# What is Kosambi-Cartan-Chern (KCC) theory?

I'm reading a paper that's basically about stability analysis of the Lane-Emden differential equation. The authors make use of "Kosambi-Cartan-Chern (KCC) theory". I've been trying to find out what this "theory" is about and haven't really gotten anywhere except for a another paper (PDF) that refers to this KCC theory as an established thing.

Given that the papers discuss the theory as an established thing, I expected there to be a Wikipedia or MathWorld article or something. But no luck. References are given to a series of papers by Kosambi, Cartan and Chern from the 1930's but I haven't hit any textbooks that describe it. For interest, the relevant references appear to be

• Cartan, E. Observations sur le m´emoir pr´ec ´edent, Math. Zeitschrift 37 (1933), 619-622.
• Chern, S.S., Sur la g´eometrie d’un syst`eme d’equations diﬀerentialles du second ordre, Bull. Sci. Math. 63 (1939), 206-212
• Kosambi, D.D. Parallelism and path-space, Math. Zeitschrift 37 (1933), 608-618
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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 $(d^2x_i/dt^2) + g_i(x,x',t) = 0$, for $i = 1, ... , n$, where $x = (x_1, ..., x_n)$, t is the time parameter, $x'$ denotes $((dx_1/dt), ..., (dx_n/dt))$, and $g_i$'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 $g_i$'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 $g_i$'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.

These review articles may be of help:

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Thanks for a great answer! You've explained both the theory and where the name (and subsequent references) come from. – Warrick Jul 6 '12 at 11:21
@Warrick, no worries, glad to be of use :D – BYS2 Jul 6 '12 at 15:38