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Dr.Harko's seminar!

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發表於 2013-9-30 10:59:57 | 顯示全部樓層 |閱讀模式
本帖最後由 Astrosing 於 2013-9-30 11:08 編輯

Dear all,

We cordially invite you to join the following Seminar.

The Kosambi-Cartan-Chern(KCC) theory and its applications in chemistry, biology and physics

Abstract
The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In the present talk we review the basic mathematical formalism of the KCC theory, and present some specific applications of this method in chemistry, biology and physics. The Jacobi stability analysis offers a powerful and simple method for constraining the physical properties of different systems, described by second order differential equations.

Date: 2 Oct 2013(Wednesday)
Time: 4:00pm-5:00pm
Venue: HJ610,   Hong Kong Polytechnic University
Guest Speaker: Dr. Tiberiu Harko, University College London, United Kingdom
發表於 2013-9-30 14:29:59 | 顯示全部樓層
本帖最後由 iheby 於 2013-9-30 15:22 編輯

CN -> HK in a week



From http://math.stackexchange.com/qu ... an-chern-kcc-theory

The name 'KCC-theory' was introduced in a book of Antonelli, Ingarden and Matsumoto entitled The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology published in 1993. It is mostly used in the physics and biology fields.

The KCC theory concerns the following. Consider a second order differential equation (d²xi/dt²)+gi(x,x′,t)=0, for i=1,...,n, where x=(x1,...,xn), t is the time parameter, x′ denotes ((dx1/dt),...,(dxn/dt)), and gi's are smooth functions of (x,x′,t) defined on a domain in the (2n+1)-dimensional Euclidean space. The aim is to understand what geometric properties of the system of integral curves - the paths associated with the system of differential equations - remain invariant under nonsingular transformation of the coordinates involved. The theory describes certain invariants, which are specific tensors depending on the gi's, which characterize the geometry of the system, in the sense that two such systems can be locally transformed into each other if and only if the corresponding invariants are equivalent tensors. In particular, a given system as above can be transformed into one for which the gi's are identically 0, so that the integral curves are all straight lines, if and only if the associated tensor invariants are all zero.

The problem may be viewed also as that of realising the integral curves of a second order differential equation as geodesics for an associated linear connection on the tangent bundle. Kosambi introduced a method using calculus of variations, which involves realizing the paths as extremals of a variational principle; this is related to finding a 'metric' for the path space.

By associating a non-linear connection and a Berwald type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system.
發表於 2013-10-3 01:35:08 | 顯示全部樓層
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