Using examples of: 1) scalar field theory, 2) electrodynamics, 3)
Pauli-Fierz theory, I will show how the Hamiltonian description of field
dynamics enables us to find appropriate mathematical tools for the
analysis of the dynamics stability. I very much hope that similar methods
can be used in General Relativity Theory, where proving stability is
extremely difficult. The reason for this difficulty is that any attempt
(there have been many of them!) to define a positive and conserved
gravitational field energy leads to serious problems. I will try to
explain these difficulties and to formulate my hopes.