scholarly journals Qualitative Investigation of Hamiltonian Systems by Application of Skew-Symmetric Differential Forms

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 25
Author(s):  
Ludmila Petrova
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Lagrange Equation ◽  
Integrability Condition ◽  
Differential Forms ◽  
Classical Mechanics ◽  
Integral Structures ◽  
Lagrange Formalism

In the present paper, a role of Hamiltonian systems in mathematical and physical formalisms is considered with the help of skew-symmetric differential forms. In classical mechanics the Hamiltonian system is realized from the Euler–Lagrange equation as the integrability condition of the Euler-Lagrange equation and discloses specific features of Lagrange formalism. In the theory of differential equations, the Hamiltonian systems reveals canonical relations that define the integrability conditions of differential equations. The Hamiltonian systems, as a self-independent equations, are an example of dynamic systems that describe a behavior of dynamical systems in phase space. The connection of the Hamiltonian systems with differential equations and dynamical systems point to the fact that dynamical systems can be generated by differential equations. Under the investigation of Hamiltonian systems, in addition to exterior skew-symmetric differential forms it is suggested to use the skew-symmetric differential forms that are defined on a nonintegrable manifolds and possess a nontraditional mathematical apparatus, such as degenerate transformations and transitions from nonintegrable manifold to integral structures.

scholarly journals Non-canonical extension of θ-functions and modular integrability of ϑ-constants

2013 ◽  
Vol 143 (4) ◽  
pp. 689-738 ◽  
Author(s):  
Yurii V. Brezhnev
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Integrability Condition ◽  
Theta Functions ◽  
Quadratic Extension ◽  
Canonical Extension ◽  
Series Expansions ◽  
Inversion Problem ◽  
Modular Inversion ◽  
Differential Properties

We present new results in the theory of the classical theta functions of Jacobi: series expansions and defining ordinary differential equations (ODEs). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta functions; they also yield an exponential quadratic extension of the canonical θ-series. An integrability condition of these ODEs explains the appearance of the modular ϑ-constants and differential properties thereof. General solutions to all the ODEs are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences.

scholarly journals A DIFFERENTIAL INTEGRABILITY CONDITION FOR TWO-DIMENSIONAL HAMILTONIAN SYSTEMS

2014 ◽  
Vol 54 (2) ◽  
pp. 139-141
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Partial Differential Equations ◽  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Harmonic Functions ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Two Dimensional ◽  
Partial Differential ◽  
Generalized Coordinates

We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two-dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form <em>H = T + V</em> where <em>V</em> and <em>T</em> are respectively harmonic functions of the generalized coordinates and the associated momenta.

The role of genericity in the history of dynamical systems theory

2017 ◽  
Author(s):  
Tatiana Roque
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Systems Theory ◽  
Singularity Theory ◽  
Dynamical Systems Theory ◽  
Henri Poincaré ◽  
New Methods ◽  
History Of ◽  
Equation Différentielle

This article examines the role of genericity in the development of dynamical systems theory. In his memoir ‘Sur les courbes définies par une équation différentielle’, published in four parts between 1881 and 1886, Henri Poincaré studied the behavior of curves that are solutions for certain types of differential equations. He successfully classified them by focusing on singular points, described the trajectories’ behavior in important particular cases and provided new methods that proved to be extremely useful. This article begins with a discussion of singularity theory and its influence on the first definitions of genericity, along with the application of the notions of structural stability and genericity to understand dynamical systems. It also analyzes the Smale conjecture and how it was proven wrong and concludes with an overview of changes in the definitions of genericity meant to describe the ‘dark realm of dynamics’.

Algebraic integrability: a survey

2007 ◽  
Vol 366 (1867) ◽  
pp. 1203-1224 ◽  
Author(s):  
Pol Vanhaecke
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Abelian Varieties ◽  
Classical Mechanics ◽  
Momentum Map ◽  
Algebraic Integrability ◽  
Algebraically Integrable

We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler–van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems.

GEOMETRIC APPROACHES TO PRODUCE PROLONGATIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

2013 ◽  
Vol 10 (04) ◽  
pp. 1350002
Author(s):  
PAUL BRACKEN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Integrability Condition ◽  
Nonlinear Partial Differential Equations ◽  
Differential Forms ◽  
Pauli Matrices ◽  
Structure Constants ◽  
Specific Equation ◽  
Prolongation Structure

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any specific equation. An integrability condition is provided which gives by construction the equation to be investigated and whose components involve the structure constants of an SU(2) Lie algebra. Along side this, a related, different kind of prolongation, a type of Wahlquist–Estabrook prolongation, over a closed differential ideal is discussed and some applications are given.

scholarly journals Gauge Functions in Classical Mechanics: From Undriven to Driven Dynamical Systems

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 425-435
Author(s):  
Zdzislaw E. Musielak ◽  
Lesley C. Vestal ◽  
Bao D. Tran ◽  
Timothy B. Watson
Keyword(s):  
Dynamical Systems ◽  
Energy Function ◽  
Physical System ◽  
Classical Mechanics ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Gauge Functions

Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and allow for converting an undriven physical system into a driven one. This is a novel phenomenon in dynamics that resembles the role of gauges in quantum field theories.

HOW INFLUENTIAL WAS MECHANICS IN THE DEVELOPMENT OF NEOCLASSICAL ECONOMICS? A SMALL EXAMPLE OF A LARGE QUESTION

2010 ◽  
Vol 32 (4) ◽  
pp. 531-581 ◽  
Author(s):  
IVOR GRATTAN-GUINNESS
Keyword(s):  
Dynamical Systems ◽  
Significant Role ◽  
Classical Mechanics ◽  
Neoclassical Economics ◽  
The Impact

It is well known that classical mechanics played a significant role in the thought of several major economists in the neoclassical tradition from the 1860s to the 1910s. Less well studied are the particular parts or features of mechanics that exercised this influence, or the depth and extent of the impact. After outlining the main traditions of mechanics and the calculus, and describing types of analogy between theories in general, I review some main pertinent features of the work of eight neoclassical economists from the 1830s to the 1910s. I argue that the influence took various forms but that in practice it was modest. Then I briefly describe a fresh set of possible influences with the development of dynamical systems in the period 1920–1950, where again the role of mechanics was limited. I end by raising a large question: does economics need some mathematics designed for its own purposes rather than that traditionally obtained by analogizing from mechanics and physics?

Limit theorems in averaging for dynamical systems

1995 ◽  
Vol 15 (6) ◽  
pp. 1143-1172 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Limit Theorems ◽  
Moderate Deviation ◽  
Suspension Flow ◽  
New Class ◽  
Hyperbolic Flow

AbstractThis paper yields diffusion and moderate deviation type asymptotics for solutions of differential equations of the form dZε(t)/dt = εB(Zε(t), fty) where ft is a suspension flow (in particular, a hyperbolic flow) over a sufficiently fast mixing transformation. Such problems emerge in the study of perturbed Hamiltonian systems. These exhibit a new class of limit theorems for dynamical systems and extend a number of previously known results.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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