entitle to call (3.1) the geoesic equation on D k, 7 For etails we refer to [8].

11 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 11 we will show that it arises from the necessary conition for a curve on D k to be locally length minimizing. For each C 1 curve γ : [a, b] D we efine its length by l(γ) = ˆ b a γ t (t) γ(t) t = ˆ b a γ t γ 1, γ t γ 1 1 t. We can exten the length to piecewise C 1 paths on D by taking the sum of the lengths of the C 1 -components of the curve. Since D is connecte cf. Section.1, any two points on D can be joine by a piecewise C 1 path. We say that a C 1 -path γ : [a, b] D has a regular parametrization if γ t at every t [a, b]. Any such curve can be reparametrize by arc length cf. [3], i.e. there is a new parametrization ϕ : [, c] D of the path such that ϕ t ϕ(t) = 1 on [, c]. The action along a path γ : [a, b] D is the quantity a(γ) = 1 ˆ b a γ t ϕ(t) t. Unlike the length l(γ), the action a(γ) epens on the parametrization. If the curve is parameterize by arc length, we have l(γ) = a(γ). This allows us to pass freely from one notion to the other. Let us now fin a necessary conition for a regularly parameterize path to be the shortest path on D between its enpoints. In view of the previous comments, we can assume the path to be parameterize by arc length, γ : [, c] D an γ is a critical point in the space of paths for the action functional, i.e. ɛ a(γ + ɛη) = ɛ= for every path η : [, c] D with enpoints at zero an such that γ + ɛη is a small variation of γ on D. But ɛ a(γ + ɛη) = ɛ= ˆ 1 c ˆ (γ t + ɛη t ) (γ + ɛη) 1 x t ɛ S ɛ= ˆ c ˆ = (γ t γ 1 ) (γ 1 ª t + ɛη t ) (γ + ɛη) x t. ɛ ɛ= Note that ɛ S (γ t + ɛη t ) (γ + ɛη) 1 ɛ= = η t γ 1 + (γ tx γ 1 ) ɛ = η t γ 1 (γ tx γ 1 ) η γ 1 γ x γ 1, (γ + ɛη) 1 as ifferentiation with respect to ɛ in the relation (γ + ɛη) (γ + ɛη) 1 = I leas to γ x (γ + ɛη) 1 an therefore We infer that ɛ a(γ + ɛη) ɛ= = ɛ (γ + ɛη) 1 + ɛ η x (γ + ɛη) 1 ˆ c ɛ ˆ S (γ + ɛη) 1 = η γ 1 ɛ= γ x γ. 1 ɛ= ɛ (γ + ɛη) 1 + η (γ + ɛη) 1 = (γ t γ 1 ) η t γ 1 (η γ 1 ) x (γ t γ 1 ) S x t. Denoting γ t γ 1 = u, we fin ˆ c ˆ ɛ a(γ + ɛη) = u [η t γ ɛ= 1 u x η γ 1 ] x t S ˆ c ˆ = u [ t (η γ 1 ) + u x (η γ 1 ) u x (η γ 1 )] x t

12 1 A. CONSTANTIN AND B. KOLEV since t (η γ 1 ) = η t γ 1 + η x γ 1 t (γ 1 ) = η t γ 1 η x γ 1 γ t γ 1 γ x γ 1. Inee, ifferentiating the relation γ γ 1 = I with respect to time, we get γ t γ 1 + γ x γ 1 t (γ 1 ) = which gives the esire expression for t (γ 1 ). Integrating by parts with respect to t an x in the above formula for the erivative of the action functional, we obtain ˆ c ˆ ɛ a(γ + ɛη) = (η γ ɛ= 1 ) [u t + 3uu x ] x t. S This calculation can be performe on D as well as on D k, k, an yiels the Euler- Lagrange equation u t + 3uu x = where u = γ t γ 1 an t γ(t) D is the curve (parameterize by arc length) yieling the critical point of the length functional to be minimize. The variational formulation gives a meaning to the geoesic equation on D k, k. To procee, we shall nee Lemma 3.1 (cf.