Hamiltonian systems on time scales
type: Item
instance of: scholarly article
title: Hamiltonian systems on time scales
zbMATH Open document ID: 0966.39010
author: Calvin D. Ahlbrandt
author: Martin Bohner
author: Jerry R. Ridenhour
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-08-08T00:00:00Z
review text: 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.
reviewed by: Ondřej Došlý
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 37J10
zbMATH DE Number: 1542464
zbMATH Keywords: Hamiltonian system on time scale
zbMATH Keywords: alpha derivative
zbMATH Keywords: chain rule
zbMATH Keywords: symplectic flow
MaRDI profile type: Publication
OpenAlex ID: W2085459820
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DOI: 10.1006/JMAA.2000.6992
description: scientific article; zbMATH DE number 1542464