Relation between the guessed and the derived super-Hamiltonians for spherically symmetric shells

The Hamiltonian dynamics of spherically symmetric massive thin shells in general relativity is studied. Two different constraint dynamical systems representing this dynamics have been described recently; the relation between these two systems is investigated. The symmetry groups of both systems are found. New variables are used, which among other things simplify the complicated system a great deal. The systems are reduced to presymplectic manifolds ${\Gamma }_1$ and ${\Gamma }_2,$ lest nonphysical aspects such as gauge fixing or embedding in extended phase spaces complicate the line of reasoning. The following facts are shown. ${\Gamma }_1$ is three and ${\Gamma }_2$ is five dimensional; the description of the shell dynamics by ${\Gamma }_1$ is incomplete so that some measurable properties of the shell cannot be predicted. ${\Gamma }_1$ is locally equivalent to a subsystem of ${\Gamma }_2$ and the corresponding local morphisms are not unique, due to the large symmetry group of ${\Gamma }_2.$ Some consequences for the recent extensions of quantum shell dynamics through the singularity are discussed.