Third Order ODEs Systems and Its Characteristic Connections
aa r X i v : . [ m a t h . DG ] A ug Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2011), 076, 15 pages Third Order ODEs Systemsand Its Characteristic Connections
Alexandr MEDVEDEVFaculty of Applied Mathematics, Belarusian State University,4, Nezavisimosti Ave., 220030, Minsk, Republic of Belarus
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Received April 20, 2011, in final form July 27, 2011; Published online August 03, 2011doi:10.3842/SIGMA.2011.076
Abstract.
We compute the characteristic Cartan connection associated with a system ofthird order ODEs. Our connection is different from Tanaka normal one, but still is uniquelyassociated with the system of third order ODEs. This allows us to find all fundamentalinvariants of a system of third order ODEs and, in particular, determine when a systemof third order ODEs is trivializable. As application differential invariants of equations oncircles in R n are computed. Key words: geometry of ordinary differential equations; normal Cartan connections
The main purpose of this article is to study geometry of systems of ordinary differential equa-tions of third order. The geometry of ordinary differential equations or, more generally, ofdifferential equation of finite type is based on the general theory of geometric structures onfiltered manifolds. First it was developed by Tanaka in [9, 10]. Recall that a filtered manifold is a smooth manifold M equipped with a filtration of the tangent bundle T M compatible withthe Lie bracket of vector fields. At any point x ∈ M the associated graded vector space gr T x M can be endowed with a Lie algebra structure. This nilpotent Lie algebra m is called a symbol ofa filtered manifold (at the point x ). In the paper we consider only the so-called filtered manifoldsof constant type , assuming that the graded nilpotent Lie algebras gr T x M are isomorphic to eachother for all points x ∈ M .By a symbol of a geometric structure on M we understand a graded Lie algebra g withthe negative part g − = P i< g i which is equal to the symbol m of the filtered manifold M of constant type. Here the Lie algebra m is the subalgebra of a so-called universal Tanakaprolongation g ( m ). Roughly speaking, this means that g ( m ) is the maximum among graded Liealgebras which satisfy the condition “for any element X ∈ g i , i ≥ X, g − ] = 0implies X = 0”.An arbitrary equation E can be viewed as a surface in jet space. The canonical restriction ofthe contact distribution on jet space defines the structure of filtered manifold on E . One of the main problems in the theory of differential equations is the problem of equivalence.Two differential equations are called equivalent if one can be transformed to another by a certain A. Medvedevchange of variables. We consider equations up to point transformations, i.e. we allow arbitrarychanges of both dependent and independent variables.First classical approach to the equivalence problem of ODEs was developed by Sophus Lie.In [5] he obtains partial results about second order ODEs. The complete answer was given laterby Tresse [11]. Invariants of the third order ODEs were computed by Chern in his paper [1].A modern approach to the equivalence problem of ODEs can be found in the papers [2] and [4],where characteristic Cartan connection was constructed for the one equation of arbitrary orderand for the system of ODEs of the second order.The general approach to the equivalence problem for the holonomic differential equationscan be find in [3]. The key fact there is the existence of a full functor from the categoryof holonomic differential equations to the category of Cartan connections. This reduces theequivalence problem for differential equations to the equivalence problem for the correspondingCartan connections.
Let P be the principal H -bundle. Let ω be a Cartan connection of type ( G, H ), where G isa Lie group with a semisimple graded Lie algebra g and H is a parabolic subgroup of G withthe Lie algebra h . In the paper [10] Tanaka built a set of normal Cartan connections on theprincipal bundle P as follows. He used the scalar product defined with the help of the Killingform to construct adjoint Lie algebra codifferential ∂ ∗ . Then a Cartan connection is normal iffthe structure function C : P → Hom( ∧ g − , g ) belongs to the kernel of the operator ∂ ∗ and thestructure function has not negative components. As usual define a Laplacian ∆ = ∂ ∗ ∂ + ∂∂ ∗ .The structure function C decomposes as C = H ( C ) + ∆( C ). The component H ( C ) is called theharmonic part of the structure function. The key fact about it is that H ( C ) is the fundamentalsystem of invariants (see Definition 5 for details). In the case of the geometry of holonomicdifferential equations the Lie algebra g is not necessarily semisimple. However in [3] is shownthat we still can find the scalar product on g such that the normal Tanaka conditions define theunique Cartan connection associated to a holonomic differential equation.In this paper we associate with every system of ODEs of third order a characteristic Cartanconnection which differ from a normal Tanaka Cartan connection. The reason for doing this isa relation between conformal geometry and geometry