Symplectic Geometry in Hamiltonian Dynamics, Hamiltonian Flows, and Poincaré-Cartan Integral Invariants

In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

Download Full-text

Geometry of Thermodynamic Processes

Entropy ◽

10.3390/e20120925 ◽

2018 ◽

Vol 20 (12) ◽

pp. 925 ◽

Cited By ~ 15

Author(s):

Arjan van der Schaft ◽

Bernhard Maschke

Keyword(s):

Symplectic Geometry ◽

Hamiltonian Dynamics ◽

Contact Geometry ◽

Thermodynamic System ◽

Lagrangian Submanifold ◽

Geometric Framework ◽

Definition Of ◽

And Control ◽

Geometric Formulation ◽

Thermodynamic Processes

Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.

Download Full-text

Canonical multi-symplectic structure on the total exterior algebra bundle

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1629 ◽

2006 ◽

Vol 462 (2069) ◽

pp. 1531-1551 ◽

Cited By ~ 11

Author(s):

Thomas J Bridges

Keyword(s):

Symplectic Geometry ◽

Symplectic Structure ◽

Hamiltonian Dynamics ◽

Partial Differential Operator ◽

Exterior Algebra ◽

Legendre Transform ◽

Natural Setting ◽

Elliptic Pde ◽

Symplectic Structures ◽

Partial Differential

The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

Download Full-text

The Symplectic Geometry of Polygons in the 3-Sphere

Canadian Journal of Mathematics ◽

10.4153/cjm-2002-002-1 ◽

2002 ◽

Vol 54 (1) ◽

pp. 30-54 ◽

Cited By ~ 5

Author(s):

Thomas Treloar

Keyword(s):

Hamiltonian System ◽

Moduli Space ◽

Moduli Spaces ◽

Symplectic Geometry ◽

Symplectic Structure ◽

Integrable Hamiltonian System ◽

Conjugacy Classes ◽

Fusion Product ◽

Hamiltonian Flows ◽

Theoretic Description

AbstractWe study the symplectic geometry of the moduli spaces Mr = Mr() of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2) by the diagonal conjugation action of SU(2). Here the fusion product of n conjugacy classes is a Hamiltonian quasi-Poisson SU(2)-manifold in the sense of [AKSM]. An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space [KMT], and for Mr(), the moduli space of n-gons with fixed side-lengths in [KM1].

Download Full-text

Constrained Hamiltonian Dynamics

10.1093/oso/9780198822370.003.0021 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Lie Algebra ◽

Symplectic Geometry ◽

Quantum Physics ◽

Vector Fields ◽

Hamiltonian Dynamics ◽

Poisson Algebra ◽

Geodesic Equation ◽

Poisson Manifolds ◽

C Algebras ◽

Poisson Reduction

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.

Download Full-text