Order reduction, projectability and constraints of second-order field theories and higher-order mechanics

Abstract

The projectability of Poincaré-Cartan forms in a third-order jet bundle $J^3\pi$ onto a lower-order jet bundle is a consequence of the degenerate character of the corresponding Lagrangian. This fact is analyzed using the constraint algorithm for the associated Euler-Lagrange equations in $J^3\pi$. The results are applied to study the Hilbert Lagrangian for the Einstein equations (in vacuum) from a multisymplectic point of view. Thus we show how these equations are a consequence of the application of the constraint algorithm to the geometric field equations, meanwhile the other constraints are related with the fact that this second-order theory is equivalent to a first-order theory. Furthermore, the case of higher-order mechanics is also studied as a particular situation.

12 pages. Section 3 has been significantly enlarged (the Hilbert-Einstein Lagrangian is explained in more detail). The proofs of theorems 1 and 2 have been slightly enlarged to better explain the constraint algorithm. Some misprints have been corrected. The bibliography has been updated and some additional references are added. To be published in Reports on Mathematical Physics

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