of the system of the third order ODEs.Conformal manifold is determined by the family of conformal circles, which was shown byYano [12]. Each conformal circle is determined by the point on it, the direction and the curvature,i.e. by the point in the third jet space. The system of appropriate differential equations ofthe third order gives us the bridge between the conformal geometry and the geometry of thedifferential equation. It is appeared that a characteristic Cartan connection, which is built inthe paper, is in close relations with the normal conformal Cartan connection. The relation ofthe conformal geometry and the geometry of third order ODEs is the topic of the next paper.The paper is organized as follows. In Section 2 we naturally associate the system of thethird order ODEs with the pair of distributions. This pair of distributions give rise to thefiltered manifold associated with the system of the third order ODEs. We write down thesymbol of the system of ODEs of the third order, the notion of adopted coframe and adoptedCartan connection. The problem of equivalence is considered in Section 3. When we workingin the case of semisimple Lie algebras and normal Cartan connections, the harmonic part of thecurvature gives us the fundamental system of differential invariants. We show that in generalcase fundamental differential invariants are contained in the Ker ∂ part of the curvature, where ∂ is the Lie algebra cohomology differential. In Section 4 we build the characteristic Cartanconnection uniquely associated to the the system of ODEs of the third order. This connectionallows us to obtain the results about equivalence third order equations and to describe thehird Order ODEs Systems and Its Characteristic Connections 3structure of the fundamental invariants of the system of third order ODEs. In particular, thisanswers the question “When is the given system trivializable?” explicitly. Consider an arbitrary system of m ordinary differential equations of third order: y ′′′ i ( x ) = f i (cid:0) y ′′ j ( x ) , y ′ k ( x ) , y l ( x ) , x (cid:1) , (1)where i, j, k, l = 1 , . . . , m and m ≥ J ( R m +1 , E in J ( R m +1 , E :( x, y , . . . , y m , p = y ′ , . . . , p m = y ′ m , q = y ′′ , . . . , q m = y ′′ m ) . There is a natural one-dimensional distribution E whose integral curves are the lifts of solu-tions of equations (1). Let π be the canonical projection from the surface E to the first jet space J ( R m +1 , π as V . In coordinates distributions E , V have the form: E = (cid:28) ∂∂x + p i ∂∂y i + q i ∂∂p i + f i ∂∂q i (cid:29) , V = (cid:28) ∂∂q i (cid:29) , where i, j = 1 , . . . , m .Define a distribution C as the direct sum of the distributions E and V . Then C and itssubsequent brackets define a filtration of a tangent bundle T E : C = C − ⊂ C − ⊂ C − = T E , where C − i − = C − i + [ C − i , C − ].It is easy to see that the symbol of the filtrated manifold E is a nilpotent Lie algebra m isomorphic to the Lie algebra of vector fields generated by m − = (cid:28) ∂∂x + p j ∂∂y j + q j ∂∂p j , ∂∂q i (cid:29) . Let Aut ( m ) be a subgroup of grading preserving elements of the group Aut( m ) . The elementsof the group Aut ( m ) which preserve the splitting E ⊕ V form subgroup G . So the splitting E ⊕ V of the distribution C defines G -structure of type m . The action of the group G on m is completely determined by its action on m − . The latter has the following form in the basis n ∂∂x + p j ∂∂y j + q j ∂∂p j , ∂∂q i o : (cid:18) a B (cid:19) , a ∈ R ∗ , B ∈ GL m ( R ) . The symbol g is the universal Tanaka prolongation of the pair ( m , g ). It has the followingform: g = ( s l ( R ) × gl m ( R )) ⋌ ( V ⊗ W ) . In other words, g is equal to the semidirect product of the Lie algebra s l ( R ) × gl m ( R ) and anAbelian ideal V . The ideal V has the form V ⊗ W , where V is an irreducible s l -module ofdimension 3 and W = R m is the standard representation of gl m ( R ). A. MedvedevLet us fix a basis of the Lie algebra s l and s l -module V . Let x , y , h be the standard basisof an algebra s l with relations:[ x, y ] = h, [ h, x ] = 2 x, [ h, y ] = − y. This basis can be represented in the following way: x = (cid:18) (cid:19) , h = (cid:18) − (cid:19) , y = (cid:18) (cid:19) . Let v , v , v be a basis of the module V such that x.v = v , x.v = v , x.v = 0.Define the grading of the Lie algebra g as follows: g = h y i , g = h h, gl m i , g − = h x i + h v ⊗ W i , g − = h v ⊗ W i , g − = h v ⊗ W i . To build a natural Cartan geometry associated to the equation (1) we will use the fact [7]that under some additional conditions (which are satisfied for geometric structures arising fromholonomic differential equations, see [3]) there exists a full functor from the category of G -structures of type m to the category of Cartan connections of type ( G, H ), where G and H arethe Lie groups with Lie algebras g and h respectively which are determined from G in naturalmanner. The group G is a semisimple product: G = ( SL ( R ) × GL m ( R )) ⋌ ( V ⊗ W ) , and the group H is the following subgroup of G : H = (cid:18) a b a − (cid:19) × A, a ∈ R ∗ , b ∈ R , A ∈ GL m ( R ) . Note that the corresponding subalgebra h is exactly the nonnegative part of the Lie algebra g : h = X i ≥ g i . Definition 1.
