Quadratic Integrals Research Articles

Erratum to “Quadratic Integral Sliding Mode Control for Nonlinear Harmonic Gear Drive Systems with Mismatched Uncertainties”

Erratum to “Quadratic Integral Sliding Mode Control for Nonlinear Harmonic Gear Drive Systems with Mismatched Uncertainties”

Quadratic first integrals of autonomous conservative dynamical systems

An autonomous holonomic dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the order of the system of differential equations and, if there are enough of them, to determine the solution. Therefore, in the study of dynamical systems, it is important that there exists a systematic method to determine the FIs of second order differential equations. On the other hand, a system of second order differential equations defines (as a rule) a kinetic energy (or Lagrangian), which provides a symmetric second order tensor that we call the kinetic metric. This metric via its symmetries (or collineations) brings into the scene the differential geometry that provides numerous results and methods concerning the determination of these symmetries. It is apparent that if one manages to provide a systematic way that will relate the determination of the FIs of a given dynamical system to the symmetries of the kinetic metric defined by this very system, then one will have at his/her disposal the powerful methods of differential geometry in the determination of the FIs and, consequently, the solution of the dynamical equations. This was also a partial aspect of Lie’s work on the symmetries of differential equations. The subject of this work is to provide a theorem that realizes this scenario. The method we follow has been considered previously in the literature and consists of the following steps: Consider the generic quadratic FI of the form I=Kab(t,qc)q̇aq̇b+Ka(t,qc)q̇a+K(t,qc), where Kab(t, qc), Ka(t, qc), and K(t, qc) are unknown tensor quantities and require dI/dt = 0. This condition leads to a system of differential equations involving the coefficients Kab(t, qc), Ka(t, qc), and K(t, qc) whose solution provides all possible quadratic FIs of this form. We demonstrate the application of the theorem in the classical cases of the geodesic equations and the generalized Kepler potential in which we obtain all the known results in a systematic way. We also obtain and discuss the time-dependent FIs that are as important as the autonomous FIs determined by other methods.

Quadratic conservation laws for equations of mathematical physics

Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the focus of interest in this paper: the existence of other quadratic integrals, the Hamiltonian property of a linear system, and the complete integrability of such a system. For non-degenerate linear systems in a finite-dimensional space essentially exhaustive answers to all these questions are known. Results of a general nature are applied to linear evolution equations of mathematical physics: the wave equation, the Liouville equation, and the Maxwell and Schrödinger equations. Bibliography: 60 titles.

Superintegrable systems on 3 dimensional conformally flat spaces

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux–Koenigs type, whose integrals are reductions of those of the 3 dimensional system.

Quadratic integrals of a multi-scalar cosmological model

In the context of Friedmann–Robertson–Walker (FRW) spacetime with zero spatial curvature, we consider a multi-scalar tensor cosmology model under the pretext of obtaining quadratic conservation laws. We propose two new interaction potentials of the scalar field. Integral to this task is the existence of dynamical Noether symmetries which are Lie–Bäcklund transformations of the physical system. Finally, analytical solutions of the field are found corresponding to each new model. In one of the models, we find that the scale factor mimics [Formula: see text]-cosmology in a special case.

On the Existence of Continuous Positive Monotonic Solutions of a Self-Reference Quadratic Integral Equation

On the Existence of Continuous Positive Monotonic Solutions of a Self-Reference Quadratic Integral Equation

On the properties of numerical solutions of dynamical systems obtained using the midpoint method

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards

The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the framework of the Fomenko–Zieschang theory of the classification of integrable systems.

Path integrals for higher derivative actions

We consider Euclidean path integrals with higher derivative actions, including those that depend quadratically on acceleration, velocity and position. Such path integrals arise naturally in the study of stiff polymers, membranes with bending rigidity as well as a number of models for electrolytes. The approach used is based on the relation between quadratic path integrals and Gaussian fields and we also show how it can be extended to the evaluation of even higher order path integrals.

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

Painlev\'e metrics are a class of Riemannian metrics which generalize the well-known separable metrics of St\"ackel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary differential equations which characterizes the St\"ackel case. Our goal in this paper is to carry out for Painlev\'e metrics the generalization of the analysis, which has been extensively performed in the St\"ackel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlev\'e metrics of the Robertson separability conditions for the Helmholtz equation which are familiar from the St\"ackel case, and a formulation thereof in terms of the vanishing of the off-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for St\"ackel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist $r<n$ linearly independent second-order differential operators which commute with the Laplace-Beltrami operator and which are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations which are compatible with the separation into blocks of variables of the Helmholtz equation for Painlev\'e metrics, leading to solutions which are $R$-separable in blocks. The paper concludes with a set of open questions and perspectives.

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold–Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

First integrals from conformal symmetries: Darboux–Koenigs metrics and beyond

On spaces of constant curvature, the geodesic equations automatically have higher order integrals, which are just built out of first order integrals, corresponding to the abundance of Killing vectors. This is no longer true for general conformally flat spaces, but in this case there is a large algebra of conformal symmetries. In this paper we use these conformal symmetries to build higher order integrals for the geodesic equations. We use this approach to give a new derivation of the Darboux–Koenigs metrics, which have only one Killing vector, but two quadratic integrals. We also consider the case of possessing one Killing vector and two cubic integrals.The approach allows the quantum analogue to be constructed in a simpler manner.

(Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry

Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we turn our attention to 2-dimensional metrics that admit one projective vector field, i.e. whose local flow sends unparametrized geodesics into unparametrized geodesics. We review and discuss the classification of these metrics, introducing special coordinates on the linear space of solutions to a certain system of partial differential equations, from which such metrics are obtained. Particularly, we discuss those that give rise to free second-order superintegrable Hamiltonian systems, i.e. which admit 2 additional, functionally independent quadratic integrals. We prove that these systems are parametrized by the 2-sphere, except for 6 exceptional points where the projective symmetry becomes homothetic.

Design and Simulation of an LQR-PI Control Algorithm for Medium Wind Turbine

In this paper, a new linear quadratic regulator (LQR) and proportional integral (PI) hybrid control algorithm for a permanent-magnet synchronous-generator (PMSG) horizontal-axis wind turbine was developed and simulated. The new algorithm incorporates LQR control into existing PI control structures as a feed-forward term to improve the performance of a conventional PI control. A numerical model based on MATLAB/Simulink and a commercial aero-elastic code were constructed for the target wind turbine, and the new control technique was applied to the numerical model to verify the effect through simulation. For the simulation, the performance data were compared after applying the PI, LQR, and LQR-PI control algorithms to the same wind speed conditions with and without noise in the generator speed. Also, the simulations were performed in both the transition region and the rated power region. The LQR-PI algorithm was found to reduce the standard deviation of the generator speed by more than 20% in all cases regardless of the noise compared with the PI algorithm. As a result, the proposed LQR-PI control increased the stability of the wind turbine in comparison with the conventional PI control.

Fuel cell and ultracapacitor energy system control using linear quadratic regulator proportional integral controller

Application of fuel cell (FC) in power generation requires efficient power converters and controllers for hybridization of energy storage devices. This paper presents the control technique in a fuel cell and ultracapacitor hybrid system to eliminate the slow dynamic problem of FC. The control technique involves a linear quadratic regulator and proportional integral (LQR–PI) controller, where the FC boost converter and the UC bidirectional converter are controlled using a double-loop PI controller and a voltage-mode LQR controller, respectively. The simulation with different types of load was performed to compare the performance of the proposed controller with both PI and LQR controllers. The results revealed that the LQR–PI controller had better overshoot and undershoot responses (7.37% and 6.43%, respectively) than the proportional integral controller (11.47% and 11.42%, respectively) and linear quadratic regulator (23.87% and 21.5%, respectively). The energy sharing performance was verified through variation in load resistance and disturbances in bus voltage. The voltage recovery time of LQR–PI controller was slightly higher (200 ms) than that of the LQR controller (23.8 ms) and, however, lower than that of the PI controller (300 ms). The results revealed that the proposed controller is efficient for load voltage as well as load current fluctuations.

The reduction of the degree of integrals of hamiltonian systems with the help of billiards

In the theory of integrable Hamiltonian systems with two degrees of freedom there are widely known integrable systems whose integrals have a high degree, namely 3 and 4: the Kovalevskaya system and its generalizations - the Kovalevskaya - Yahya system and the Kovalevskaya system on the Lie algebra so(4), Goryachev-Chaplygin-Sretensky, Sokolov and Dullin-Matveyev. The article shows that using integrable billiards bounded by arcs of confocal quadrics decreases the degree of integrals 3 and 4 of these systems fo some isoenergy 3-surfaces. Moreover, the integrals of degree 3 and 4 reduce to the same canonical quadratic integral on billiards.

Correction to “On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension” [Axioms 7(3), 58, 2018

The authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order p + r , where r = 2 and p is the order of the method. This generalization of conjugate-symplecticity states that the methods conserve quadratic first integrals and the Hamiltonian function over time intervals of length O ( h − r ) . Theorem 1 of the above mentioned paper is then replaced by a new one. All the other results in the paper do not change. Two new figures related to the already considered Kepler problem are also added.

Reducing the Degree of Integrals of Hamiltonian Systems by Using Billiards

In the theory of integrable Hamiltonian systems with two degrees of freedom, widely known are integrable systems having integrals of high degrees, namely, 3 and 4. Examples are the Kovalevskaya system and its generalizations—the Kovalevskaya–Yehia system and the Kovalevskaya system on the Lie algebra so(4) the Goryachev–Chaplygin–Sretensky, Sokolov, and Dullin–Matveev systems. It is shown that, at a number of isoenergy 3-surfaces, the third and fourth degrees of integrals of these systems can be reduced by using integrable billiards bounded by arcs of confocal quadrics. Moreover, the integrals of degree 3 and 4 are reduced to the same canonical quadratic integral on a billiard.

Eisenhart lift of 2-dimensional mechanics

The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy–momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux–Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented.

Quantum cosmology of Bianchi VIII, IX LRS geometries

In the present work we revisit the axisymmetric Bianchi VIII and IX models. At the classical level we reproduce the known analytic solution, in a novel way making use of two quadratic integrals of motion, the constraint equation, as well as a linear non-local integral of motion. These quantities correspond to two second rank Killing tensors and a homothetic vector field existing on the relevant configuration space. On the corresponding phase space the two quadratic charges commute with the Hamiltonian constraint but not among themselves. Thus, after turning these charges into operators we obtain two different solutions to the Wheeler DeWitt equation utilizing each of the quadratic operators. The homothetic vector is then used, as a natural guide line, to define a normalizable conditional probability which assigns zero to the classically collapsed configurations.