Abstract
In this chapter, we study symplectic manifolds. We start with the Theorem of Darboux, which states that all symplectic structures of a given dimension are locally equivalent. Thus, in sharp contrast to the situation in Riemannian geometry, symplectic manifolds of the same dimension can at most differ globally. The second important observation is that a symplectic structure provides a duality between smooth functions and certain vector fields, called Hamiltonian vector fields. As a consequence, one obtains the notion of Poisson structure. Given the great importance of Poisson structures both in mathematics and in physics, we go beyond the symplectic case and give a brief introduction to general Poisson manifolds, including a proof of the Symplectic Foliation Theorem. Two classes of symplectic manifolds are discussed in detail: cotangent bundles, because they serve as a mathematical model of phase space, and orbits of the coadjoint representation of a Lie group, because they show up in the study of systems with symmetries. Moreover, we show that the coadjoint orbits coincide with the symplectic leaves of the Lie-Poisson structure. Next, we discuss coisotropic submanifolds, present a number of natural generalizations of the Darboux Theorem and give an introduction to general symplectic reduction. We introduce the concept of generating function and make some elementary remarks on the group of symplectomorphisms. The last section is devoted to an introduction to Morse theory, which can be naturally formulated in the language of symplectic geometry. Methods of Morse theory are of special importance in the study of Hamiltonian systems, in particular, for the discussion of qualitative dynamics.
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Notes
- 1.
There is a huge field of research called symplectic topology, which deals with the study of global invariants of symplectic manifolds. For a nice intuitive introduction to this field we refer the reader to an article of Arnold [22]. There is a number of detailed expositions of this subject, see e.g. [206].
- 2.
A symplectic vector bundle is a real vector bundle E endowed with a section ω in ⋀2 E ∗ which is fibrewise symplectic.
- 3.
See Sect. 3.4 for the notation.
- 4.
A vertical vector bundle morphism of TM with this property is called an almost complex structure on M.
- 5.
With the index LH standing for locally Hamiltonian.
- 6.
The notion of Φ-relation extends in an obvious way from vector fields to multivector fields, cf. Definition 2.3.6/1.
- 7.
Note that the statement of the proposition is not about the maximal integral manifolds of \(D^{\omega_{N}}\).
- 8.
That is, an embedded submanifold of codimension 1.
- 9.
A hyperplane distribution on P is a regular distribution E⊂TP of codimension one.
- 10.
We omit the natural projections to the factors of the direct product.
- 11.
U 0 is the union of the subsets of V obtained by applying the Tube Lemma to every point of N.
- 12.
This notion has been already announced in Remark 6.1.3.
- 13.
A sequence of mappings φ i :N→P converges to a mapping φ:N→P iff for every compact K⊂N and every ε>0 there exists n 0 such that sup m∈K d(φ n (m),φ(m))<ε for all n>n 0; here d is some metric on P, compatible with the topology.
- 14.
Equivalently, a sequence {φ i } converges to φ in the C 1-topology iff \(\varphi_{i}'\) converges to φ′ in the C 0-topology on C ∞(TN,TP).
- 15.
- 16.
A complete metrizable locally convex vector space.
- 17.
A Lie group modelled on a Fréchet space.
- 18.
With Φ t playing the role of φ in the proof of Proposition 8.8.4.
- 19.
- 20.
That is, ∫ M fΩ=0.
- 21.
A group G which does not contain normal subgroups besides
and G. - 22.
That is, symplectomorphisms which outside a compact set coincide with the identical mapping.
- 23.
That is, f is a Morse function and
is a Riemannian metric on M such that for every pair of critical points m
1, m
2 the stable manifold of m
1 with respect to the gradient vector field ∇f is transversal to the unstable manifold of m
2. - 24.
See e.g. [55] for this notion.
- 25.
This is equivalent to the condition \(\ker (\operatorname{Hess}_{m}(f) ) = \mathrm {T}_{m} N\).
and G.
is a Riemannian metric on M such that for every pair of critical points m
1, m
2 the stable manifold of m
1 with respect to the gradient vector field ∇f is transversal to the unstable manifold of m
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Rudolph, G., Schmidt, M. (2013). Symplectic Geometry. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_8
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