By José F. Cariñena, Visit Amazon's Alberto Ibort Page, search results, Learn about Author Central, Alberto Ibort, , Giuseppe Marmo, Giuseppe Morandi

This booklet describes, through the use of uncomplicated innovations, how a few geometrical constructions customary this present day in lots of parts of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. it truly is assumed that what will be accessed in real reviews whilst learning a given process is simply its dynamical habit that's defined through the use of a family members of variables ("observables" of the system). The publication departs from the primary that ''dynamics is first'' after which attempts to respond to in what feel the only real dynamics determines the geometrical constructions that experience proved so invaluable to explain the dynamics in such a lot of very important cases. during this vein it's proven that almost all of the geometrical buildings which are utilized in the normal displays of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are decided, notwithstanding commonly now not uniquely, via the dynamics by myself. an identical application is complete for the geometrical buildings correct to explain quantum dynamics. eventually, it really is proven that extra houses that permit the specific description of the dynamics of sure dynamical platforms, like integrability and great integrability, are deeply relating to the former improvement and may be lined within the final a part of the publication. The mathematical framework used to provide the former software is saved to an trouble-free point through the textual content, indicating the place extra complicated notions could be had to continue additional. A relations of suitable examples is mentioned at size and the required rules from geometry are elaborated alongside the textual content. despite the fact that no attempt is made to provide an ''all-inclusive'' creation to differential geometry as many different books exist already out there doing precisely that. despite the fact that, the advance of the former software, regarded as the posing and answer of a generalized inverse challenge for geometry, results in new methods of considering and referring to essentially the most conspicuous geometrical buildings showing in Mathematical and Theoretical Physics.

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**Geometry from Dynamics, Classical and Quantum**

This e-book describes, by utilizing undemanding concepts, how a few geometrical buildings everyday this present day in lots of components of physics, like symplectic, Poisson, Lagrangian, Hermitian, and so on. , emerge from dynamics. it's assumed that what could be accessed in genuine stories whilst learning a given procedure is simply its dynamical habit that's defined through the use of a relations of variables ("observables" of the system).

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**Example text**

65) After an essentially trivial reshuffling of variables, it can be shown that the matrix A˜ is precisely the realified counterpart of the complex matrix B (see Sect. 3). 18 1 Some Examples of Linear and Nonlinear Physical . . We can now adapt to the complex formalism the discussion of symmetries and constants of motion for the harmonic oscillator. , b ∈ ker B. Of course, unless some of the frequencies vanish, ker B = 0. Leaving aside this case, complex linear symmetries will be homogeneous transformations with [M, B] = 0 .

The radius of the (circular) orbit is the Larmor radius R L = R = v0 / . 33) More generally, the standard formulae for transformation of the fields under Lorentz boosts [Ja62] show that, if (in units c = 1): v D = E / B < 1, a Lorentz boost with velocity v D leads to E = 0, B = B + O (E/B)2 . 34) So, if E · B = 0 and E < B (actually under normal experimental conditions E B ) there is a frame in which the electric field can be boosted away. 12 1 Some Examples of Linear and Nonlinear Physical . .

Eq. 50) is a symmetry. As long as taking powers generates independent matrices, this procedure will generate new symmetries at each step. Notice however that only a finite number of them, will be independent, because of the celebrated Hamilton–Cayley theorem [Ga59], according to which any matrix satisfies its characteristic equation. 52) and the infinitesimal symmetry Eq. 53) leads to δq = λ(Cv + b) and δv = λC(Cv + b). Let us discuss now briefly the constants of motion. 54) aq and using Eqs.