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Quadratic homogeneous ODE systems of Jordan-rigid body type. (English) Zbl 1147.34304

From the text: Trying to yield a Lax representation with the Lax pair of so(3)-type for some quadratic differential systems but with the Jordan bracket instead of usual Lie bracket, a class of homogeneous systems with remarkable properties is obtained.
These types of deformation brackets appear in the first section where we add the so-called Brockett perturbation, a notion useful in rigid body control. Let us note that we use the form of Lax pair given by the rigid body, which admits a classical Lax representation i.e. with respect to the Lie multiplication. For our aim, we must restrict to the upper-triangular part from a matricial differential system and then a quadratic homogeneous ODE system is obtained called Jordan-rigid body type system.
In the third section, using the Brockett perturbation for a system of elliptic Jacobi functions, we obtain a so-called Brockett perturbation of trigonometric functions and a rational first integral of this last system is derived via Bernoulli type equations.
In the following section, a Jordan counterpart of two systems, rigid body and Maxwell-Bloch, with usual Lax representation, is obtained. In the next section to a Jordan-rigid body type system we associate (i) two particular solutions, (ii) two quadratic first integrals and (iii) a numerical integrator preserving these first integrals. A search for an so(4) version of our result and an appendix with the Maple computation of some matrices, useful for our models, and the Maple plot of three examples are inserted at the end of paper.
An important remark is that in this new approach the above mentioned physical significance of Lax pairs is lost. Our initial aim was to point at an algebraic character of Jordan-Lax type for some important systems but this searching for Lax yields a veritable round through algebra, geometry, differential equations and mechanics.

MSC:

34A26 Geometric methods in ordinary differential equations
17B80 Applications of Lie algebras and superalgebras to integrable systems
34A34 Nonlinear ordinary differential equations and systems

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