Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods

Abstract Linear systems of differential equations with an invariant in the form of a positive definite quadratic form in a real Hilbert space are considered. It is assumed that the system has a simple spectrum and the eigenvectors form a complete orthonormal system. Under these assumptions, the linear system can be represented in the form of the Schrödinger equation by introducing a suitable complex structure. As an example, we present such a representation for the Maxwell equations without currents. In view of these observations, the dynamics defined by some linear partial differential equations can be treated in terms of the basic principles and methods of quantum mechanics.

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On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

Discrete and Continuous Models and Applied Computational Science ◽

10.22363/2658-4670-2020-28-4-327-345 ◽

2020 ◽

Vol 28 (4) ◽

pp. 327-345

Author(s):

Yu Ying ◽

Mikhail D. Malykh

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Conservation Law ◽

Numerical Experiments ◽

Kutta Method ◽

Runge Kutta Method ◽

Quadratic Invariant ◽

Quadratic Invariants ◽

Runge Kutta ◽

Autonomous Dynamical Systems

We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

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Dynamical Invariant Applied on General Time-Dependent Three Coupled Nano-Optomechanical Oscillators

Journal of Nanomaterials ◽

10.1155/2021/6903563 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Sara Hassoul ◽

Salah Menouar ◽

Hamid Benseridi ◽

Jeong Ryeol Choi

Keyword(s):

Invariant Operator ◽

Time Dependent ◽

Unitary Matrix ◽

Optomechanical System ◽

Von Neumann ◽

Classical Counterpart ◽

Quadratic Invariant ◽

Fock State ◽

The Matrix ◽

General Time

A quadratic invariant operator for general time-dependent three coupled nano-optomechanical oscillators is investigated. We show that the invariant operator that we have established satisfies the Liouville-von Neumann equation and coincides with its classical counterpart. To diagonalize the invariant, we carry out a unitary transformation of it at first. From such a transformation, the quantal invariant operator reduces to an equal, but a simple one which corresponds to three coupled oscillators with time-dependent frequencies and unit masses. Finally, we diagonalize the matrix representation of the transformed invariant by using a unitary matrix. The diagonalized invariant is just the same as the Hamiltonian of three simple oscillators. Thanks to such a diagonalization, we can analyze various dynamical properties of the nano-optomechanical system. Quantum characteristics of the system are investigated as an example, by utilizing the diagonalized invariant. We derive not only the eigenfunctions of the invariant operator, but also the wave functions in the Fock state.

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Decreasing height along continued fractions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.55 ◽

2018 ◽

Vol 40 (3) ◽

pp. 763-788

Author(s):

GIOVANNI PANTI

Keyword(s):

Continued Fractions ◽

Number Fields ◽

Gauss Map ◽

Projective Line ◽

Triangle Groups ◽

Quadratic Invariant ◽

Weil Height ◽

Symbolic Sequences ◽

Commensurability Class ◽

Quadratic Integer

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map$x\mapsto x^{-1}-\lfloor x^{-1}\rfloor$eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of non-cocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that non-cocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of non-negative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

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