Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method

Abstract

We consider differential equations with quadratic right-hand sides that admit two quadratic first integrals, one of which is a positivedefinite quadratic form. We indicate conditions of general nature under which a linear change of variables reduces this system to a certain 'canonical' form. Under these conditions, the system turns out to be divergenceless and can be reduced to a Hamiltonian form, but the corresponding linear Lie-Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case, the equations can be reduced to the classical equations of the Euler top, and in four-dimensional space, the system turns out to be superintegrable and coincides with the Euler-Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplying by which the Poisson bracket satisfies the Jacobi identity. In the general case for n > 5 we prove the absence of a reducing multiplier. As an example we consider a system of Lotka-Volterra type with quadratic right-hand sides that was studied by Kovalevskaya from the viewpoint of conditions of uniqueness of its solutions as functions of complex time. Bibliography: 38 titles.

The Hojman Construction and Hamiltonization of Nonholonomic Systems

“…It turns out that such a conclusion cannot be drawn in the general case, when the properties of solutions (5) are unknown to us. As already noted, a full set of global Casimir functions can be absent in J (some of them are defined only locally).…”

“…There are various well-known obstructions to Hamiltonization, which are examined in detail in [7] (see also [5]). This problem has several aspects: local, semilocal, and global.…”