Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), ISSN

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1 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis , wwwemisde/journals ISSN ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACES MARCEL ROMAN Abstract Considering a Lagrangian of order k, we determine a nonlinear connection N on T k M such that the horizontal and vertical distributions to be Lagrangian subbundles for the presymplectic structure given by the Cartan-Poincaré two-form ω k L 1 Introduction We denote by T k M, π k, M, k 1, the space of tangent bundle of order k over a smooth, real, n-dimensional manifold M, [5] Local coordinates x i on M induce local coordinates x i, y i,, y i on T k M, where for a k-jet j0 k ρ T k M, the coordinate functions are defined as follows y αi j k 0 ρ 1 α! d α x i ρ dt α, α {1,, k} t0 The tangent structure of order k, J is defined as follows, [6], J y i dxi + y 2i dyi + + y i dyk 1i The foliated structure of T k M allows for k regular, integrable, vertical distributions, V k α+1 Ker J α Im J k α+1, α {1,, k} 2000 Mathematics Subject Classification 53B40, 53C20, 53C60 Key words and phrases higher order Lagrange spaces, nonlinear connection, Cartan- Poincaré forms This work has been supported by CEEX Grant 31742/2006 and by Grant CNCSIS 1158/2006 from the Romanian Ministry of Education 115

2 116 MARCEL ROMAN The following k vertical vector fields are globally defined on T k M and they are called Liouville vector fields: Γ k y i + 2y2i + + kyi y i y 2i y, i Γ α J k α Γ k, α {1, 2,, k} A semispray is a globally defined vector field S on T k M that satisfies the equation JS Γ k Therefore, a semispray S, which is a vector field of order k + 1, which can be expressed as follows 11 S y i x i + 2y2i + + kyi y i y k 1i k + 1G i x, y,, y y i and it is perfectly determined by its coefficient functions G i x, y,, y A nonlinear connection, or a horizontal distribution on T k M is a regular distribution N : u T k M Nu T u T k M such that the following direct sum holds true: 12 T u T k M Nu N 1 u N k 1 u V k u, where N 1 JN, N α 1 J α 1 N, α {3,, k} The adapted basis to this decomposition is given by { } x i, y,, i y,, k 1i y i where, [7]: 13 x i y i y k 1i x i N j i y N j i i y N j k 1i i y N j j i y N j 2j i y j y, j k 1 y, j The functions N j i connection N, N j i,, N j i are called the local coefficients of the nonlinear 2

3 ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE 117 The dual basis of the previous adapted basis is given by { dx i, y i,, y i}, where: 14 y i dy i + M i jdx j, y 2i dy 2i + M i j y i dy i + M i j dy j + M i jdx j, 2 dy k 1j + + M i jdx j The functions M j i, M j i,, M j i are called the dual coefficients of the nonlinear 2 connection N 2 Cartan-Poincaré forms for a higher order Lagrangian Let us consider a regular Lagrangian of order k, k > 1, Lx i, y i,, y i The metric tensor is given by the symmetric d-tensor field: 21 g ij 1 2 L 2 y i y, j which has maximal rank on T k M, rank g ij n For a regular Lagrangian of order k, we consider the following Cartan-Poincaré one-forms, [4]: 22 θl α d J αl L y αi dxi + + L y k αi dyαi, We consider also the following Cartan-Poincaré two-forms: L ωl α dθl α d dx i + + d 23 y αi α {1,, k} L y k αi α {1,, k} dy αi, We remark here that for k > 1, the regularity of the Lagrangian L implies the fact that rankωl k 2n < k + 1n dimt k M We refer to ωl k as to the canonical presymplectic structure of the Lagrangian L 3 Nonlinear connection Now, our question is: Can we find an adapted basis such that the horizontal distribution to be Lagrangian subbundle for the presymplectic structure ω k L? Taking into account 13 this is equivalent with the determination of the coefficients of a nonlinear connection N

