Control theory has found several applications in Physics in the last decades, from Statistical and Classical Mechanics to Chaos Theory. This thesis is focused upon a specific topic of this mathematical theory, known as Controllability. We give the two main results in this field, the Frobenius and ChowRashevsky theorems, first in a differentiable environment and then in a less smooth one, with the last chapter focused on some examples in Classical Mechanics. Precisely speaking: we’ll start giving the definition of small time locally controllable system; then, basic Differential geometry notions are given (tangent bundle, vector fields and their fluxes, Lie brackets) and we state the Frobenius and Chow Theorems in this differentiable context; we then examine the nonsmooth case, particularly focusing on redefining Lipschitz vector fields and introducing objects like Setvalued maps and Generalized Differential Quotients with their properties, all of which are necessary to give a new precise definition of Lie brackets in this nondifferentiable context. This will allow an immediate restatement of the Frobenius Theorem; an exact formula for the composition of the fluxes of vector fields, then, will let a generalization of the Chow’s theorem in the nonsmooth case. The last chapter shows some applications of the two theorems in Classical Mechanics, as well as an interesting connection between Lie and Poisson brackets, which allows the use of this work in Hamiltonian mechanics.
Teoremi di Frobenius e Chow per campi vettoriali lipschitziani
Simonetti, Paolo
2016/2017
Abstract
Control theory has found several applications in Physics in the last decades, from Statistical and Classical Mechanics to Chaos Theory. This thesis is focused upon a specific topic of this mathematical theory, known as Controllability. We give the two main results in this field, the Frobenius and ChowRashevsky theorems, first in a differentiable environment and then in a less smooth one, with the last chapter focused on some examples in Classical Mechanics. Precisely speaking: we’ll start giving the definition of small time locally controllable system; then, basic Differential geometry notions are given (tangent bundle, vector fields and their fluxes, Lie brackets) and we state the Frobenius and Chow Theorems in this differentiable context; we then examine the nonsmooth case, particularly focusing on redefining Lipschitz vector fields and introducing objects like Setvalued maps and Generalized Differential Quotients with their properties, all of which are necessary to give a new precise definition of Lie brackets in this nondifferentiable context. This will allow an immediate restatement of the Frobenius Theorem; an exact formula for the composition of the fluxes of vector fields, then, will let a generalization of the Chow’s theorem in the nonsmooth case. The last chapter shows some applications of the two theorems in Classical Mechanics, as well as an interesting connection between Lie and Poisson brackets, which allows the use of this work in Hamiltonian mechanics.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.12608/24374