Pedro L. García

Stress–energy–momentum tensors in higher order variational calculus

Abstract Given a variational problem defined by a natural Lagrangian density L ω on the k-jet extension Jk(Y/X) of a natural bundle p:Y→X over an n-dimensional manifold X, oriented by a volume element ω, a stress–energy–momentum tensor T(s) is constructed for each section s∈Γ(X,Y) from the multimomentum map μ Θ :Γ(X,Y)→ Hom R ( X (X),Ω n−1 (X)) associated to any Poincare–Cartan form Θ and to the natural lifting of vector fields X (X) to the bundle Y→X. The characterization made for T(s) gives an intrinsic expression of this tensor as well as a generalization of the classical Belinfante–Rosenfeld formula. This tensor satisfies the typical properties of a stress–energy–momentum tensor: Diff(X)-covariance, Hilbert formula, conservation law, etc. Furthermore, it plays the expected role in the theory of minimal gravitational interactions.

Stress–energy–momentum tensors for natural constrained variational problems

Abstract Under certain parameterization conditions for the “infinitesimal admissible variations”, we propose a theory for constrained variational problems on arbitrary bundles, which allows us to introduce in a very general way the concept of multi-momentum map associated to the infinitesimal symmetries of the problem. For natural problems with natural parameterization, a stress–energy–momentum tensor is constructed for each “admissible section” from the multi-momentum map associated to the natural lifting of vector fields on the base manifold. This tensor satisfies the typical properties of a stress–energy–momentum tensor (Diff( X )-covariance, Belinfante–Rosenfeld type formulas, etc.), and also satisfies corresponding conservation and Hilbert type formulas for natural problems depending on a metric. The theory is illustrated with several examples of geometrical and physical interest.

Critical principal connections and gauge-invariance

Abstract Let p : P → X be a principal G -bundle over an oriented manifold. As suggested by the classical Yang-Mills field theory, a certain class of variational problems for connections on P is defined. Global equations are derived for the corresponding critical connections and for the Jacobi fields along a critical connection, which are a very adequate tool for the study of the symplectic theory associated with the said variational problems.

Gauge-invariant covariant Hamiltonians

Given an Ehresmann connection γ on a fibered manifold p:E→M, a covariant Hamiltonian density Hγ is then associated to each Lagrangian density L on J1E. Assume E is the bundle of connections of a principal bundle and that L is gauge invariant. Our goal in this paper is to determine conditions on γ under which Hγ is also gauge invariant. The general conclusion is that there is no gauge-invariant Ehresmann connection but there is plenty of such connections providing gauge-invariant covariant Hamiltonians. The relevant cases of U(1) bundles and SU(2) bundles are discussed in detail.

Euler–Poincaré Reduction by a Subgroup of Symmetries as an Optimal Control Problem

Given a Lagrangian density Ldt defined in the 1-jet bundle \(J^{1}P\) of a principal G-bundle \(P\rightarrow \mathbb {R}\), invariant with respect to the action of a closed subgroup \(H\subset G\), its Euler–Poincare reduction in \((J^{1}P)/H=C(P)\times _{\mathbb {R}}(P/H)\) (C(P): the bundle of connections, P / H: the bundle of H-structures) induces an optimal control problem. The control variables of this problem are connections \(\sigma \), the dynamical variables \(\bar{s}\) are H-structures, the Lagrangian density \(l(t,\sigma ,\bar{s})dt\) is the reduction of Ldt and the dynamical equations are \(\nabla ^{\sigma }\bar{s}=0\). We prove that the solution of this problem are solutions of the original reduction problem. We study the Hamilton–Cartan–Pontryagin formulation of the problem under an appropriate regularity condition. Finally, the theory is illustrated with the example of the heavy top, for which the symplectic structure of the set of solutions with zero vertical component of the angular momentum is also provided.