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Zbl 0912.58002
Crampin, M.; Martínez, E.; Sarlet, W.
Linear connections for systems of second-order ordinary differential equations. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 65, No.2, 223-249 (1996). ISSN 0246-0211

In this interesting and important paper the authors develop a geometrical approach to the study of systems of second-order ordinary differential equations of the form $$\ddot x^i= f^i(t,x^j,\dot x^j).$$ Such a system of equations may be represented as a certain type of vector field on a differentiable manifold of the form $\bbfR\times TM$, where $M$ is a manifold and $TM$ is its tangent bundle. This geometrical approach to tackling many problems is encountered in the study of systems of second-order ODE's, for example, in problems concerning conditions for the existence of coordinates with respect to which the equations take a special form -- in which the right-hand sides vanish, or are linear, or in which the equations decouple, and also in problems concerned with the qualitative behaviour of families of solutions. It is shown that a convenient linear connection is a very effective tool for the investigation of problems of the kind described above. In particular, the vanishing of the curvature of the connection is a necessary and sufficient condition for the existence of coordinates with respect to which the solution curves of the equations are straight lines.\par The paper is well written.
[A.Klíč (Praha)]

MSC 2000:: *58A20 Jets
37-99 Dynamic systems and ergodic theory
34A26 Geometric methods in differential equations
53C07 Special connections and metrics on vector bundles

Keywords: tangent bundle; jet bundle; horizontal distribution; systems of second-order ordinary differential equations; connections

Cited in: Zbl 1150.70011 Zbl 1078.58005 Zbl 1003.58005

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