[6]). Let F C([, T ); H k (S)) with k. Then the ifferential equation ϕt = F (t, ϕ) has a unique solution ϕ C 1 ([, T ); D k ). ϕ() = I The consierations in Section 3.1 show that for any u H k (S), k, the system (3.) efines a unique C 1 -curve t ϕ(t) D k on a maximal time interval [, T ) with T > is given by (3.4). Note that we obtaine the geoesic equation (3.1) - a geoesic for the L right-invariant metric being efine to be a C 1 -curve satisfying (3.1). The iscussion in Section.4 woul suggest to efine geoesics as C -curves t ϕ(t) D k satisfying (3.1) whereas our approach yiels only a C 1 -epenence on time. It is not possible to require ϕ C ([, T ); H k (S)) as this assumption woul lea by (3.) to ϕ tt = u t ϕ+u ϕ u x ϕ = u ϕ u x ϕ C([, T ); H k (S)). Letting in this relation t we woul obtain that u u H k (S) for all u H k (S), a contraiction. Inspecting the previous consierations it becomes clear that we prove Proposition 3.. For every u T I D k H k (S), k, there exists a unique geoesic on D k starting at I in the irection of u. This geoesic is efine for some finite maximal time T > unless u is constant. From the etaile iscussion of the equation (3.1) we know that if u C (S), then the unique solution u of (3.1) with ata u belongs to C 1 ([, T ); D) with T given by (3.4). By a recursive argument using (3.1) we euce that u C ([, T ); D). Applying Lemma 3 for all k we obtain again by a recursive argument that Theorem 3.3. For every u T I D C (S), there exists a unique geoesic ϕ C ([, T ); D) starting at I in the irection of u. The only geoesic that can be continue inefinitely in time is the one in the constant irection. For every u H k (S), k, we efine a geoesic curve ϕ C 1 ([, T ), D k ) starting at I. On the other han, the metho of characteristics also associates a C 1 -curve t q(t) on D k starting at I by q(t, x) = x + 3t u (x), t [, T ), x S. Generally the two curves o not coincie (if u is constant we have the same curve). While t q(t) satisfies (3.3), note that (3.5) u(t, ϕ(t, x)) ϕ x(t, x) = u (x), t [, T ), x S,

13 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 13 as one can see ifferentiating both sies with respect to time 8. Observe that the solution to (3.1) is given by u = ϕ t ϕ 1 = u q 1 up to the maximal existence time given by (3.4): the geometric approach iffers from the metho of characteristics The exponential map. These results enable us to efine the Riemannian exponential map exp of the L right-invariant metric. Let ϕ(t; u ) be the geoesic starting at I in the irection u on D k, k, or on D. For later use, let us first observe that, using (3.), it is easy to obtain (3.6) ϕ(t; su ) = ϕ(ts; u ) for t, s such that both geoesics are well-efine. On the other han, note that u H k < 1 ensures that the maximal existence time of ϕ(t; u 4 ) is strictly larger than one. Inee, by the inequality ˆ u (x) (u ) + (u ) x u H k max x S S we obtain that max x S { 3u (x)} 3 so that the assertion follows from relation (3.4). 