Quadratic Invariant Research Articles

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes.

High-order linearly implicit schemes conserving quadratic invariants

In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be maintained relatively easily compared with general invariants, and can be preserved by canonical Runge–Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge–Kutta methods and prove some properties involving accuracy.

Dynamical invariants for time-dependent real and complex Hamiltonian systems

The Struckmeier and Riedel (SR) approach is extended in real space to isolate dynamical invariants for one- and two-dimensional time-dependent Hamiltonian systems. We further develop the SR-formalism in zz̄ complex phase space characterized by z = x + iy and z̄=x−iy and construct invariants for some physical systems. The obtained quadratic invariants contain a function f2(t), which is a solution of a linear third-order differential equation. We further explore this approach into extended complex phase space defined by x = x1 + ip2 and p = p1 + ix2 to construct a quadratic invariant for a time-dependent quadratic potential. The derived invariants may be of interest in the realm of numerical simulations of explicitly time-dependent Hamiltonian systems.

Linear System of Differential Equations with a Quadratic Invariant as the Schr\xf6dinger Equation

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.

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

Symplectic geometry of a linear transformation with a quadratic invariant

A linear transformation with an invariant being a nondegenerate quadratic form is symplectic. The geometric properties of such transformations are discussed. A complete set of quadratic invariants which are pairwise in involution is explicitly specified. The structure of the isotropic cone on which all these integrals simultaneously vanish is investigated. Applications of the general results to the problem on the stability of a fixed point of a linear transformation with a quadratic invariant are discussed.

Construction of time-dependent dynamical invariants: A new approach

We propose a new way to obtain polynomial dynamical invariants of the classical and quantum time-dependent harmonic oscillator from the equations of motion. We also establish relations between linear and quadratic invariants, and discuss how the quadratic invariant can be related to the Ermakov invariant.

Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant

We consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be anisotropic and the magnetic field is applied along z-axis. A canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators. Two classes of real and complex quadratic invariants (constants of motion) are obtained. We employ the Lie algebraic technique to find the most general solution for the wave-function for both real and complex invariants. The quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave-functions in the coherent states and number states are obtained. Some discussion related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself is also given.

Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant

In this paper we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be isotropic and the magnetic field is applied along the z-axis. The canonical transformation is invoked to remove the interaction term and the system is reduced to a model containing the second harmonic generation. Two classes of the real and complex quadratic invariants (constants of motion) are obtained. We have employed the Lie algebraic technique to find the most general solution for the wave function for both real and complex invariants. Some discussions related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself are also given.

Further classification of 2D integrable mechanical systems with quadratic invariants

Four new integrable classes of mechanical systems on Riemannian 2D manifolds admitting a complementary quadratic invariant are introduced. Those systems have quite rich structure. They involve 11–12 arbitrary parameters that determine the metric of the configuration space and forces with scalar and vector potentials. Interpretations of special versions of them are pointed out as problems of motions of rigid body in a liquid or under action of potential and gyroscopic forces and as motions of a particle on the plane, sphere, ellipsoid, pseudo-sphere and other surfaces.

Lie algebraic treatment of the quadratic invariants for a quantum system

We consider the problem of the time-dependent degenerate parametric amplifier. We obtain the quadratic invariant and use it to derive the wave function via its su(1, 1) algebraic basis and a unitary transformation to the time-dependent Schrodinger equation for the parametric amplifier. We obtain the real and the complex invariants, which we use to solve the time-dependent Cauchy problem. Using different integrability conditions, we find the most general solution, which we analyze extensively, providing details of the calculations.

Quadratic invariants of elastic moduli

A quadratic invariant is defined as a quadratic form in the elements of a tensor that remains invariant under a group of coordinate transformations. It is proved that there are seven quadratic invariants of the 21-element elastic modulus tensor under SO(3) and 35 under SO(2). This answers some open questions raised by Ting and Ahmad in this journal.

Wigner functions for time-dependent coupled linear oscillators via linear and quadratic invariant processes

At exact resonance we derive the linear and quadratic invariants for time-dependent coupled linear oscillators in the presence of second harmonic generation. Employing these invariants we introduce an accurate definition of the Dirac operators from which the wavefunctions in both coherent and Fock (number) states representations are calculated. We also derive the W-Wigner function of an arbitrary state at time t, evolving in the system under consideration. Moreover we calculate the correlation coefficient between position and momentum and we find that it is identical in both fields whatever the state used.

