Related papers: Variational principles for involutive systems of v…
A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both…
Graded vector bundles over a given $\mathbb{Z}$-graded manifold can be defined in three different ways: certain sheaves of graded modules over the structure sheaf of the base graded manifold, finitely generated projective graded modules…
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…
We show how to "concatenate" variational principles over different bases into one over a single base, thereby providing a unified Lagrangian treatment of interacting systems. As an example we study a Klein-Gordon field interacting with a…
The principle of stationary variance is advocated as a viable variational approach to quantum field theory. The method is based on the principle that the variance of energy should be at its minimum when the state of a quantum system reaches…
In the first part of the paper we define a perturbative (pre-formal) geometry and formulate a theorem on the relation between the construction of a perturbative neighborhood of affine varieties and the higher tangent bundles. In the second…
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields.…
This note aims at obtaining a variational characterization of complex structures by means of a calculus of variations for real vector bundle valued differential forms, and outlines a perspective to study existence questions via functionals…
We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new…
This paper shows that various relevant dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some dissipative systems,…
We show the existence of formal equivalences between reversible and Hamiltonian vector fields. The main tool we employ is the normal form theory.
Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…
We apply the notion of parametrized vector field on a manifold M, where the parameters are also in M, to the study of the zero-curvature condition that arises in the context of integrable systems.
In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given…
We give sufficient Gordin-type criteria for the iterated (enhanced) weak invariance principle to hold for deterministic dynamical systems. Such an invariance principle is intrinsically related to the interpretation of stochastic integrals.…
We study the applicability of the time-dependent variational principle in matrix product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time…
We consider the bifurcation problem u'' + \lambda u = N(u) with two point boundary conditions where N(u) is a general nonlinear term which may also depend on the eigenvalue \lambda. A new derivation of a variational principle for the lowest…
Recall that a vector field on an n-dimensional differentiable manifold M is a mapping X defined on M with values in the tangent bundle TM that assigns to each point $x\in M$ a vector X(x) in the tangent space $T_x M$. A vector field may be…
We define the concepts of topological particles and topological radiation. These are nothing more than connected components of defects of a vector field. To each topological particle we assign an index which is an integer which is conserved…