The relation between the Jacobi morphism and the Hessian in gauge-natural field theories. (English. Russian original) Zbl 1186.70020
Theor. Math. Phys. 152, No. 2, 1191-1200 (2007); translation from Teor. Mat. Fiz. 152, No. 2, 377-389 (2007).
Summary: We generalize a classic result, due to Goldschmidt and Sternberg, relating the Jacobi morphism and the Hessian for first-order field theories to higher-order gauge-natural field theories. In particular, we define a generalized gauge-natural Jacobi morphism where the variation vector fields are Lie derivatives of sections of the gauge-natural bundle with respect to gauge-natural lifts of infinitesimal principal automorphisms, and we relate it to the Hessian. The Hessian is also very simply related to the generalized Bergmann-Bianchi morphism, whose kernel provides necessary and sufficient conditions for the existence of global canonical superpotentials. We find that the Hamilton equations for the Hamiltonian connection associated with a suitably defined covariant strongly conserved current are satisfied identically and can be interpreted as generalized Bergmann-Bianchi identities and thus characterized in terms of the Hessian vanishing.
MSC:
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
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