Geometry of
plasma dynamics II: Lie algebra of Hamiltonian vector fields

International audience It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.

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LIFTS, JETS AND REDUCED DYNAMICS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005166 ◽

2011 ◽

Vol 08 (02) ◽

pp. 331-344 ◽

Cited By ~ 3

Author(s):

OĞUL ESEN ◽

HASAN GÜMRAL

Keyword(s):

Incompressible Fluid ◽

Euler Equations ◽

Continuum Mechanics ◽

Vector Fields ◽

Kinetic Equations ◽

Ideal Incompressible Fluid ◽

Hamiltonian Vector ◽

Plasma Dynamics ◽

Hamiltonian Vector Fields ◽

Vlasov Equations

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and momentum-Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie–Poisson reductions and with the techniques of present framework.

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INFINITESIMAL TAKESAKI DUALITY OF HAMILTONIAN VECTOR FIELDS ON A SYMPLECTIC MANIFOLD

Reviews in Mathematical Physics ◽

10.1142/s0129055x00000460 ◽

2000 ◽

Vol 12 (12) ◽

pp. 1669-1688 ◽

Cited By ~ 1

Author(s):

KATSUNORI KAWAMURA

Keyword(s):

Lie Algebra ◽

Symplectic Manifold ◽

Vector Fields ◽

Type Theorem ◽

Hamiltonian Vector ◽

Crossed Product ◽

Hamiltonian Vector Fields ◽

Symplectic Action

For an infinitesimal symplectic action of a Lie algebra [Formula: see text] on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and [Formula: see text]. We obtain its second crossed product in the case where [Formula: see text] and obtain an infinitesimal version of a Takesaki duality type theorem.

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Universal Central Extension of the Lie Algebra of Hamiltonian Vector Fields: Table 1.

International Mathematics Research Notices ◽

10.1093/imrn/rnv301 ◽

2015 ◽

Vol 2016 (16) ◽

pp. 4996-5047 ◽

Cited By ~ 1

Author(s):

Bas Janssens ◽

Cornelia Vizman

Keyword(s):

Lie Algebra ◽

Central Extension ◽

Vector Fields ◽

Hamiltonian Vector ◽

Universal Central Extension ◽

Hamiltonian Vector Fields

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POISSON-NIJENHUIS STRUCTURES AND SATO HIERARCHY

Reviews in Mathematical Physics ◽

10.1142/s0129055x91000151 ◽

1991 ◽

Vol 03 (04) ◽

pp. 403-466 ◽

Cited By ~ 24

Author(s):

G. MAGNANO ◽

F. MAGRI

Keyword(s):

Finite Number ◽

Lie Algebra ◽

Vector Fields ◽

Hamiltonian Structure ◽

Differential Operators ◽

Hamiltonian Vector ◽

Poisson Brackets ◽

Moody Algebra ◽

Hamiltonian Vector Fields ◽

Direct Sum

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

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