Journal ArticleDOI
Lifts, jets and reduced dynamics
Oğul Esen,Hasan Gümral +1 more
TLDR
In this paper, complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics.Abstract:
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.read more
Citations
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Journal ArticleDOI
A hierarchy of Poisson brackets in non-equilibrium thermodynamics
TL;DR: In this article, a new infinite grand-canonical hierarchy of Poisson brackets is proposed, which leads to Poisson bracket expressing non-local phenomena such as turbulent motion or evolution of polymeric fluids.
Journal ArticleDOI
Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy
TL;DR: A purely geometric pathway is proposed, which establishes a link from particle motion to evolution of the field variables, and is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets.
Posted Content
Tulczyjew's Triplet for Lie Groups II: Dynamics
Oğul Esen,Hasan Gümral +1 more
TL;DR: In this article, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialised Tulczyjew's symplectic space.
Journal ArticleDOI
Geometry ofplasma dynamics II: Lie algebra of Hamiltonian vector fields
Oǧul Esen,Hasan Gümral +1 more
Abstract: We introduce natural differential geometric structures underlying the
Poisson-Vlasov equations in momentum variables. First, we decompose the
space of all vector fields over particle phase space into a semi-direct
product algebra of Hamiltonian vector fields and its complement. The latter
is related to dual space of the Lie algebra. We identify generators of
homotheties as dynamically irrelevant vector fields in the complement. Lie
algebra of Hamiltonian vector fields is isomorphic to the space of all
Lagrangian submanifolds with respect to Tulczyjew symplectic structure. This
is obtained as tangent space at the identity of the group of canonical
diffeomorphisms represented as space of sections of a trivial bundle. We
obtain the momentum-Vlasov equations as vertical equivalence, or
representative, of complete cotangent lift of Hamiltonian vector field
generating particle motion. Vertical representatives can be described by
holonomic lift from a Whitney product to a Tulczyjew symplectic space. We
show that vertical representatives of complete cotangent lifts form an
integrable subbundle of this Tulczyjew space. We exhibit dynamical relations
between Lie algebras of Hamiltonian vector fields and of contact vector
fields, in particular; infinitesimal quantomorphisms on quantization bundle.
Gauge symmetries of particle motion are extended to tensorial objects
including complete lift of particle motion. Poisson equation is then
obtained as zero value of momentum map for the Hamiltonian action of gauge
symmetries for kinematical description.
Journal ArticleDOI
Lagrangian dynamics on matched pairs
Oğul Esen,Serkan Sütlü +1 more
TL;DR: In this paper, the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as Euler and Poincare equations on matched pairs of Lie algebras were obtained.
References
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Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
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Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
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Introduction to mechanics and symmetry
TL;DR: A basic exposition of classical mechanical systems; 2nd edition Reference CAG-BOOK-2008-008 Record created on 2008-11-21, modified on 2017-09-27 as mentioned in this paper.
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Introduction to symplectic topology
Dusa McDuff,Dietmar Salamon +1 more
TL;DR: In this article, the authors present a survey of the history of classical and modern manifold geometry, from classical to modern, including linear and almost complex structures, and the Arnold conjecture of the group of symplectomorphisms.
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Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.