Invariants for the time-dependent harmonic oscillator. I

Two equivalent families of linear invariants and two equivalent families of quadratic invariants are obtained for a harmonic oscillator with variable mass or with variable frequency. Explicit formulae are obtained for the linear and quadratic invariants in the case of a damped oscillator and the Hamiltonian is seen as a special quadratic invariant.

Quadratic Hamiltonians: The four classes of quadratic invariants, their interrelations and symmetries

The quadratic invariants of the three basic quadratic Hamiltonian systems-attractive oscillator, repulsive oscillator, and free particle- are shown to be the same. These invariants are divided into two categories, useful and nonuseful. The definition of useful is in terms of contributing to the (quadratic invariant based) symmetry group of the appropriate Hamiltonian. Usefulness is not invariant under a time-dependent linear canonical transformation. Hence different classes of invariants produce the different symmetry groups for the three different types of quadratic Hamiltonian considered here. The paper concludes with a consideration of a useful transformation of arbitrary quadratic Hamiltonians.

Quadratic Hamiltonians, quadratic invariants and the symmetry group SU(n)

We show that any 2n-dimensional quadratic Hamiltonian may be transformed by a (usually time-dependent) linear canonical transformation into any other 2n-dimensional quadratic Hamiltonian, in particular that of the isotropic harmonic oscillator. This latter Hamiltonian possesses the symmetry group SU(n) and n2−1 linearly independent quadratic invariants which provide a basis for the generators of the group. Every other quadratic Hamiltonian is shown to have a quadratic invariant possessing SU(n) symmetry. The free particle structure is given explicitly. The anisotropic oscillator is shown not to possess SU(3) symmetry based on quadratic invariants. However, its wavefunctions and energy levels may be obtained directly from those of the isotropic oscillator whether the frequencies are commensurable or not.

The Quadratic Invariances of a Generalized Network

The Manley-Rowe relation, as applied in the small signal linearized approximation, may be stated as a quadratic form that is invariant under the operation of the system. It is, however, only one of the set of such forms that is invariant through a given type of system. It is shown that the existence of quadratic invariances is a consequence of the eigenvalues of the system operator being either of unit magnitude or else grouped in pairs such that one is the conjugate reciprocal of the other. If this condition applies, then there exists at least n such linearly independent forms, where n is the number of degrees of freedom of the system. Each form then specifies a quantity that is conserved by the system. Methods of determining the quadratic invariant forms from the matrix operation of the system are developed. Application is made to certain simple two-port networks to illustrate the analysis and the significance of the resulting invariances. Parametric circuits are also studied. The Manley-Rowe relation is found, as expected. Other relations, applicable to subclasses of such networks are also found. Finally, application is made to a lossy parametric shunt element, such as an imperfect nonlinear capacity. The quadratic invariances for such a device, for the two-frequency case, are derived.

Non-Linear Invariants and the Problem of Motion

The effect of adding quadratic invariants to the integrand of the usual variation principle in general relativity is considered in the light of a new approximation method. The case in which the quadratic invariant is a constant multiple of the square of the scalar curvature is treated up to the Newtonian approximation, insofar as the problem of motion is concerned. The Newtonian equations result for sufficiently large "distances" between particles regardless of the order of magnitude of the multiplying constant. The requirement that the "force function" be finite everywhere places restrictions on the number of "particles" $p$, comprising the system. When $p=1,2$, the requirement is fulfilled. The less stringent requirement that the "force function" be finite when all "particles" coincide restricts $p$ less severely. If the absolute values of the "masses" are not all equal there is no restriction on $p$, but if they are all equal $p$ is restricted to certain integral values.

Collineation Groups in a Finite Space with a Linear and a Quadratic Invariant

Collineation Groups in a Finite Space with a Linear and a Quadratic Invariant

Determination of the Structure of All Linear Homogeneous Groups in a Galois Field Which are Defined by a Quadratic Invariant

Determination of the Structure of All Linear Homogeneous Groups in a Galois Field Which are Defined by a Quadratic Invariant