Geometric decomposition, potential-based representation and integrability of non-linear systems

This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

Topological gauge fixing II: A homotopy formulation

We revisit the implementation of the metric-independent Fock–Schwinger gauge in the Abelian Chern–Simons field theory defined in ℝ3 by means of a homotopy condition. This leads to the Lagrangian [Formula: see text] in terms of curvatures F and of the Poincaré homotopy operator h. The corresponding field theory provides the same link invariants as the Abelian Chern–Simons theory. Incidentally the part of the gauge field propagator which yields the link invariants of the Chern–Simons theory in the Fock–Schwinger gauge is recovered without any computation.

Poincaré Inequalities for Composition Operators withLφ-Norm

We establish the Poincaré-type inequalities for the composition of the homotopy operator, exterior derivative operator, and the projection operator withLφ-norm applied to the nonhomogeneousA-harmonic equation inLφ(Ω)-averaging domains.

Recent Advances inLp-Theory of Homotopy Operator on Differential Forms

The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study ofLp-theory of the homotopy operator applied to differential forms. Specifically, we will discuss various local and global norm estimates for the homotopy operatorTand its compositions with other operators, such as Green’s operator and potential operator.

Variations of Integrals in Diffeology

We establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator K, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.