Connections on Tangent Bundles of Higher Order

Connections on tangent bundles of higher order associated to regular Lagrangians

“…A connection F of type r has an associated semispray 4 of the same type (which is a spray if F is homogeneous); 4 is given by 4 = h4', where 4' is an arbitrary semispray of type r. Moreover F and 4 have the same paths (see [2,11])). Conversely, let 4 be a semispray of type 1 on TkM.…”

“…a connection in the fibration TkM -~ T k-IM (see [2,11])) such that (i) its associated semispray is 41 = 4 + (1/(k+ 1))4", (ii) its tension is H1 = -(1/(k+ 1))[4", J1], (iii) its weak torsion tl vanishes, (iv) its strong torsion is 7"1 = (1/(k + 1))[4", J1]-Therefore, if 4 is a spray, then ¢1=4, Hi=T1=0…”

“…From (5) and (6) we have But we may consider connections of higher order, say r defined only on TrM--{zero section} and the decomposition theorem still holds (see [2], [11]). …”

“…A natural generalization of tangent bundles are the tangent bundles of higher order (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] for instance). These bundles are adequate manifolds for the study of Lagrangian functions involving derivatives greater than 1.…”

“…This is the purpose of the paper. In a previous work [2] the authors have examined some properties of generalized connections in higher order tangent bundles. Here we show how to construct a connection having the higher order EulerLagrange vector field as associated semispray (a 2k-th order differential equation).…”