4 For u H k < 1 we efine the Riemannian exponential map exp as the time one map of 4 the geoesic flow, i.e. exp(u ) = ϕ(1; u ). For strong Riemannian manifols, the Riemannian exponential map always efines charts cf. [3]. This is not the case for the (weak) L right-invariant metric. Proposition 3.4. The Riemannian exponential map of the L right-invariant metric on D k, k, is not a C 1 map from a neighborhoo of zero in T I D k H k (S) to D k. Proof. Assuming the contrary, we will reach a contraiction by showing that although the erivative of exp is the ientity at zero, it fails to be invertible at nearby points. This will prove the assertion, for if exp were C 1, the inverse function theorem woul prevent this egeneracy. We assume that exp is a C 1 map. Let t tv be a curve in T I D k. For t > small enough, we have by (3.6) that exp(tv) = ϕ(1; tv) = ϕ(t; v) so that t exp(tv) = t= t ϕ(t; v) = v, v T I D t= k. This shows that Dexp() is the ientity. We shall now compute the erivative of exp at a point v T I D k near I by consiering an infinitesimal change w of v. Denoting we will show that for t [, 1] we have (3.7) ψ(t, x) = ψ(t) = t Dexp(tv) w H k (S), t [, 1], ˆ t w(x) ϕ x(s, x) s ˆ t v(x) ϕ 3 x(s, x) ψ x(s, x) s, x S. From its efinition we know that ψ(t, x) epens continuously on time while the Sobolev imbeing H k 1 (S) C(S) shows the C 1 epenence on the spatial variable. Differentiating the above equation with respect to time we obtain the linear partial ifferential equation (3.8) ψ t (t, x) = w(x) ϕ x(t, x) v(x) ϕ 3 x(t, x) ψ x(t, x), t [, 1], x S. 8 Relation (3.5) is an expression of the conservation of momentum: we refer to the en of Section 4. for a etaile iscussion of this aspect in the context of the H 1 right-invariant metric, refraining from repeating here the proceure.

14 14 A. CONSTANTIN AND B. KOLEV In the special case v(x) = c >, x S, it is easy to see by (3.) that ϕ(t, x) = x + ct an (3.8) becomes ψ t = c ψ x + w, t [, 1], x S. Since ψ(, x) =, x S, we euce that thus, if v(x) = c >, we have (3.9) ψ(t, x) = 1 c ˆ x x ct Dexp(v) w (x) = 1 c w(y) y, x S, ˆ x x c w(y) y, x S. This relation shows that, uner the assumption that exp is locally C 1, the erivative Dexp of the exponential map at v n (x) = 1, x S, annihilates the functions w n n(x) = sin(πnx), x S, an is therefore not invertible. This yiels the esire contraiction. To complete the proof, we have to check (3.7). Let ϕ ɛ be the geoesic on D k starting at I in the irection (v + ɛw). Using (3.) an (3.5) we euce that for ɛ > small enough, ϕ(t, x) = x + ϕ ɛ (t, x) = x + ˆ t ˆ t For t [, 1], x S, we obtain that (3.1) ϕ ɛ (t, x) ϕ(t, x) ɛ = ˆ t v(x) s, t [, 1], x S, [ϕ x (s, x)] v(x) + ɛ w(x) [ϕ ɛ x(s, x)] s, t [, 1], x S. w(x) [ϕ ɛ x(s, x)] s ˆ t v(x) [ϕ ɛ x(s, x) + ϕ x (s, x)] ϕ ɛ x(s, x) ϕ x (s, x) s. [ϕ x (s, x)] [ϕ ɛ x(s, x)] ɛ We woul like to let ε in (3.1) an in oing so, we seek to apply the Lebesgue ominate convergence theorem. For the pointwise convergence, by (3.6) we have so that, (3.11) 8 > < > : ϕ ɛ (t) ϕ(t) = exp(t(v + ɛ w)) exp(tv) ϕ ɛ (t, x) ϕ(t, x) lim ɛ = ψ(t, x), uniformly on S, ɛ ϕ ɛ lim x(t, x) ϕ x (t, x) ɛ = ψ x (t, x), uniformly on S. ɛ in view of the compact imbeing of H (S) in C 1 (S). To obtain a uniform boun uner the integral sign in (3.1), we procee as follows. Fix t [, 1] an ɛ > small. For ɛ (, ɛ ) we efine F [, 1] H k exp(tv + ɛ sw) exp(tv) (S), F (s) = s Dexp(tv) w. ɛ By the mean-value theorem an the fact that by assumption exp is C 1, we infer that F (s) H k = F (s) F () H k max ξ [,1] F (ξ) H k = max ξ [,1] Dexp(tv + ɛ ξw) w Dexp(tv) w H k M,

15 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 15 for some M > that is inepenent of t [, 1] an of ε (, ɛ ). We euce that for all t [, 1], ɛ (, ɛ ), This relation yiels sup t [,1], x S exp(tv + ɛ tw) exp(tv) ɛ ϕ ɛ x(t, x) ϕ x (t, x) ɛ t Dexp(tv) w ψ x (t, x) H k H k M. M, ɛ (, ɛ ). Taking into account the previous relation an (3.11) while letting ɛ in (3.1) leas to ψ(t, x) = ˆ t w(x) ϕ x(s, x) s ˆ t v(x) ϕ 3 x(s, x) ψ x(s, x) s, t [, 1], x S, in view of the Lebesgue ominate convergence theorem. The proof is complete. Let us now prove that the Riemannian exponential map for the (weak) L right-invariant metric on D oes not efine charts. Theorem 3.5. The Riemannian exponential map of the L right-invariant metric on D is not a C 1 iffeomorphism from a neighborhoo of zero in T I D C (S) to D. Proof. Assume exp is a local C 1 iffeomorphism. Note that in the proof of Proposition we compute irectional erivatives. Take v, w C (S) an fix k. The same arguments show that Dexp(v) w is given precisely by (3.9) if v(x) = c >, x S. As v n, w n efine above happen to belong to C (S) with v n in C (S), we conclue that Dexp(v n ) annihilates w n an is therefore not invertible in any neighborhoo of C (S). The obtaine contraiction completes the proof Breakown of the geoesic flow. We saw that most of the geoesics have a finite life-span T < given by (3.4). Let us prove that it is not possible to consier a weaker epenence on time of the geoesic that coul allow us to continue each geoesic past this time T <. Take u = sin(πx), x [, 1]. In view of (3.4), the maximal existence time of the corresponing solution u(t, x) to (3.1) is T = 1. Using (3.1) it is easy to see 6π that an o initial ata yiels spatially o solutions. Differentiating (3.3) with respect to x, we get u x (t, x 3t sin(πx)) = π cos(πx), x [, 1], 1 6πt cos(πx) so that min u x(t, x) = u x (t, ) = π 1 as t x S 1 6πt 6π. If ϕ(t) is the geoesic on D starting at I in the irection u, note that ϕ t = u(t, ϕ) leas to ϕ tx = u x (t, ϕ) ϕ x. Therefore ˆ t 1 ϕ x (t, x) = exp u x (s, ϕ(s, x)) s, x [, 1], t [, 6π ). Evaluating at x =, we obtain ϕ x (t, ) = (1 6πt) 1 1 3, t [, 6π ). Inee, u(t, ) = on [, 1 1 ), ensure by spatial oness, forces ϕ(t, ) = on [, ) in 6π 6π view of the orinary ifferential equation ϕ(t, ) = u(t, ϕ(t, )) with a locally Lipschitz t right-han sie. We see that ϕ x (t, ) as t 1. Therefore, letting t T on the 6π geoesic t ϕ(t), we o not obtain a C (S) iffeomorphism in the limit.