We say that a coframe { ω i − , ω i − , ω i − , ω x } on E is adapted to equation (1) if: • the annihilator of forms ω i − , ω i − , ω x is V ; • the annihilator of forms ω i − , ω i − , ω i − is E ; • the annihilator of forms ω i − is C − .Let π : P → E be a principle H -bundle and let ω be and arbitrary Cartan connection of type( G, H ) on P . Connection ω can be written as: ω = ω i − v ⊗ e i + ω i − v ⊗ e i + ω i − v ⊗ e i + ω x x + ω h h + ω ij e ji + ω y y. Definition 2.
We say that a Cartan connection ω on a principal H -bundle π is adapted toequations (1), if for any local section s of π the set (cid:8) s ∗ ω x , s ∗ ω i − , s ∗ ω i − , s ∗ ω i − (cid:9) is an adapted co-frame on E .We have described the set of Cartan connection adapted to the system of third order ODEs.However, we can chose the representative in different ways. The next two sections are devotedto the building of a canonical connection which we call characteristic.hird Order ODEs Systems and Its Characteristic Connections 5 As in Section 2 let π : P → E be a principle H -bundle and let ω be and arbitrary Cartanconnection of type ( G, H ) on P : ω = ω i − v ⊗ e i + ω i − v ⊗ e i + ω i − v ⊗ e i + ω x x + ω h h + ω ij e ji + ω y y. Let Ω = d ω + [ ω, ω ] be the curvature of the Cartan connection ω :Ω = Ω i − v ⊗ e i + Ω i − v ⊗ e i + Ω i − v ⊗ e i + Ω x x + Ω h h + Ω ij e ji + Ω y y. Definition 3.
The structure function of a Cartan connection ω is a function C : P → Hom (cid:0) ∧ g − , g (cid:1) , which is defined by C ( p )( g , g ) = Ω p (cid:0) ω − p ( g ) , ω − p ( g ) (cid:1) . We can obtain the structure function of a Cartan connection explicitly. Let { e , . . . , e n + k } bea basis of Lie algebra g such that { e n +1 , . . . , e n + k } form a basis of the subalgebra h . In our case { e n +1 , . . . , e n + k } = { h, y, e ji } . An arbitrary element ϕ ∈ Hom( ∧ g − , g ) defined by constants C kij ,where ϕ ( e i , e j ) = n + k X k =1 C kij e k , ≤ i, j ≤ n. The structure function C : P → Hom( ∧ g − , g ) defines functions C kij ( p ) . If ω = X ω i e i , Ω = X Ω k e k , then the functions C kij ( p ) can be found from the decomposition of the curvature tensor Ω interms of forms ω i :Ω k = X C kij ω i ∧ ω j . Let Ω i be one of the 2-forms Ω i − , Ω i − , Ω i − , Ω x , Ω h , Ω ij . We can write it explicitly as:Ω i = X p,q =1 Ω i (cid:2) ω j − q , ω k − p (cid:3) ω j − q ∧ ω k − p + X p =1 Ω i (cid:2) ω x , ω k − p (cid:3) ω x ∧ ω k − p . Then Ω i [ ω j − q , ω k − p ] and Ω i [ ω x , ω k − p ] are the coefficients of the structure function of the Cartanconnection ω. The grading of Lie algebra g induces degree of the coefficients Ω i [ ω j − q , ω k − p ] andΩ i [ ω x , ω k − p ]. Definition 4.
We say that Cartan connection associated with the equation (1) is characteristicif the following conditions on a curvature is satisfied: • all coefficients of degree ≤ • in degree 2 we have Ω h [ ω x ∧ ω i − ] = 0, Ω ij [ ω x ∧ ω k − ] = 0, Ω x [ ω x ∧ ω i − ] = 0, Ω i − [ ω x ∧ ω i − ] = 0; A. Medvedev • in degree 3 we have Ω y [ ω x ∧ ω i − ] = 0, Ω h [ ω x ∧ ω i − ] = 0, Ω ij [ ω x ∧ ω k − ] = 0; • in degree 4 we have Ω y [ ω x ∧ ω i − ] = 0.In other worlds these conditions define the subspace U and Cartan connection is characteristicif and only if it belongs to U . Theorem 1.
There exists a unique characteristic Cartan connection associated with the equa-tion (1) . Proof .
We will proceed with parametric computations of characteristic Cartan connection inthe forth section of the paper. We will fix a section s : E → P and prove that locally forevery equation there exists a unique Cartan connection ω with structure function pullback s ∗ C : E →
Hom( ∧ g − , g ) takes values in the space U . Now we show that the characteristicCartan connection is uniquely globally defined with this data.Take a covering U α of the space E and construct a Cartan connection ω α on each trivialfibre bundle π α : U α × H → U α . Let s α and s β be the trivial sections of the fibre bundles π α and π β . Let e ω α = s ∗ α ω α and e ω β = s ∗ β ω β . Since forms ω α and ω β are uniquely defined thereexists a unique function ϕ αβ : U α ∩ U β → H, such that ω β = Ad (cid:0) ϕ − αβ (cid:1) ω α + ϕ ∗ αβ ω H , where ω H is Maurer–Cartan form of the Lie group H . The functions ϕ αβ uniquely definea principle H -bundle with the Cartan connection ω .In order to prove that the structure function C of the Cartan connection ω takes values inthe space U it is sufficient to show that U is Ad( H )-invariant.Note that the action of G preserves the zero condition on the structure function of thecharacteristic connection. We need only to check that the space U is exp( y )-invariant or equallyad( y ) invariant. The action of the element y has degree one. Conditions on the curvature ofthe Theorem 1 are ad( y )-invariant up degree 2, since all components of degree less than 2 areequal to zero. Finally, the conditions of degree 3 and 4 are ad( y )-invariant, since the coefficientsΩ y [ ω x ∧ ω i − ], Ω h [ ω x ∧ ω i − ], Ω ij [ ω x ∧ ω k − ] and Ω y [ ω x ∧ ω i − ] can be obtained only from Ω h [ ω x ∧ ω i − ],Ω x [ ω x ∧ ω i − ], Ω ij [ ω x ∧ ω k − ], Ω h [ ω x ∧ ω i − ] and Ω y [ ω x ∧ ω i − ] which all are zero for characteristicCartan connection. This ends the proof of a global existence of the form ω . (cid:4) Let V be an arbitrary finite-dimensional vector space and let f be a smooth function f : P → V . Denote by L ( f ) the space of all functions of the form h f, v ∗ i , where v ∗ ∈ V ∗ and by L ( f ) the algebra generated by elements from L ( f ) and all their covariant derivatives.For example, the algebra L ( C ), where C is structure function of the Cartan connection ω ,consists of local invariants of the connection ω . Definition 5.
We say that functions f i are the fundamental system of differential invariantsfor the structure with Cartan connection ω if L ( f i ) = L ( C ).The key to calculation of the fundamental system of differential invariants is to determinewhich parts of the curvature are expressed through another. In [3] it is shown that fundamentalinvariants of holonomic differential equation lie in non-negative harmonic part of the curvatureof the normal Cartan connection. In general we have approximately the same situation: thereis one to one correspondence between fundamental differential invariants of the characteristicCartan connection and H ( g − , g ) part of the structure function. Here H ( g − , g ) is the non-negative part of the second Lie algebra cohomology group.hird Order ODEs Systems and Its Characteristic Connections 7 Proposition 1.
Let ω be a Cartan connection of type ( G, H ) on a principal H -bundle P , where ( G, H ) is an arbitrary pair of Lie group and its subgroup. Assume that the Lie algebra g isa graded Lie algebra of the Lie group G with the negative part g − . Assume that a structurefunction of ω takes values in subspace W ⊂ Hom( ∧ g − , g ) and has only components of posi-tive degree. Then a Ker ∂ ∩ W part of the structure function forms a system of fundamentaldifferential invariants. Proof .
The algebra of differential invariants is generated by the structure function coefficients.We will use the Bianchi identity to show that some coefficients of the characteristic Cartanconnection curvature are obtained from the image of the operator ∂ .Let e i be the basis of the Lie algebra g , X i be the corresponding fundamental vector fieldson P and ω i be the dual coframe. We can write the Cartan connection ω in the form: ω = ω i e i . Assume that the Lie algebra g has structure constants A kij . That means that:[ e i , e j ] = A kij e k . Write the curvature of the Cartan connection ω in coordinates:Ω = C kij ω i ∧ ω j e k . (2)Then the following equality is fulfilled:d ω k = (cid:0) C kij − A kij (cid:1) ω i ∧ ω j . Now apply the Bianchi identity d Ω = [Ω , ω ] to the equation (2): ∂C kij ∂X l ω l ∧ ω i ∧ ω j e k + C kij d ω i ∧ ω j e k + C kij ω i ∧ d ω j e k ! = C kij [ e k , e l ] ω i ∧ ω j ∧ ω l . Express the covariant derivative of the structure function: ∂C pij ∂X l ω l ∧ ω i ∧ ω j e p = (cid:0) − C pkl d ω k ∧ ω l − C pkl ω k ∧ d ω l + C kij A pkl ω i ∧ ω j ∧ ω l (cid:1) e p = C pkl (cid:0) C kij − A kij (cid:1) ω i ∧ ω j ∧ ω l e p + C kij A pkl ω i ∧ ω j ∧ ω l e p . We get that: ∂C pij ∂X l ω l ∧ ω i ∧ ω j e p − C pkl C kij ω i ∧ ω j ∧ ω l e p = C pkl A kij ω i ∧ ω j ∧ ω l e p + C kij A pkl ω i ∧ ω j ∧ ω l e p . (3)If we take the Hom( ∧ g − , g ) part of (3) (i.e. assume that ω l ∈ g ∗− ) we get that the right side ofthe (3) is exactly the Lie cohomology differential.On the right side of (3) coefficients have the same degree as in the curvature. On theother hand coefficients on the left side have an increased degree. So, we have obtained thatcoefficients which are mapped to the im ∂ can be expressed through the covariant derivative ofthe coefficients of the lower degree. This proves the proposition. (cid:4) Remark 1.
