Related papers: Hamel's Formalism and Variational Integrators on a…
In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular…
Functionals that explicitly depend on occupied, unoccupied, or fractionally-occupied orbitals are rigorously formalized using Clifford algebras, and a variational principle is established that facilitates orbital (and occupation)…
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
A systematic construction of St\"{a}ckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is derived. One…
In this article one introduces a formalism of classical mechanics where complex Lagrangian functions are admitted. The results include complex versions of the Lagrangian function, of the Euler-Lagrange equation, of the Hamilton principle, a…
This paper details a generalization of the formalism presented in the author's 2024 paper, "The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory",…
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with…
Dependable numerical results from long-time simulations require stable numerical integration schemes. For Hamiltonian systems, this is achieved with symplectic integrators, which conserve the symplectic condition and exactly solve for the…
Retractions maps are used to define a discretization of the tangent bundle of the configuration manifold as two copies of the configuration manifold where the dynamics take place. Such discretization maps can be conveniently lifted to a…
The present article introduces a generalization of the (multisymplectic) Hamiltonian field theory for a Lagrangian density, allowing the formulation of this kind of field theories for variational problem of more general nature than those…
Combinatorial design theory studies set systems with certain balance and symmetry properties and has applications to computer science and elsewhere. This paper presents a modular approach to formalising designs for the first time using…
The non-Hermitian formalism is used at present in many papers for the description of open quantum systems. A special language developed in this field of physics which makes it difficult for many physicists to follow and to understand the…
Autonomous systems are highly complex and present unique challenges for the application of formal methods. Autonomous systems act without human intervention, and are often embedded in a robotic system, so that they can interact with the…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
There is a lack of formalism for some key foundational concepts in systems engineering. One of the most recently acknowledged deficits is the inadequacy of systems engineering practices for engineering intelligent systems. In our previous…
Accurate modeling of spin-orbit coupling and noncollinear magnetism requires noncollinear density functionals within the two-component generalized Kohn-Sham (GKS) framework, yet constructing and implementing noncollinear functionals remains…
In this paper, we study the Lagrangian functions for a class of second-order differential systems arising from physics. For such systems, we present necessary and sufficient conditions for the existence of Lagrangian functions. Based on the…
Following the techniques of [4], we formulate a Normal Form Lemma suited to close to be integrable Hamiltonian systems where not all the coordinates are action angles. The Lemma turns to be useful in the theory of KAM tori of…
In electronic structure theory, variational methods offer a valuable paradigm for approximating electronic ground states. However, for historical reasons, this principle is mostly restricted to model chemistries in pre-defined fixed basis…
In earlier work, my colleagues and I developed a formalism for using information theory to understand scales of organization and structure in multi-component systems. One prominent theme of that work was that the structure of a system…