5. Analysis of Hamiltonian Systems

After a first section dealing with the Euler–Lagrange equations, modeling mechanical systems, the chapter deals with the natural extension of the Euler–Lagrange systems constituted by the Hamiltonian ones. The Hamiltonian systems are classically defined by using the Poisson brackets, which are strictly connected with first integrals; several properties of the Poisson brackets, as their relation with changes of coordinates, are explored in detail. In view of the historical importance of the Hamiltonian systems, their symmetries and their normal forms were studied independently and have therefore names referring to researchers that studied them; e.g., a Hamiltonian symmetry of a Hamiltonian system is named a Noether symmetry and the Poincaré–Dulac normal form of a Hamiltonian system is named the Birkhoff–Gustavson form. The chapter is completed by a study about how wide is the class of the Hamiltonian systems; it is shown that every system having an inverse Jacobi last multiplier equal to one can be written as a Hamiltonian system, if the Poisson bracket is properly defined (this can be done in general with the help of the Nambu bracket).