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3 - A Note on the Commutator of Hamiltonian Vector Fields
- By Henryk Żoła̡dek
Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
Book:

Integrable Systems and Algebraic Geometry

Published online:

19 March 2020

Print publication:

02 April 2020, pp 57-62
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Summary

We present two proofs of the Jacobi identity for the Poisson bracket on a symplectic manifold.

4 - Pontryagin Extremals: Characterization and LocalMinimality
Andrei Agrachev, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Davide Barilari, Université de Paris VII (Denis Diderot), Ugo Boscain, Centre National de la Recherche Scientifique (CNRS), Paris
Book:

A Comprehensive Introduction to Sub-Riemannian Geometry

Published online:

28 October 2019

Print publication:

31 October 2019, pp 109-148
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Summary

This chapter is devoted to the study of the geometricproperties of Pontryagin extremals. We then provethat small pieces of normal extremal trajectoriesare length-minimizers. To this aim, all through thischapter, we develop the language of symplecticgeometry, starting from the key concept of thePoisson bracket.

9 - Poisson Brackets for Field Theory and Equations of Motion: Applications
from Part I - General Properties of Fields; Scalars and Gauge Fields
Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
Book:

Classical Field Theory

Published online:

04 March 2019

Print publication:

14 March 2019, pp 81-88
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Summary

We examine Poisson brackets in field theory and the symplectic formulation of Hamiltonian dynamics. We start by describing the symplectic formulation of classical mechanics. Then we generalize it and Poisson brackets to field theory. As examples of the formalism, we consider a scalar field with canonical kinetic term and the nonlinear sigma model.

1 - Short Review of Classical Mechanics
from Part I - General Properties of Fields; Scalars and Gauge Fields
Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
Book:

Classical Field Theory

Published online:

04 March 2019

Print publication:

14 March 2019, pp 3-13
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Summary

In the first chapter, the most important concepts of classical mechanics are quickly reviewed. The Lagrangian and Hamiltonian formalism are described. The way to deal with systems with constraints is described. Poisson brackets and the use of canonical transformations in the Hamiltonian formalism, as well as the basics of Hamilton–Jacobi theory complete this chapter.

Poisson Brackets with Prescribed Casimirs
Pantelis A. Damianou, Fani Petalidou
Journal:

Canadian Journal of Mathematics / Volume 64 / Issue 5 / 01 October 2012

Published online by Cambridge University Press:

20 November 2018, pp. 991-1018

Print publication:

01 October 2012
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We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.

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