Symmetries and Hamiltonian Formalism for Complex Materials

Abstract

Preliminary results toward the analysis of the Hamiltonian structure of multifield theories describing complex materials are mustered: we involve the invariance under the action of a general Lie group of the balance of substructural interactions. Poisson brackets are also introduced in the material representation to account for general material substructures. A Hamilton-Jacobi equation suitable for multifield models is presented. Finally, a spatial version of all these topics is discussed without making use of the notion of paragon setting.

Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media

“…was discussed in its various aspects in [20,21,22] in the theory of a single affine body, certain more general aspects were also discussed by Capriz [3,4]. It is not only theoretically interesting, but very often just computationally effective to use Hamiltonian formalism.…”

Geometry of interactions in complex bodies

“…Corollaries [9]. The following statements hold true for any Lagrangian density of the type (32) in the sense of Definition 1:…”

“…The detailed proof is contained in [9]. Here we remind only that it is based on a direct explicit calculation of the terms in (38) on the basis of (36) and (37), and on the exploitation of the relations…”

“…By following the general unified framework of multifield theories proposed by Capriz in 1989 [4] (see also [3]) and then developed further on in its abstract structure (see [32], [23], [8], [9]), we do not specify the nature of ν. We require only that ν be an element of a finite-dimensional differentiable paracompact manifold without boundary M to cover a class as large as possible of special theories.…”

Classical Dynamics of Fullerenes

“…The corresponding Cauchy and Green deformation tensors C ∈ V * ⊗ V * , G ∈ U * ⊗ U * are given by [12,18,[46][47][48] 6) i.e., in components:…”