Note that if intersection of W and Im ∂ is zero then subspace Ker ∂ ∩ W is generatedby representatives of H ( g − , g ). A. Medvedev Theorem 2.
The following invariants are fundamental differential invariants for the system ofthird order ODEs: ( W ) ij = tr (cid:18) ∂f i ∂p j − ddx ∂f i ∂q j + 13 ∂f i ∂q k ∂f k ∂q j (cid:19) , ( I ) ij,k = tr (cid:18) ∂ f i ∂q j ∂q k (cid:19) , ( W ) ij = ∂f i ∂y j + 13 ∂f i ∂q k ∂f k ∂p j − ddx ∂f i ∂p j + 16 d dx ∂f i ∂q j − (cid:18) ∂f i ∂q k (cid:19) − ∂f i ∂q k ddx ∂f k ∂q j − ddx (cid:18) ∂f i ∂q k (cid:19) ∂f k ∂q j , ( I ) j,k = − ∂H − k ∂p j + ∂∂q j ∂∂q k H x − ∂∂q k ddx H − j − ∂∂q k (cid:18) H − l ∂f l ∂q j (cid:19) + 2 H − j H − k , where H − j = 16( m + 1) (cid:18) ∂ f i ∂q i ∂q j (cid:19) and H x = − m (cid:18) ∂f i ∂p i − ddx ∂f i ∂q i + 13 ∂f i ∂q k ∂f k ∂q i (cid:19) . Proof .
We will use Proposition 1. The fundamental differential invariants is in one to onecorrespondence with the cohomology group H ( g − , g ). For the case of the system of ODEs ofthe third order the Lie cohomology group H ( g − , g ) was studied in [6]. The main result of thatwork is that the space H ( g − , g ) has the following decomposition:Degree Space − v ⊗ ∧ ( W ∗ ) ⊗ W v ⊗ S ( W ∗ ) ⊗ W v ⊗ ∧ ( W ∗ ) ⊗ W v ⊗ ∧ W ∗ ⊗ W/V ⊗ W ∗ x ∗ ⊗ R y ⊗ s l ( W ) v ⊗ S ( W ∗ ) ⊗ W x ∗ ⊗ R y ⊗ gl ( W ) v ⊗ S ( W ∗ ) v if m = 2 Here v k is the lowest vector of corresponding ( k + 1)-dimensional s l -module V k .Now we list the result table with the corresponding invariant. We start from degree 2 sinceall part of curvature of degree less than 2 is zero.Degree Space Part of the curvature Invariant2 x ∗ ⊗ R y ⊗ s l ( W ) Ω i − [ ω x ∧ ω j − ] W v ⊗ S ( W ∗ ) ⊗ W Ω i − [ ω j − ∧ ω k − ] I x ∗ ⊗ R y ⊗ gl ( W ) Ω i − [ ω x ∧ ω j − ] W v ⊗ S ( W ∗ ) Ω y [ ω x ∧ ω j − ] I v if m = 2 Ω y [ ω − ∧ ω − ] ≡ (cid:4) Corollary 1.
The system (1) is equivalent to the trivial one via point transformations if andonly if all invariants I , W , W , I vanish identically. Example 1 (Differential equations on circles in R n ) . As application of the previous results wecompute invariants of the system of third order ODEs on circles in Euclidean space.hird Order ODEs Systems and Its Characteristic Connections 9
Lemma 1.
Let E be the ( m + 1) -dimensional Euclidean space with the orthonormal basis { e , . . . , e n } and the coordinates { r , r , . . . , r n } . Then the equation of circles in E parametrizedby the coordinate r is: ... r i = 3¨ r i m P j =1 ˙ r j ¨ r j m P j =1 ˙ r j , i = 1 , . . . , m. This equation is invariant under conformal transformations of E . Proof .
Let the curve R ( t ) = ( r ( t ) , . . . , r n ( t )) be a circle. Assume now that r ( t ) = t . We have... R ( t ) = a ( t ) ¨ R ( t ) + b ( t ) ˙ R ( t ) , (4)since R ( t ) is 2-dimensional curve. Next, b ( t ) = 0 in our parametrization, since0 = ... r ( t ) = a ( t )¨ r ( t ) + b ( t ) ˙ r ( t ) = b ( t ) . To determine a ( t ) note that( R ( t ) − C, R ( t ) − C ) = d for some constant d and C ∈ E . Differentiating, we get:( ˙ R ( t ) , R ( t ) − C ) = 0 , ( ¨ R ( t ) , R ( t ) − C ) = − ( ˙ R ( t ) , ˙ R ( t )) , (... R ( t ) , R ( t ) − C ) + 3( ¨ R ( t ) , ˙ R ( t )) = 0 . Now substitute (4) into previous formula:( a ( t ) ¨ R ( t ) + b ( t ) ˙ R ( t ) , R ( t ) − C ) = −
3( ¨ R ( t ) , ˙ R ( t )) , ( a ( T ) ¨ R ( t ) , R ( t ) − C ) = − a ( t )( ˙ R ( t ) , ˙ R ( t )) = −
3( ¨ R ( t ) , ˙ R ( t )) . We get that a ( t ) = 3 ( ¨ R ( t ) , ˙ R ( t ))( ˙ R ( t ) , ˙ R ( t )) . Substituting a ( t ) into (4) we get our equations. (cid:4) Proposition 2.
