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Multi-Hamiltonian property of a linear system with quadratic invariant. (English. Russian original) Zbl 1479.37058

St. Petersbg. Math. J. 30, No. 5, 877-883 (2019); translation from Algebra Anal. 30, No. 5, 159-168 (2018).
The paper is organized in three sections. In Section 1 the author defines a multi-Hamiltonian system associated to a linear system of differentiable equations \(\dot x= Ax\), \(x\in\mathbb{R}^k\), with a quadratic invariant \(f=\frac12(Bx,x)\), where \(A\) and \(B\) are nondegenerate operators and \(B\) is selfadjoint w.r.t. the scalar product \((\cdot,\cdot)\) in \(\mathbb{R}^k.\) This linear system is endowed with a natural symplectic structure \(\Omega(\xi,\eta)=(\Gamma\xi,\eta)\) with \(\Gamma=BA^{-1}\) and \(\xi,\eta\in\mathbb{R}^k\). The author proves the existence of a chain of different symplectic structures \(\Omega_m(\xi,\eta)=(\Gamma_m\xi,\eta)\) with \(\Gamma_m=A^{*m}\Gamma A^m\) and Hamiltonians \(f_m=\frac12(B_mx,x)\) with \(B_m=A^{*m}B A^m\) in involution w.r.t. all \(\Omega_m\), that are independent when the spectrum of the operator \(A\) is simple.
In Section 2, the author proves that a monotone nonincreasing or nondecreasing quadratic form \(f=\frac12(Bx,x)\) along the solutions of the linear system \(\dot x= Ax\), i.e., \(\dot f=BA^*+B^*A\) is nonpositive or nonnegative, implies that the quadratic form \(f_m=\frac12(B_mx,x)\) is respectively monotone nonincreasing or nondecreasing along the solutions of the linear system, for each integer \(m\). In this section, the author also generalizes the results of the previous section to the case of a time-periodic linear system of differential equations that admit a time-periodic quadratic integral with the same period. The author uses a linear transformation \(x\mapsto Fx\) that admits an invariant quadratic form \((Bx,x)\) to define the chain of symplectic forms and Hamiltonians. The operators \(B\) and \(F\) satisfy two nondegeneracy conditions that enable to define the Cayley transformation \(F\mapsto A=(F+I)(F-I)^{-1}\), essential to prove the last theorem.
Finally, in the last section the author gives references and explains the connection between the results given in the paper and known results about multi-Hamiltonian systems.

MSC:

37J06 General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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