Equivalence problem for systems of second-order ordinary differential equations
Let ,
, and
be
coordinates in an open connected subset
of the Euclidean
-dimensional space
. Suppose that there is given a second-order system
![]() | (a1) |
for which each is
in a neighbourhood of initial conditions
.
Following D. Kosambi [a8], one wishes to solve the problem of finding the intrinsic geometric properties (i.e., the basic differential invariants) of (a1) under non-singular coordinate transformations of the type
![]() |
A similar problem was solved by E. Cartan and S.S. Chern [a4], [a5], but in the real-analytic case with transformations replaced by
![]() |
Below, only is considered; see the references for
.
Define the KCC-covariant differential of a contravariant vector field on
by
![]() | (a2) |
where the semi-colon indicates partial differentiation with respect to . Note that the Einstein summation convention (cf. Einstein rule) on repeated upper and lower indices is used throughout. Using (a2), equation (a1) can be re-expressed as
![]() | (a3) |
The quantity is a contravariant vector field on
and constitutes the first KCC-invariant of (a1). It represents an "external force" .
If the trajectories of (a1) are varied into nearby ones according to
![]() | (a4) |
where denotes a constant with
small and the
are the components of some contravariant vector defined along
, substitution of (a4) into (a1) and taking the limit as
results in the variational equations
![]() | (a5) |
where the comma indicates partial differentiation with respect to . Using the KCC-covariant differential (a2), this can be re-expressed as
![]() | (a6) |
where
![]() | (a7) |
![]() |
The tensor is the second KCC-invariant of (a1). The third, fourth and fifth invariants are:
![]() | (a8) |
The main result of KCC-theory is the following assertion: Two systems of the form (a1) on are equivalent relative to
if and only if the five KCC-invariant tensors
,
,
,
, and
are equivalent. In particular, there exist coordinates
for which the
all vanish if and only if all KCC-invariants are zero.
Remarks.
if and only if
are positively homogeneous of degree two in the variable
. In this case, the structure of
must accommodate possible non-differentiability in
. This happens in Finsler geometry, but not in affine and Riemannian geometries, where (a1) are geodesics or autoparallels of a linear connection whose coefficients are
. The latter are known as the coefficients of the Berwald connection in Finsler geometry, and of the Levi-Civita connection for Riemannian theory [a3], [a9], [a10]. Furthermore, in the Finsler case,
,
are the Berwald torsion and curvature tensors. Also,
is the Douglas tensor, whose vanishing is necessary and sufficient for all
to be quadratic in the variables
. The latter is always zero in Riemannian and affine geometries, and also for Berwald spaces in Finsler theory [a6], [a3].
Finally, the KCC-invariants can be readily computed in each of the two following cases:
1)
![]() |
where ,
are
functions of
only. In this case
and
![]() |
The trajectories of this equation are Lyapunov stable if , and unstable if
;
2)
![]() |
where are the coefficients of the Levi-Civita connection of a two-dimensional Riemannian metric,
are fixed constants and where the bracket on the right-hand side indicates no summation, [a3]. The KCC-invariants in the case where
are close to Riemannian, but
has a significant effect on Lyapunov stability.
Further applications of KCC-theory can be found in [a2], [a1]. The equivalence problem can be found in a more general context in [a7], [a11].
References
[a1] | P.L. Antonelli, P. Auger, "Aggregation and emergence in population dynamics" Math. Compt. Mod. , 27 : 4 (1998) (Edited volume) |
[a2] | P.L. Antonelli, R.H. Bradbury, "Volterra–Hamilton models in the ecology and evolution of colonial organisms" , World Sci. (1996) |
[a3] | P.L. Antonelli, R.S. Ingarden, M. Matsumoto, "The theory of sprays and finsler spaces with applications in physics and biology" , Kluwer Acad. Publ. (1993) pp. 350 |
[a4] | E. Cartan, "Observations sur le mémoire précédent" Math. Z. , 37 (1933) pp. 619–622 |
[a5] | S. Chern, "Sur la géométrie d'un système d'équations differentielles du second ordre" Bull. Sci. Math. II , 63 (1939) pp. 206–212 (Also: Selected Papers, Vol. II, Springer 1989, 52–57) |
[a6] | J. Douglas, "The general geometry of paths" Ann. of Math. , 29 (1928) pp. 143–169 |
[a7] | R.B. Gardner, "The method of equivalence and its application" , CBMS , 58 , SIAM (Soc. Industrial Applied Math.) (1989) |
[a8] | D. Kosambi, "Parallelism and path-spaces" Math. Z. , 37 (1933) pp. 608–618 |
[a9] | E. Kreyszig, "Introduction to differential and Riemannian geometry" , Univ. Toronto Press (1968) |
[a10] | D. Laugwitz, "Differential and Riemannian geometry" , Acad. Press (1965) |
[a11] | P.J. Olver, "Equivalence, invariants, and symmetry" , Cambridge Univ. Press (1995) |
[a12] | D. Kosambi, "Systems of differential equations of second order" Quart. J. Math. Oxford , 6 (1935) pp. 1–12 |
Equivalence problem for systems of second-order ordinary differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_problem_for_systems_of_second-order_ordinary_differential_equations&oldid=37608