For differential equation on conformal circles invariants W , I , W vanishidentically. Invariant I has the following form: ( I ) ij = 12 δ ij
11 + m P k =1 ˙ r k −
12 ˙ r i ˙ r j (cid:18) m P k =1 ˙ r k (cid:19) . Proof .
The proof is straightforward applying of the formulas from Theorem 2. (cid:4)
Remark 2.
There are other equations satisfying W = I = W = 0. For example, it is anunion of a system on circles in R n − k and a system of k trivial equations. It would be interestingto characterize geometrically the class of such equations.0 A. Medvedev Consider a system of third-order ordinary differential equations of the form( y i ) ′′′ = f i (cid:0) x, y j , ( y k ) ′ , ( y l ) ′′ (cid:1) , where i, j = 1 , . . . , m with m ≥
2. It determines a holonomic differential equation
E ⊂ J ( R m +1 , E : x, y , . . . , y m , p = y ′ , . . . , p m = y ′ m , q = y ′′ , . . . , q m = y ′′ m . We choose a coframe θ on the surface E : θ x = dx ; θ i − = dq i − f i ( x, y, p, q ) dx, i = 1 , . . . , m ; θ i − = dp i − q i dx, i = 1 , . . . , m ; θ i − = dy i − p i dx, i = 1 , . . . , m. To connect our computation on the surface E with the principle bundle P let us use the followinguniquely defined section s : E → P with relations: s ∗ ω i − = θ i − ,s ∗ ω h ≡ h θ i − , θ i − , θ i − i ,s ∗ ω x ≡ − θ x mod h θ i − , θ i − , θ i − i . Define a pullback ω : T E → g by the formula ω = s ∗ ω . Let Ω be a curvature tensor of ω , andlet Ω = s ∗ Ω. We see thatΩ = Ω i − v ⊗ e i + Ω i − v ⊗ e i + Ω i − v ⊗ e i + Ω x x + Ω h h + Ω ji e ij + Ω y y = ( dω i − + ω x ∧ ω i − + 2 ω h ∧ ω i − + ω ij ∧ ω j − ) v ⊗ e i + ( dω i − + ω x ∧ ω i − + ω ij ∧ ω j − + 2 ω y ∧ ω i − ) v ⊗ e i + ( dω i − − ω h ∧ ω i − + ω ij ∧ ω j − + 2 ω y ∧ ω i − ) v ⊗ e i + ( dω x + 2 ω h ∧ ω x ) x + ( dω h + ω x ∧ ω y ) h + ( dω ij + ω ik ∧ ω kj ) e ji + ( dω y − ω h ∧ ω y ) y. An arbitrary Cartan connection adapted to equation (1) has the form: ω i − = θ i − ,ω i − = α ij θ j − + A ij θ j − ,ω i − = β ij θ j − + B ij θ j − + C ij θ j − ,ω x = − θ x + D j θ j − + E j θ j − ,ω h = F − j θ j − + F − j θ j − + F − j θ j − ,ω ij = G i,xj θ x + G i, − jk θ k − + G i, − jk θ k − + G i, − jk θ k − ,ω y = H x θ x + H − j θ j − + H − j θ j − + H − j θ j − . In degree 0 of the curvature we have two nonzero components:Ω i − mod h θ − ∧ θ − , θ − i = θ x ∧ θ i − − α ij θ x ∧ θ j − , hird Order ODEs Systems and Its Characteristic Connections 11Ω i − mod h θ − , θ − i = θ x ∧ θ i − − β ij θ x ∧ θ j − . Assume these two equalities is zero and get α ij = δ ij and β ij = δ ij .We have three nonzero components in degree 1. The first component is:Ω i − mod h θ − ∧ θ − , θ − ∧ θ − i = − θ x ∧ A ij θ j − + D j θ j − ∧ θ i − + G i,xi θ x ∧ θ j − + G i, − jk θ k − ∧ θ j − + 2 F − j θ j − ∧ θ i − . The second component is:Ω i − mod h θ − ∧ θ − , θ − i = A ij θ x ∧ θ j − + D j θ j − ∧ θ i − − θ x ∧ B ij θ j − + G i,xj θ x ∧ θ j − + G i, − jk θ k − ∧ θ j − . The third component is:Ω i − mod h θ − , θ − i = ∂f i ∂q j θ x ∧ θ j − + B ij θ x ∧ θ j − − F − j θ j − ∧ θ i − + G i,xj θ x ∧ θ j − + G i, − jk θ k − ∧ θ j − . After applying zero conditions to these parts of the curvature we obtain