Hamiltonian Structures and Generating Families by Sergio Benenti

By Sergio Benenti

This e-book is an stronger model of an previous Russian variation. along with thorough revisions, extra emphasis used to be wear reordering the subjects based on a category-theoretical view. this enables the mathematical effects to be acknowledged, proved, and understood in a far more uncomplicated and stylish way.

From the experiences of the Russian edition:

"The major accessory is shifted to the appliance . . . in geometrical optics, thermostatics and regulate concept, and never to the Hamiltonian mechanics simply. . . . To make the publication really self-contained, complete information of uncomplicated definitions and all proofs are incorporated. during this manner, nearly all of the textual content might be learn with out the prerequisite of a direction in geometry. the superb number of examples illustrates the fairly challenging and hugely summary mathematical concept and its hidden problems. . . . The ebook can upward thrust genuine curiosity for experts . . . . The . . . ebook is an important enter within the glossy symplectic geometry and its applications."
(Andrey Tsiganov, St. Petersburg nation University)

Sergio Benenti is a professor of mathematical physics at Università di Torino, Italy. His present fields of analysis contain symplectic geometry with functions to actual theories, Riemannian geometry with purposes to the idea of the separation of variables within the Hamilton-Jacobi equation and in different appropriate differential equations of physics, and mathematical versions of the dynamics of non-holonomic platforms.

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Example text

A basic property of linear relations between symplectic vector spaces is the following symplectic functorial rule. 1. 4) A proof of this fundamental formula is given in Sect. 4 By using this formula we can easily prove the following two fundamental statements. 2. 5 Proof. If R§ = R and S § = S, then (S ◦ R)§ = S § ◦ R§ = S ◦ R. 3. The image R ◦ K of an isotropic (coisotropic, Lagrangian) subspace K by a linear symplectic relation R is an isotropic (coisotropic, Lagrangian) subspace. 4 It is taken from (Benenti 1988) and it is rather cumbersome.

V ω(u, v) = 0 ι I u K isotropic (M, ω) ι ω=0 Fig. 2. 12) The left-hand side of this equation is the precursor of the so-called Lagrange bracket. ♦ This suggests the following extension of the definition of Lagrangian submanifold: a Lagrangian immersion is an immersion ι : K → M into a symplectic manifold such that dim K = 12 dim M and ι∗ ω = 0. An immersed Lagrangian submanifold is the image Λ = ι(K) of a Lagrangian immersion.

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