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Original Articles
Jacobi Stability of Linearized Geometric Dynamics
Constantin Udrişteudriste@mathem.pub.roDepartment of Mathematics-Informatics IFaculty of Applied SciencesUniversity Politehnica of BucharestSplaiul Independentei 313, Bucharest, 060042, RomaniaView full profile → , Ileana Rodica Nicolanicola_rodica@yahoo.comFaculty of Mathematics and InformaticsUniversity ”Spiru Haret” of BucharestIon Ghica Str. 13, RO-030045 Bucharest, RomaniaView full profile →
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- Published Online:
- 03 Jun 2013
- Article type:
- Original Articles
- Language:
- EN
- Article no.:
- 1726037X.2009.10698571
- Pages:
- 161–173
Abstract
This paper is dedicated to the study of geometric dynamics (an Euler-Lagrange prolongation of a flow on a Riemannian manifold) from the point of view of KCC theory, Jacobi stability and Lyapounov stability. Section 1 recalls the geometrical roots of Jacobi stability and announces the subject of the paper. Section 2 introduces the variational ODEs (Jacobi fields ODEs), the KCC differential invariants for a second order ODE system, and defines the Jacobi stability. Section 3 studies the KCC differential invariants associated to geometric dynamics. Section 4 describes various linearizations of geometric dynamics. Section 5 studies the Jacobi stability for linearized geometric dynamics around a stationary point of the field. Section 6 shows that the linearized geometric dynamics around a critical point of the energy can be Jacobi stable or unstable. Section 7 proves the Lyapounov instability of Jacobi fields ODEs along geometric dynamics trajectories.
Keywords
References
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