A ij = G i,xj = 12 B ij = − ∂f i ∂q j , D j = F − j = G i, − jk = 0 . Proceed now to the second degreeΩ i − mod h θ − ∧ θ − , θ − i = ∂f i ∂p j θ x ∧ θ j − + 2 dA ij dx θ x ∧ θ j − + 2 ∂A ij ∂q k θ k − ∧ θ j − + C ij θ x ∧ θ j − − F − j θ j − ∧ θ i − + G i, − jk θ k − ∧ θ j − + 2 H x θ x ∧ θ i − + 2 H − j θ j − ∧ θ i − + G i,xk θ x ∧ B kj θ j − . We have:Ω i − (cid:2) θ x ∧ θ j − (cid:3) = ∂f i ∂p j + 2 dA ij dx + C ij + 2 H x + 2 A ik A kj . Assuming the previous tensor is zero, we obtain: C ij = − ∂f i ∂p j + 2 dA ij dx + 2 H x + 2 A ik A kj ! . Next curvature component contains all second order invariants:Ω i − mod h θ − ∧ θ − , θ − ∧ θ − i = dA ij dx θ x ∧ θ j − + ∂A ij ∂q k θ k − ∧ θ j − − θ x ∧ C ij θ j − + E j θ j − ∧ θ i − + G i,xj θ x ∧ A jk θ k − + 2 H x θ x ∧ θ i − + 2 H − j θ j − ∧ θ i − + G i, − jk θ k − ∧ θ j − + G i, − jk θ − k ∧ A jl θ l − . In coefficient Ω i − [ θ k − ∧ θ j − ] we get invariant I Ω i − (cid:2) θ k − ∧ θ j − (cid:3) = ∂A ij ∂q k − E j δ ik + 2 H − k δ ij = ∂A ij ∂q k + 2 H − k δ ij + 2 F − j δ ik . I is the following: I = tr (cid:18) ∂ f i ∂q j ∂q k (cid:19) , where tr is a traceless part of the tensor.In the coefficientΩ i − (cid:2) θ x ∧ θ j − (cid:3) = − C ij dA ij dx + A ik A kj + 2 H k δ ij we obtain a so-called generalized Wilczynski invariant. As shown in [2], a part of differentialinvariants of systems of ODEs comes from its linearisation. As in [2], we call them generalizedWilczynski invariants. In our case we have two Wilczynski invariants of degree 2 and 3. Wedenote them as W and W respectively. The second degree generalized Wilczynski invariant isthe following: W = tr (cid:18) ∂f i ∂p j − ddx ∂f i ∂q j + 13 ∂f i ∂q k ∂f k ∂q j (cid:19) . Normalizing the trace of previous tensor to zero we obtain: H x = − m (cid:18) ∂f i ∂p i + 3 dA ii dx + 3 A ik A ki (cid:19) . It remains to compute only s l × gl m part of the curvature in degree 2.Ω x mod h θ − ∧ θ − , θ − i = E j θ x ∧ θ j − + 2 F j θ x ∧ θ j − . Assuming that it vanishes identically we get the following condition: E j = − F − j . We have:Ω h mod h θ − , θ − i = F − j θ x ∧ θ j − − θ x ∧ θ j − H − j . The condition Ω ih [ θ x ∧ θ i − ] = 0 gives equality F − j = H − j . Assuming the trace of the tensor Ω i − [ θ j − ∧ θ k − ] is equal to zero we get: F − k = H − k = − m + 1) ∂A ii ∂q k . The last part of degree 2 calculation is:Ω ij mod h θ − , θ − i = ∂A ij ∂q k θ x ∧ θ − k + G i, − jk θ x ∧ θ − k . We obtain G i, − jk = ∂A ij ∂q k from condition Ω ij [ ω x ∧ ω k − ] = 0 . Proceed now to the degree 3. The first part of degree 3 we need to compute is Ω i − :Ω i − mod h θ − ∧ θ − , θ − ∧ θ − i = ∂f i ∂y i θ x ∧ θ j − + ∂B ij ∂p k θ k − ∧ θ j − + ∂C ij ∂x θ x ∧ θ j − + ∂C ij ∂q k θ k − ∧ θ j − − F − j θ j − ∧ θ i − − F − j θ j − ∧ B ik θ k − hird Order ODEs Systems and Its Characteristic Connections 13+ G i, − jk θ k − + G i, − jk θ k − ∧ B ij θ j − + G i,xj θ x ∧ C jk θ k − + 2 H − j θ j − ∧ θ i − . Wilczynski invariant W appears as the Ω i − [ θ x ∧ θ j − ] coefficient: ∂f i ∂y j + dC ij dx + A ik C kj + 2 H x A ij . Direct computation shows that:Ω i − [ θ x ∧ θ j − ] = ∂f i ∂y j + 13 ∂f i ∂q k ∂f k ∂p j − ddx ∂f i ∂p j + 23 d dx ∂f i ∂q j − (cid:18) ∂f i ∂q j (cid:19) − ∂f i ∂q k ddx ∂f k ∂q j − ddx (cid:18) ∂f i ∂q k (cid:19) ∂f k ∂q j − δ ij H x . Denote invariant Ω i − [ θ x ∧ θ j − ] + ddx W as W . Invariant W is equivalent to the fundamentalinvariant Ω i − [ θ x ∧ θ j − ]. It means that after replacing Ω i − [ θ x ∧ θ j − ] with W the system wouldremain fundamental. Explicitly the Wilczynski invariant W is: W = ∂f i ∂y j + 13 ∂f i ∂q k ∂f k ∂p j − ddx ∂f i ∂p j + 16 d dx ∂f i ∂q j − (cid:18) ∂f i ∂q j (cid:19) − ∂f i ∂q k ddx ∂f k ∂q j − ddx (cid:18) ∂f i ∂q k (cid:19) ∂f k ∂q j . An expression ( ∂f i ∂q j ) here is the third power of the matrix ∂f i ∂q j . Note that invariant W hasknown analogue in the case of one differential equation of third order: ∂f∂y + 13 ∂f∂q ∂f∂p − ddx ∂f∂p + 16 d dx ∂f∂q − (cid:18) ∂f∂q (cid:19) − ∂f∂q ddx ∂f k ∂q j . The reader can find this invariant for example in Chern work [1]; also see Sato and Yoshikawa [8].Let us compute the third degree normalization conditions.Ω h mod h θ − ∧ θ − , θ − i = F − j θ x ∧ θ j − + dF − j dx θ x ∧ θ j − + ∂F − j ∂q k θ k − ∧ θ j − − θ x ∧ A − j θ j − . Thus:Ω h [ θ x ∧ θ j − ] = − H − j + F − j + dF − j dx . Normalizing this coefficient to 0 we obtain: F − j = H − j − dF − j dx . Next, Ω ij mod h θ − ∧ θ − , θ − i = ∂A ij ∂p k θ k − ∧ θ x + dG i, − jk dx θ x ∧ θ k − + G i, − jk θ x ∧ θ k − + G i,xk θ x ∧ G k, − jl θ l − + G i, − kl θ k − ∧ G l,xj θ x . ij (cid:2) θ x ∧ θ k − (cid:3) = − ∂A ij ∂p k + dG i, − jk dx + G i, − jk + G i,xl G l, − jk − G i, − lk G l,xj . Assuming this coefficient is equal to 0 we get: G i, − jk = ∂A ij ∂p k − dG i, − jk dx − G i,xl G l, − jk + G i, − lk G lxj . Finally,Ω y mod h θ − , θ − i = ∂H x ∂q i θ j − ∧ θ x + dH − j dx θ x ∧ θ j − + ∂H − j ∂q k θ k − ∧ θ j − + H − j ∂f j ∂q k θ x ∧ θ k − + H − j θ x ∧ θ j − . The coefficient Ω y [ θ j − ∧ θ x ] is the following: ∂H x ∂q j − dH − j dx − H − k ∂f k ∂q j − H − j . Normalizing it to 0 we obtain: H − j = ∂H x ∂q j − dH − j dx − H − k ∂f k ∂q k . The last coefficient we need in degree 3 is Ω y [ θ k − ∧ θ j − ]: ∂H − j ∂q k − ∂H − k ∂q j = 0 . In the degree 4 we need to compute only one coefficient of curvature:Ω y mod h θ − ∧ θ − , θ − i = ∂H x ∂p j θ j − ∧ θ x + ∂H − j ∂p k θ k − ∧ θ j − + dH − j dx θ x ∧ θ j − + ∂H − j ∂q k θ − k ∧ θ j − + H − j ∂f j ∂p k θ x ∧ θ k − + H − j θ x ∧ θ j − − F − j θ j − ∧ (cid:0) H x θ x + H − k θ k − (cid:1) . The Cartan connection coefficient Ω y [ θ x ∧ θ j − ] has the following form: − ∂H x ∂p j + dH − j dx + H − k ∂f k ∂q j − H − j . Assuming it is equal to 0 we get: H − j = − ∂H x ∂p j + dH − j dx + H − k ∂f k ∂q j . Finally, invariant I is the tensor Ω y [ θ k − ∧ θ j − ]: − ∂H − k ∂p j + ∂H − j ∂q k + 2 H − j H − k . hird Order ODEs Systems and Its Characteristic Connections 15 References [1] Chern S.-S., The geometry of the differential equation y ′′′ = F ( x, y, y ′ , y ′′ ), Sci. Rep. Nat. Tsing HuaUniv. (A) (1940), 97–111.[2] Doubrov B., Contact trivialization of ordinary differential equations, in Differential Geometry and Its Ap-plications (Opava, 2001), Math. 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