Teorema della media per equazioni differenziali alle derivate parziali
Autore
Lorenzo Lerra - Università degli Studi di Milano - [2001-02]
Documenti
Abstract
According to his supervisor, Prof. D. Bambusi, he managed to discover new quantitative estimations for the motion of a slightly forced continuous mechanical system near its equilibrium point.
For discrete systems there are significative results that give an approximate description for “long time” (but not endless) of the perturbed system in terms of its “normal form”, which is obtained via a canonical coordinate transformation.
Basically this is equivalent to evaluate the functional increment between the perturbed Hamiltonian and its normal form through averaging techniques.
For system having a finite number of degrees of freedom such a transformation can always be built following a standard algorithm, but for continuous mechanical systems the method fails: indeed, even though a function can be always built, heavy resonance phenomenon prevent the ability to build its reverse, making it no more a coordinate transformation and in particular no more a canonical transformation.
The main idea followed to escape from this “no way out” state has implied a dramatic change in perspective: no more attempts to exactly transform the perturbed Hamiltonian system into its normal form via a canonical transformation, but rather the choice to quasi-transform its normal form into the former Hamiltonian via a quasi-canonical transformation, this one built as an approximation of the formal inverse of the canonical transformation if such a function would exist. The exact inverse doesn’t exist, a formal approximation would be. The final step would be estimating the errors induced by the procedure.
For discrete systems there are significative results that give an approximate description for “long time” (but not endless) of the perturbed system in terms of its “normal form”, which is obtained via a canonical coordinate transformation.
Basically this is equivalent to evaluate the functional increment between the perturbed Hamiltonian and its normal form through averaging techniques.
For system having a finite number of degrees of freedom such a transformation can always be built following a standard algorithm, but for continuous mechanical systems the method fails: indeed, even though a function can be always built, heavy resonance phenomenon prevent the ability to build its reverse, making it no more a coordinate transformation and in particular no more a canonical transformation.
The main idea followed to escape from this “no way out” state has implied a dramatic change in perspective: no more attempts to exactly transform the perturbed Hamiltonian system into its normal form via a canonical transformation, but rather the choice to quasi-transform its normal form into the former Hamiltonian via a quasi-canonical transformation, this one built as an approximation of the formal inverse of the canonical transformation if such a function would exist. The exact inverse doesn’t exist, a formal approximation would be. The final step would be estimating the errors induced by the procedure.
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