4 118 MARCEL ROMAN As we know, [1], in the k 1 case, we consider the canonical nonlinear connection and we have ω L 2g ji y j dx i It follows that in the basis dx i, y i of the cotangent bundle, the matrix of ω L is 0 2gji 2g ji 0 For k > 1, we consider the Cartan-Poincaré two-form: L ωl k d dx i y i 1 [ L 2 x j ] L y i x i dx j dx i y j + 31 L y j dx i + y j y i + y k 1j L y i + 2g ji y j dx i y k 1j dx i We are looking for a nonlinear connection on T k M such that the presymplectic structure of the lagrangian L to be: ωl k 2g jiy j dx i, ie to have the following matrix: g ji 0 O kn 2g ji k+1n k+1n This is equivalent with the vanishes of all coefficients from 31, except the last one We will obtain the coefficients of the nonlinear connection N We have: y k 1j L 0 y i 2 L y k 1j y N m i j 2g mi 0 Therefore, we find the first coefficient for the nonlinear connection: 32 N m j Now, we take y k 2j the nonlinear connection: 33 N m j 2 L y i 1 2 gmi 2 L y k 1j y i 0 and we obtain the second coefficient of 1 2 L 2 gmi y k 2j y N r 1 2 L i j 2 gmi y k 1r y i

5 ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE 119 Finally, from y j nonlinear connection N: 34 L y i 0 we obtain the k 1 th coefficient of the N m j 1 2 L 2 gmi y k 1 j y N r 1 2 L i j 2 gmi y 2r y i N r j k L 2 gmi y k 1r y i Consequently, the coefficients N m j, N m j,, N m j are unique determined 2 k 1 We have: Theorem 31 With respect to a transformation of the local coordinates x i, y i,, y i x i, ỹ i,, ỹ i on T k M, the coefficient of the nonlinear connection N are transformed by the rule: 35 Ñ i x m m x j xi x m N m j ỹi x j, Ñ i x m m 2 x j xi x m N m j + ỹi x 2 m N m j ỹ2i x j, Ñ i x m m k 1 x j xi x m N m j k 1 + ỹi x m N m j k ỹk 2i x m N m j ỹk 1i x j Proof A transformation of local coordinates x i, y i,, y i x i, ỹ i,, ỹ i on the manifold T k M is given by, [7]: x i x i x 1,, x n, rank xi x j n, ỹ i xi x j yj, 2ỹ 2i ỹi x j ky i ỹk 1i x j y j + 2 ỹi y j y2j, y j + 2 ỹk 1i y j y 2j + + k ỹk 1i y k 1j yj

6 120 MARCEL ROMAN Also, we have the identities, [7]: ỹ αi x j ỹα+1i y j ỹi y k αj, α 0,, k 1; y0i x i With respect to the previous transformation of local coordinates, the natural basis is changed by the following rule: x i xj x i ỹj y i y i x j + ỹj x i ỹj y i y i ỹ j ỹj + + ỹ j y i Taking into account last formulas, we have and consequently, we obtain 2 L ỹj ỹ k 1r y i y k 1m y i y k 1m But, L ỹj y i y i ỹj + + ỹ j x i L ỹ j ỹ j, ỹ j, 2 L ỹ j ỹ + ỹr ỹ j k 1r y k 1m y i ỹ r ỹr y k 1m x m and 2g 2 L jr ỹ j ỹ r Now, contracting by 1 2 gij, we obtain and finally, N j m xj x r x i x m Ñ i r + ỹr x j x m x r Ñ r i x i x m N j x r m x j ỹr x m 2 L ỹ j ỹ r Similarly, in order to check the second relation from 35, we calculate 2 L, for the third relation we calculate 2 L y i y k 2m y i y k 3m and so on, for the last relation we calculate 2 L y i y m Now, considering the following notations: 36 M r j α 1 2 L 2 gri, α {1,, k 1}, y k αj y i