16 16 A. CONSTANTIN AND B. KOLEV 4. The H 1 right-invariant metric The results of the previous section raise the question whether another right-invariant metric coul provie D with a nice local geometric structure. We consier now on T I D C (S) the H 1 inner prouct ˆ u, v H 1 = u(x)v(x) + u (x)v (x) x, u, v C (S), S that is move by right translation to efine a smooth right-invariant metric on D, cf. Section.. A straightforwar calculation yiels B(u, v) = (1 x) 1 v x (1 x)u + v(1 x)u x, u, v C (S), so that Theorem 1 ensures the existence of a Riemannian connection. equation for the H 1 right-invariant metric is (4.1) u t + uu x + x (1 x) 1 u + 1 u x =, The geoesic where t ϕ(t, ) is the geoesic curve starting at time t = at the ientity I in the irection u T I D an u = ϕ t T ϕ(t) D. We write this as the system (4.) 8 < : ϕ t = u(t, ϕ), with initial ata ϕ() = I, u C (S). u t + uu x + x (1 x) 1 u + 1 u x 4.1. The geoesic equation. Fokas an Fuchssteiner [4] obtaine (4.1) as a bi-hamiltonian abstract equation by the metho of recursion operators. In imensionless space-time variables (x, t), (4.1) arises in several physical contexts. Accoring to Camassa an Holm [8], it is a moel for the uniirectional propagation of waves uner the influence of gravity at the free surface of a shallow layer of water 9 over a flat bottom [8] with u(t, x) representing the horizontal component of the velocity or, equivalently, the water s free surface [9]. Equation (4.1) is a moel for finite-length an small-amplitue axial-raial eformation waves in cylinrical ros compose of a compressible hyperelastic material [18] with u(t, x) representing the raial stretch relative to a pre-stresse state. We woul also like to point out that the viscous three-imensional generalization of (4.1) can be use as the basis for a turbulence closure moel [1] an was consiere an stuie in the theory of secon grae fluis [11] (examples of secon grae fluis inclue molten asphalt, honey, paints; relevant for such a flui is that it will climb up a ro which is rotating in an open vat [1]). ˆ In the expression 1 (u + u x) x that it conserve along the flow of (4.1), the first S term represents the kinetic energy inuce by the horizontal component of the velocity while the secon part stans for the kinetic energy ue to vertical motion [7]. Since the propagation is uniirectional, the transversal horizontal motion is neglecte. Let us iscuss some aspects of the partial ifferential equation (4.1). The methos of [13] show that for every u H k (S), k, there exists a maximal time T = T (u ) > such that (4.1) has a unique solution u C([, T ); H k 1 (S)) C 1 ([, T ); H k (S)). The solution epens continuously on the ˆ initial ata in the H k (S) norm. Note that the conservation of the energy functional 1 (u + u x) x ensures that all solutions to (4.1) remain uniformly boune. Moreover, the only way that this solution =, 9 For an alternative erivation of this moel in the context of water waves, we refer to [9]. S

17 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 17 fails to exist for all time is that the wave breaks cf. [14]. This means that the solution remains boune while its slope becomes unboune at a finite time T >. As an obvious consequence we infer that the maximal existence time oes not epen on the egree of smoothness of u H k (S), k. Uner some conitions the solution is global. Associate to each initial profile u H k (S), k, the expression y := u u,xx. If y oes not change sign properly, the solution is global [16]. This conition is also necessary for global existence, cf. [34]. In case of wave breaking, the rate of blow-up is given by lim t T inf x S {u x(t, x)} (T t) =, cf. [14]. For a large class of initial profiles it is also possible to etermine the exact blow up set. If y H 1 (S) is such that y (x) for x [, 1], y is o with y for x x with x (, 1), y, then the blow-up set consists of the three points {, 1, 1}. More precisely, we have u x (t, ) = u x (t, 1 ) = u x(t, 1) as t T < while (recall that u remains uniformly boune) sup u x (t, x) < for every x (, 1 t [,T ) ) (1, 1). An interesting aspect of equation (4.1) is its integrability in the sense of the infiniteimensional extension of Liouville s theorem for classical completely integrable Hamiltonian systems: there is a transformation which converts the equation into an infinite sequence of linear orinary ifferential equations which can be trivially integrate 1. Equation (4.1) is integrable provie the initial ata u is regular an the associate y has no zeros - for etails we refer to [15]. Let us mention that (4.1) is a counterexample to a conjecture on the complete integrability of nonlinear partial ifferential