*A. A. Ivanov*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198527596
- eISBN:
- 9780191713163
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527596.003.0001
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses concrete group theory. Topics covered include symplectic and orthogonal GF(2)-forms, transvections and Siegel transformations, Witt's theorem, and the space of forms.

This chapter discusses concrete group theory. Topics covered include symplectic and orthogonal GF(2)-forms, transvections and Siegel transformations, Witt's theorem, and the space of forms.

*Dusa McDuff and Dietmar Salamon*

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0004
- Subject:
- Mathematics, Analysis, Geometry / Topology

The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser ...
More

The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser isotopy, with its various applications such as Darboux’s theorem and the Lagrangian neighbourhood theorem. The chapter now includes a brief discussion of the Chekanov torus and Luttinger surgery. The last section on contact structures has been significantly expanded.Less

The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser isotopy, with its various applications such as Darboux’s theorem and the Lagrangian neighbourhood theorem. The chapter now includes a brief discussion of the Chekanov torus and Luttinger surgery. The last section on contact structures has been significantly expanded.

*Dusa McDuff and Dietmar Salamon*

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0002
- Subject:
- Mathematics, Analysis, Geometry / Topology

The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the ...
More

The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the Legendre transformation leads to Hamilton’s equations, symplectic forms, symplectomorphisms, and the symplectic action. It ends with a brief overview of some modern results in the subject on the symplectic topology of Euclidean space, such as the Weinstein conjecture and the Gromov nonsqueezing theorem.Less

The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the Legendre transformation leads to Hamilton’s equations, symplectic forms, symplectomorphisms, and the symplectic action. It ends with a brief overview of some modern results in the subject on the symplectic topology of Euclidean space, such as the Weinstein conjecture and the Gromov nonsqueezing theorem.

*Peter Mann*

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0021
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), ...
More

This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), tangent bundles, cotangent bundles, vector fields, the Poincaré–Cartan 1-form and Darboux’s theorem. It covers symplectic transforms, the Marsden–Weinstein symplectic quotient, presymplectic and symplectic 2-forms, almost symplectic structures, symplectic leaves and foliation. It also discusses contact structures, musical isomorphisms and Arnold’s theorem, as well as integral invariants, Nambu structures, the Nambu bracket and the Lagrange bracket. It describes Poisson bi-vector fields, Poisson structures, the Lie–Poisson bracket and the Lie–Poisson reduction, as well as Lie algebra, the Lie bracket and Lie algebra homomorphisms. Other topics include Casimir functions, momentum maps, the Euler–Poincaré equation, fibre derivatives and the geodesic equation. The chapter concludes by looking at deformation quantisation of the Poisson algebra, using the Moyal bracket and C*-algebras to develop a quantum physics.Less

This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), tangent bundles, cotangent bundles, vector fields, the Poincaré–Cartan 1-form and Darboux’s theorem. It covers symplectic transforms, the Marsden–Weinstein symplectic quotient, presymplectic and symplectic 2-forms, almost symplectic structures, symplectic leaves and foliation. It also discusses contact structures, musical isomorphisms and Arnold’s theorem, as well as integral invariants, Nambu structures, the Nambu bracket and the Lagrange bracket. It describes Poisson bi-vector fields, Poisson structures, the Lie–Poisson bracket and the Lie–Poisson reduction, as well as Lie algebra, the Lie bracket and Lie algebra homomorphisms. Other topics include Casimir functions, momentum maps, the Euler–Poincaré equation, fibre derivatives and the geodesic equation. The chapter concludes by looking at deformation quantisation of the Poisson algebra, using the Moyal bracket and C*-algebras to develop a quantum physics.