Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1997-07-03
J.Math.Phys. 39 (1998) 4578-4603
Physics
High Energy Physics
High Energy Physics - Theory
New sections on integrability of Multivector Fields and applications to Field Theory (including some examples) are added. The
Scientific paper
10.1063/1.532525
The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable multivector fields in $E$ are equivalent to integrable connections in the bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is applied to the particular case of multivector fields in the manifold $J^1E$ and connections in the bundle $J^1E\to M$ (that is, jet fields in the repeated jet bundle $J^1J^1E$), in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections. These results allow us to set the Lagrangian evolution equations for first-order classical field theories in three equivalent geometrical ways (in a form similar to that in which the Lagrangian dynamical equations of non-autonomous mechanical systems are usually given). Then, using multivector fields; we discuss several aspects of these evolution equations (both for the regular and singular cases); namely: the existence and non-uniqueness of solutions, the integrability problem and Noether's theorem; giving insights into the differences between mechanics and field theories.
Echeverría-Enríquez Arturo
Muñoz-Lecanda Miguel C.
Roman-Roy Narciso
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