equations, the Painlevé test - see [5]. The equation (4.1) amits traveling wave solutions, i.e. solutions of the form u(t, x) = φ(x ct) which travel with fixe spee c. Further, these traveling wave solutions are solitons 11 : two traveling waves reconstitute their shape an size after interacting with each other, as iscovere by Camassa an Holm [8]. For a iscussion of the soliton interaction for (1.) we refer to [5]. The solitons are stable, the appropriate notion of stability being orbital stability [17]. That is, a wave starting close to a solitary wave always remains close to some translate of it at all later times. Thus the shape of the wave remains approximately the same for all times. The fact that equation (4.1) is formally a re-expression of the geoesic flow in the group of compressible iffeomorphisms of the circle enowe with the H 1 right-invariant metric was alreay note in [36]. As we will see below, the rigorous stuy of the geoesic flow leas to a proof of the Least Action Principle. It is quite natural to view (4.1) as the geoesic equation for the right-invariant H 1 -metric on the Hilbert manifols D k, k 3. However, this nees further justification since, in contrast to the case of D, we can not start from the notion of covariant erivative to efine the geoesics. Just like in the situation encountere in Section 3., the allege covariant erivative given by Theorem 1 is not well-efine on D k ue to loss of smoothness. We woul also like to point out that if X X (D k ), k 3, then the map η X(η) η 1 is 1 An introuction to the ieas of integrability, couple with a escription of some examples, is provie by [33]. 11 A clear exposition of most of the essential features of soliton theory is []; see also the survey paper [39].

18 18 A. CONSTANTIN AND B. KOLEV only continuous on D k so that the H 1 right-invariant metric on D k is not smooth whereas the H 1 right-invariant metric on D is smooth. To fully justify why we are entitle to call (4.1) the geoesic equation on D k, we will show that it arises from the necessary conition for a regularly parameterize path to be locally the shortest path on D k between its fixe enpoints. In view of the comments on a similar issue mae in Section 3., we can assume the path to be parameterize by arc length, γ : [, c] D, an the necessary conition for γ to be locally the shortest path on D k between its fixe enpoints is that γ is a critical point in the space of paths for the action functional, i.e. ɛ a(γ + ɛη) ɛ= = for every path η : [, c] D k with enpoints at zero an such that γ + ɛη is a small variation of γ on D k. A lengthy calculation, similar to the one presente in Section 3., shows that ɛ a(γ + ɛη) ɛ= = ˆ c This yiels the Euler-Lagrange equation ˆ S (η γ 1 ) u t u txx + 3uu x u x u xx uu xxx u t u txx + 3uu x u x u xx uu xxx =, x t. where u = γ t γ 1 an t γ(t) D k is the curve (parameterize by arc length) yieling the critical point of the length functional to be minimize. Applying the operator (1 x) 1 to the above form of the Euler-Lagrange equation we obtain (4.1). The variational formulation can therefore be use to give a meaning to (4.1) as the geoesic equation on D k, k 3. From the state analytical results on (4.1) an Lemma 3 we raw some first conclusions about the geoesic flow on D k. Proposition 4.1. For every u T I D k H k (S), k 3, there exists a unique geoesic on D k, starting at I in the irection of u. Certain geoesics are efine for some finite maximal time T > while others can be continue inefinitely in time. In the above result, a geoesic is a solution to (4.1) with a C 1 -epenence on time, as ensure by Lemma 3. As a byprouct of Proposition 4 below we will see that the time epenence of the geoesic is actually C. Note the contrast to the case of the L right-invariant metric on D k where the epenence can not generally be C in view of the comments preceing Proposition The exponential map. We efine now the Riemannian exponential map exp of the H 1 right-invariant metric an stuy some of its properties. If ϕ(t; u ) is the geoesic on D or on D k, k 3, starting at I in the irection u, note that (4.3) ϕ(t; su ) = ϕ(ts; u ) for t, s such that both sies are well-efine. The continuous epenence on initial ata of the solutions to (4.1) an Lemma 3 show that there is some δ > so that all geoesics ϕ(t; u ) are all efine on the same time interval [, T ] with T >, provie u H < δ. For u H k < δ, we efine exp(u T ) = ϕ(1; u ). In contrast to the case of the L right-invariant metric, we have Proposition 4.. The Riemannian exponential map of the H 1 right-invariant metric on D k, k 3, is a C 1 local iffeomorphism from a neighborhoo of zero on T I D k to a neighborhoo of I on D k.

