scholarly journals Helmoltz problem for the Riccati equation from an analogous Friedmann equation

2022 ◽  
Vol 82 (1) ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Differential Equation ◽  
General Relativity ◽  
Riccati Equation ◽  
Friedmann Equation ◽  
Riccati Differential Equation ◽  
Physical Systems ◽  
First Order ◽  
The Many

AbstractWe report a solution of the inverse Lagrangian problem for the first order Riccati differential equation by means of an analogy with the Friedmann equation of a suitable Friedmann–Lemaître–Robertson–Walker universe in general relativity. This analogous universe has fine-tuned parameters and is unphysical, but it suggests a Lagrangian and a Hamiltonian for the Riccati equation and for the many physical systems described by it.

scholarly journals A special property of the matrix Riccati equation

1974 ◽  
Vol 10 (2) ◽  
pp. 245-253 ◽  
Author(s):  
A.N. Stokes
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Symmetric Matrix ◽  
Special Property ◽  
Positive Definiteness ◽  
Symmetric Matrices ◽  
Unusual Property ◽  
Riccati Differential Equation ◽  
The Matrix ◽  
Independent Variable

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

scholarly journals A CONVENIENT COORDINATIZATION OF SIEGEL–JACOBI DOMAINS

2012 ◽  
Vol 24 (10) ◽  
pp. 1250024 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Half Plane ◽  
Order Differential Equation ◽  
Quantum Evolution ◽  
Classical Motion ◽  
Matrix Riccati Equation ◽  
First Order ◽  
Upper Half Plane ◽  
Linear Differential

We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi ball [Formula: see text] as the sum of the Kähler two-form on ℂn and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a hermitian linear Hamiltonian in the generators of the Jacobi group [Formula: see text] are described by a matrix Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂn, with coefficients depending also on [Formula: see text]. Hn denotes the (2n+1)-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on [Formula: see text]. When the transform FC : (η, W) → (z, W) is applied, the first-order differential equation in the variable [Formula: see text] becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel–Jacobi upper half plane [Formula: see text], where [Formula: see text] denotes the Siegel upper half plane.

scholarly journals Perturbations of Half-Linear Euler Differential Equation and Transformations of Modified Riccati Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Ondřej Došlý ◽  
Hana Funková
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Riccati Differential Equation ◽  
Oscillatory Properties

We investigate transformations of the modified Riccati differential equation and the obtained results we apply in the investigation of oscillatory properties of perturbed half-linear Euler differential equation. A perturbation is also allowed in the differential term.

scholarly journals THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

1998 ◽  
Vol 13 (21) ◽  
pp. 3601-3627 ◽  
Author(s):  
J. F. CARIÑENA ◽  
G. MARMO ◽  
J. NASARRE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Order Differential Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
First Order ◽  
Second Order Differential Equations

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

scholarly journals On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

2021 ◽  
Author(s):  
Gunawan Nugroho ◽  
Purwadi Agus Darwito ◽  
Ruri Agung Wahyuono ◽  
Murry Raditya
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Variable Coefficients ◽  
Higher Order ◽  
First Order ◽  
Generalized Riccati Equation ◽  
Polynomial Differential Equation

The simplest equations with variable coefficients are considered in this research. The purpose of this study is to extend the procedure for solving the nonlinear differential equation with variable coefficients. In this case, the generalized Riccati equation is solved and becomes a basis to tackle the nonlinear differential equations with variable coefficients. The method shows that Jacobi and Weierstrass equations can be rearranged to become Riccati equation. It is also important to highlight that the solving procedure also involves the reduction of higher order polynomials with examples of Korteweg de Vries and elliptic-like equations. The generalization of the method is also explained for the case of first order polynomial differential equation.

scholarly journals Null scrolls with prescribed curvatures in Lorentz-Minkowski 3-space

2020 ◽  
Vol 28 (3) ◽  
pp. 229-240
Author(s):  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
First Order ◽  
Order Nonlinear Differential Equation ◽  
Base Curve

AbstractIn Lorentz-Minkowski 3-space, null scrolls are ruled surfaces with a null base curve and null rulings. Their mean, as well as their Gaussian curvature, depends only on a parameter of a base curve. In the present paper, we obtain the first-order nonlinear differential equation (Riccati equation) which relates curvatures of a base curve to curvatures of a null scroll. Conditioned by this equation, we can determine a family of null scrolls with a given null base curve and prescribed curvatures, in particular, a family of minimal and constant mean curvature null scrolls.

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

scholarly journals An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

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