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Jet fields, connections and second-order differential equations. (English) Zbl 0627.70013
The study of second-order differential equation fields is central to the theory of time-dependent Lagrangian particle dynamics. In general, these fields may be defined as vector fields \(\Lambda\) on the manifold TE\(\times {\mathbb{R}}\) (where E is the configuration manifold) which both annihilate all the contact forms and also satisfy the normalisation condition \(\Lambda (t)=1\). In local coordinates \((t,q^{\alpha},\dot q^{\alpha})\) such a field would be written \(\Lambda =\partial /\partial t+\dot q^{\alpha}(\partial /\partial q^{\alpha})+\Lambda^{\alpha}(\partial /\partial \dot q^{\alpha})\), where \(\Lambda^{\alpha}\) are functions on TE\(\times {\mathbb{R}}.\)
The main argument of this paper is that it is profitable to consider, not the vector field \(\Lambda\) as such, but the associated type (1,1) tensor field \(\Lambda\otimes dt\). The reason for introducing this apparent complexity is that there is a natural construction of the tensor field which generalizes immediately to jet bundles, and may therefore be used in the study of Lagrangian field theories. The tensor field itself is essentially a Cartan-Ehresmann connection, but a definition in terms of ‘jet fields’ clarifies the analogy with mechanics and motivates this use of second-order jet fields. To demonstrate the use of the construction we prove three theorems which generalize results from the modern theory of tangent bundle geometry. Although we shall only consider first-order Lagrangian systems in the present work, the ideas may in principle be extended to higher-order systems.
The structure of this paper is as follows. We summarize the notation used, and we give a general definition of jet fields and develop some of their properties. We apply these ideas to second-order differential equation fields.

MSC:
70Sxx Classical field theories
70H03 Lagrange’s equations
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