19 ON THE GEOMETRIC APPROACH TO THE MOTION OF INERTIAL MECHANICAL SYSTEMS 19 Proof. We recast (4.) as a ifferential system ϕt = v, v t = P ϕ (v), where v = u(t, ϕ) an the operator P ϕ is given by P ϕ (v) = x (1 x) 1 (v ϕ 1 ) + 1 (v ϕ 1 ) x Note that P ϕ is a composition of the two operators with D ϕ = R ϕ x R ϕ 1, Q ϕ = R ϕ (1 x) 1 R ϕ 1, E ϕ (w) = R ϕ (w + 1 w x) R ϕ 1, i.e. P ϕ (v) = (D ϕ Q ϕ E ϕ )(v) for v H k (S). We will prove that the map (ϕ, v) (v, P ϕ (v)) is C 1 from a small neighborhoo of (I, ) D k H k (S) to H k (S) H k (S). The theorem on the epenence on initial ata for solutions of ifferential equations in Banach spaces (see [3]) ensures then that exp is of class C 1. Observe that Dexp is the ientity. Inee, let t tv be a curve in T I D k. For t > small enough, we have by (4.3) that exp(tv) = ϕ(1; tv) = ϕ(t; v) so that t exp(tv) t= = t ϕ(t; v) t= = v, v T I D k. This shows that Dexp is the ientity. Therefore, if the map (ϕ, v) (v, P ϕ (v)) is locally C 1, the assertion of Proposition 4 follows from the inverse function theorem. To complete the proof, let us show that (ϕ, v) (v, P ϕ (v)) is C 1 from a small neighborhoo of (I, ) D k H k (S) to H k (S) H k (S). Note that the map ª ϕ. (ϕ, v) (ϕ, E ϕ (v)) is C 1 from D k H k (S) to D k H k 1 (S), on a small neighborhoo of (I, ), while (ϕ, w) (ϕ, D ϕ (w)) is C 1 from D k H k+1 (S) to D k H k (S), on a small neighborhoo of (I, ), as one can see by explicit calculations. If we show that on a small neighborhoo of (I, ) D k H k 1 (S), (4.4) (ϕ, w) (ϕ, Q ϕ (w)) is C 1 to D k H k+1 (S), the proof is complete. Inee, combining the previous three assertions we infer that the map (ϕ, v) P ϕ (v) is C 1 from a small neighborhoo of (I, ) D k H k (S) to H k (S). Clearly the map (ϕ, v) (v, P ϕ (v)) will be then C 1 from a small neighborhoo of (I, ) D k H k (S) to H k (S) H k (S) an we are one. The inverse of the map in (4.4) is the map S given by (ϕ, w) (ϕ, R ϕ (1 x) R ϕ 1 (w)). By explicit calculation (see below) it is easy to see that S is of class C 1 from D k H k+1 (S) to D k H k 1 (S). Therefore, to conclue that (4.4) hols, in view of the inverse function theorem, it will be enough to check that the Fréchet ifferential of S at (I, ) is invertible. The C 1 regularity of S ensures that the Fréchet ifferential can be compute by calculating irectional erivatives (it is actually the other way aroun that we showe S to be C 1 ). Clearly, consiering partial erivatives D i, i = 1,, of the components S i, i = 1,, we have D 1 S 1 = I, D S 1 =,