Even if the Euler-Lagrange equations are basic in analysis of
many problems in physics or geometry, their invariants are well
understood in Riemannian, pseudo-Riemannian and (to less extend) Finsler
geometry, only. Invariants of general second order ODEs (SODE) were
introduced already in the 30-ties of last century (Kosambi, E. Cartan,
Chern) but are little known, perhaps of their obscure coordinate
presentation.
We will propose a geometric framework which allows to define invariants,
analogous to classical ones, in more general settings. The basic objects
of study are pairs (X,V), where X is a vector field on a manifold M and
V is a distribution of constant rank, both satisfying some regularity
conditions. Using the Lie bracket one assigns to (X,V) a canonical
connection, a Jacobi endomorphism and equation, an invariant metric
along trajectories, etc. The framework includes canonical classes of
control systems. The behaviour of the trajectories of X can be partially
understood studying the Jacobi endomorphism, its eigenvalues,
eigenvectors, and the flag curvature. We will give criteria for
existence/non-existence of conjugate points on tra-
jectories of X, analogous to classical Bonet-Myers and Cartan-Hadamard.
A semi-Hamiltonian setting will also be treated.
We will present formulas for the Jacobi endomorphism K (curvature) in
the classical case of Euler-Lagrange equations defining the motion of a
charged particle in electromagnetic field. In the case of motion of N
particles with Newtonian interactions in Rn one gets that tr(K)=(3-n)F,
where F is an explicit positive function of positions. This shows that
conjugate points always exist for n=1,2, (and n=3 if there is no
symmetry) but, in general, are absent for n greater than 3. Joint work
with Wojciech Kryński.