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Linear connections for systems of higher order differential equations. (English) Zbl 1078.58005

Summary: For a system of \((k+1)\) order differential equations (or a semi-spray of order \(k\) on the tangent bundle of order \(k)\) we determine a nonlinear connection induced by it. This nonlinear connection induces a linear connection \(D\) on the total space of the tangent bundle of order \(k\), that is called the Berwald connection. Using the Cartan’s structure equations of the Berwald connection, we determine the conditions by which a system of \((k+1)\) order differential equations is linearizable with respect to the accelerations of order \(k\). This is a generalization for the \(k=1\) case presented in the first part of the paper, that was studied also, but using different techniques, in [M. Crampin, E. Martínez and W. Sarlet, Ann. Inst. Henri Poincaré, Phys. Théor. 65, No. 2, 223–249 (1996; Zbl 0912.58002) and M. de Leon and P. R. Rodrigues, Generalized classical mechanics and field theory. North-Holland (1985; Zbl 0581.58015)].

MSC:

58A99 General theory of differentiable manifolds
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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