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Hamiltonian systems on time scales. (English) Zbl 0966.39010

The authors investigate linear Hamiltonian systems on time scales \[ x^{\Delta }={\mathcal H}(t)x, \] where \({}^{\Delta }\) denotes the generalized (time scale) derivative, \(x\in \mathbb{R}^{2n}\), \({\mathcal H}\) is a \(2n\times 2n\) matrix satisfying \({\mathcal H}^*(t){\mathcal J}+{\mathcal J}{\mathcal H}(t)+ \mu(t){\mathcal H}^*(t){\mathcal J}{\mathcal H}(t)=0\), \(\mu\) is the graininess of the time scale under consideration and \({}^*\) denotes the conjugate transpose of the matrix indicated. In recent years, the calculus on time scales was developed in such a way that it incorporates the differential and difference calculus as special cases. Hence, Hamiltonian systems on time scales cover both linear Hamiltonian differential systems and symplectic difference systems as particular cases.
In this paper, a chain rule for differentiation on time scales is introduced and using this rule various aspects of transformation theory of time scale Hamiltonian systems are investigated.

MSC:

39A12 Discrete version of topics in analysis
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
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