7 ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE 121 we obtain: Proposition 31 The relationship between the coefficients N m i, N m i,, N m i 2 k 1 of the nonlinear connection N and the coefficents given in 36 is expressed by: N m i M m i 37 N m i 2 M m i 2 M m r N r i N m i M m i M m r k 1 k 1 k 2 N r i M m r N r i k 2 Indeed, replacing 36 in 32,33 { and 34 we} have the conclusion There- is the system of dual coef- fore, the system of functions M m i, M m i,, M m i 2 k 1 ficients of the nonlinear connection N Example 31 Let R M, g ij x be a Riemannian space and Prol 2 R n its prolongation of order 2, [8] We consider the Liouville d-vector fields, [7]: 38 z m y m, z 2m 1 [ Γz m + γij m z i z j], 2 where γij m x are the Christoffel symbols and the operator Γ is given by: Γ y i x i + 2y2i y i Considering the Lagrangian L g ij xz i z j, we remark that the first coefficient N m i of our nonlinear connection is the same with the first coefficient from nonlinear connections determined by I Bucataru, [2], M de Leon, [4], and R Miron, [7], ie is expressed by: N i j M i j γ i jhxy h Now, for the last coefficient of the nonlinear connection N i j, we have:

8 122 MARCEL ROMAN Theorem 32 The skew symmetric part of the coefficient N i j follows: 39 where M ji N [ji] g im M m j L x i y 2 L j x j y i 1 k 1 2 gsm α1 N js M im α k α N is M jm α k α is expressed as Proof For the last coefficient of the nonlinear connection N i j the first coefficient from 31: 310 We obtain: 1 2 [ x j L y i ] L x i 0 y j we have to consider 311 N [ji] 1 N ji N ij L 4 x i y 2 L j x j y i 1 N m 2 L j 2 y m y N m 2 L i i y m y j 1 N m 2 L j 2 y k 1m y N m 2 L i i, y k 1m y j k 1 k 1 where N ji g im N m j The condition 310 determines uniquely the skew symmetric part of the coefficient N m j, only For the symmetric part we need a supplementary condition In the k 1 case, I Bucataru proved that the symmetric part is uniquely determined by a metric condition, [3] So far, we have a whole family of nonlinear connections that are determined by the presymplectic structure ωl k These connections are derived directly from the Lagrangian L and does not use the canonical semispray

9 ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE 123 References [1] M Anastasiei Symplectic structures and Lagrange geometry In Finsler and Lagrange geometries Iaşi, 2001, pages 9 16 Kluwer Acad Publ, Dordrecht, 2003 [2] I Bucătaru Sprays and homogeneous connections in the higher order geometry Stud Cerc Mat, 505-6: , 1998 [3] I Bucataru Metric nonlinear connections Differential Geom Appl, 253: , 2007 [4] M de León and P R Rodrigues Generalized classical mechanics and field theory, volume 112 of North-Holland Mathematics Studies North-Holland Publishing Co, Amsterdam, 1985 A geometrical approach of Lagrangian and Hamiltonian formalisms involving higher order derivatives, Notes on Pure Mathematics, 102 [5] C Ehresmann Les prolongements d une variété différentiable I Calcul des jets, prolongement principal C R Acad Sci Paris, 233: , 1951 [6] H A Eliopoulos On the general theory of differentiable manifolds with almost tangent structure Canad Math Bull, 8: , 1965 [7] R Miron The geometry of higher-order Lagrange spaces, volume 82 of Fundamental Theories of Physics Kluwer Academic Publishers Group, Dordrecht, 1997 Applications to mechanics and physics [8] R Miron and G Atanasiu Prolongation of Riemannian, Finslerian and Lagrangian structures Rev Roumaine Math Pures Appl, 413-4: , 1996 [9] W M Tulczyjew The Lagrange differential Bull Acad Polon Sci Sér Sci Math Astronom Phys, 2412: , 1976 Marcel Roman, Technical University Gh Asachi, Department of Mathematics, Iaşi, , Romania address: marcelroman@gmailcom