Singularities of Functions on the Martinet Plane, Constrained Hamiltonian Systems and Singular Lagrangians

We consider here the analytic classification of pairs $(\omega,f)$ where $\omega$ is a germ of a 2-form on the plane and $f$ is a quasihomogeneous function germ with isolated singularities. We consider only the case where $\omega$ is singular, i.e. it vanishes non-degenerately along a smooth line $H(\omega)$ (Martinet case) and the function $f$ is such that the pair $(f,H(\omega))$ defines an isolated boundary singularity. In analogy with the ordinary case (for symplectic forms on the plane) we show that the moduli in the classification problem are analytic functions of 1-variable and that their number is exactly equal to the Milnor number of the corresponding boundary singularity. Moreover we derive a normal form for the pair $(\omega,f)$ involving exactly these functional invariants. Finally we give an application of the results in the theory of constrained Hamiltonian systems, related to the motion of charged particles in the quantisation limit in an electromagnetic field, which in turn leads to a list of normal forms of generic singular Lagrangians (of first order in the velocities) on the plane.