Seminarium KMMF "Teoria Dwoistości"

Even if the Euler-Lagrange equations are basic in analysis ofmany 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) inthe 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.