Prof. Dr. Hasan Gümral

 

Research

General interests

Research Summary

References to Mathematical Reviews

Publication List

Works in Progress

General interests:

Geometric mechanics: symplectic, multisymplectic, Poisson, contact, Jacobi
and multi-Hamiltonian structures with applications to non-linear di erential
equations of physical importance, in particular, to fluids and plasma. Higher
order and degenerate Lagrangian theories and their (covariant) Hamiltonian formalisms.


Research Summary:

The published works may be grouped under four headings: 1) geometry of
plasma dynamics and kinetic theories, 2) generalized Hamiltonian formulation
for non-autonomous dynamical systemswith applications to (magneto-
)hydrodynamic motions in Lagrangian description, 3) local and global aspects
of Poisson structures in three dimensions and, 4) multi-Hamiltonian
structure of equations of hydrodynamic type in one space dimension.

  • In J. Math. Phys. 51 (2010) 083501 (23pp), The dynamics of collisionless
    plasma described by the Poisson{Vlasov equations is connected
    to the Hamiltonian motions of particles and their symmetries.
    The Poisson equation is obtained as a constraint arising from the
    gauge symmetries of particle dynamics. Lie{Poisson reduction for the
    group of canonical di eomorphisms gives the momentum-Vlasov equations.
    Plasma density is given a momentum map description associated
    with the action of additive group of functions of particle phase
    space. Equivalence of Hamiltonian functionals in momentum and density
    formulations is shown. An alternative formulation in momentum
    variables is described. A comparison of one-dimensional plasma and
    two-dimensional incompressible fluid is presented. In the second paper
    J. Geom. Mech. (2012) the underlying geometry of the momentum-
    Vlasov equations is elaborated. These equations are obtained as vertical
    equivalence of complete cotangent lift of Hamiltonian vector eld
    generating the particle motion. This technique is also applied, in Int.
    J. of Geom. Meth. in Mod. Phys. 8 (2011) 331-344, to kinetic theories
    of particles whose congiguration space is the group of contact
    diffeomorphisms.

  • A Hamiltonian formulation for non-autonomous dynamical systems in
    the framework of contravariant geometry is given in Phys. Lett. A
    218 (1996) 235-239 (MR 97e: 58080). This is further elaborated to
    show that any Poisson bi-vector in time-extended space possesses two
    in nitesimal automorphism and applied to the investigations of the
    phase space geometry and of the invariants of hydrodynamical motions
    in Phys. Lett. A 257 (1999) 43-52 (MR 00e: 37078). The timeextended
    Hamiltonian formalism is used to investigate the invariants
    of hydrodynamical equations in three dimensions. In a series of articles
    (Phys. Lett. A 232 (1997) 417-424 (MR 98i: 76003), Physica D 135
    (2000) 117-136, Physica D 139 (2000) 335-359 (MR 01i: 76027)) the
    symplectic geometry of Lagrangian motions are given. The implication
    of this structure to kinematical symmetries and invariants of motion are
    discussed. The equivalence of Lagrangian and Eulerian types of helicity
    conservation laws are shown and, this equivalence is characterized in the
    framework of symplectic and conformally symplectic, or more generally,
    in the language of Jacobi structures. Time-extended formalism is also
    applied for Aristotelian model of three-body motion in J. Phys. A:
    Math. Theor. 44 (2011) 325204 (15pp) where it is also shown to be an
    autonomous bi-Hamiltonian system.

  • A survey of Poisson structures in three dimensions and a discussion of
    integrability in this framework is given in J. Math. Phys. 34 (1993)
    5691-5723 (MR 94i: 58066). The problem in Lichnerowicz complex is
    converted into the one on de Rham complex. Frobenius theorem is
    used to characterize integrable bi-Hamiltonian systems. Godbillon-Vey
    invariants are obtained as obstruction to global integrability in three
    dimensions. Applications to dynamical systems from general relativity,
    biology, epidemiology, atomic physics, etc. are given. In Physica D 238 6
    (2009) 526-530, it is shown that in three dimensions, the construction
    of bi-Hamiltonian structure can be reduced to the solutions of a Riccati
    equation with the arclength coordinate of a Frenet-Serret frame
    being the independent variable. All explicitly constructed examples in
    the literature are exhausted by constant solutions. It is proved that
    vector elds which are not eigenvectors of the curl operator are locally
    bi-Hamiltonian. Based on this, the work Adv. in Dyn. Sys. and Appl.,
    5 (2010) 159-171 shows that explicit integration of conserved quantities
    are connected with the coecients of Riccati equation which are
    elements of the third cohomology class. The Darboux-Halphen system,
    as the only non-trivial example of locally bi-Hamiltonian system in the
    literature, is revisited and it is concluded that the Godbillon-Vey invariant
    arises as obstruction to integrability of integrating factor for
    Hamiltoian functions. In J. Phys. A: Math. Theor. 44 (2011) 325204
    (15pp) Aristotelian model of three-body motion on the line is shown
    to be bi-Hamiltonian for all physical parameters involved.

  • Integrability of equations of hydrodynamic type in one space dimension
    is studied in the framework of multi-Hamiltonian structure. Gas
    dynamics hierarchy is completed with the inclusion of new conserved
    quantities and a continuum limit of Toda lattice J. Math. Phys. 31(11)
    1990, 2606-2611 (MR 91j: 35229). In J. Phys. A: Math. Gen. 25
    (1992) 5141-5149 (MR 93h: 35164) a combinatorial method is developed
    for the construction of bi-Hamiltonian structures and is applied to
    N-component Kodama equations. Some transformations are used in J.
    Phys. A: Math. Gen. 27 (1994) 193-200 (MR 95e: 58085) for the same
    purpose and the integrability of dispersionless-Boussinesq, Benney-Lax
    equations are proved.

References to Mathematical Reviews:

1. 91j: 35229, 35Q35, 58F05, 76B15, 76N15.
2. 93h: 35164, 35Q35, 58F07.
3. 94i: 58066, 58F05, 34A26, 58F07, 70H05.

4. 95e: 58085, 58F07, 35Q58.
5. 97e: 58080, 58F05, 70H05, by Charles-Michel Marle.
6. 98i: 76003, 76A02, 58D05, 58F05, by Yuri E. Gliklikh.
7. 00e: 37078, 37J05, 37C60, 53D17, 70H05, by David Martin de Diego.
8. 01(kin sym) 37K65, 37N10, 76M60,
9. 01i: 76027, 76B99, 37K05, 37N10, 76A25, 76W05, by Hans-Peter Kruse.

Publication List:

(Articles in International Refereed Journals)

1. H. Gümral, Y. Nutku, Multi-Hamiltonian structure of equations of hydrodynamic
type, J. Math. Phys. 31(11) 1990, 2606-2611.

2. H. Gümral, Bi-Hamiltonian structure of N-component Kodama equations,
J. Phys. A: Math. Gen. 25 (1992) 5141-5149.

3. H.Gümral, Y. Nutku, Poisson structure of dynamical systems with
three degrees of freedom, J. Math. Phys. 34 (1993) 5691-5723.

4. H.Gümral, Y. Nutku, Bi-Hamiltonian structures of dispersionless-Boussinesq
and Benney equations, J. Phys. A: Math. Gen. 27 (1994) 193-200.

5. H.Gümral, Contravariant geometry of time-dependent dynamical systems,
Phys. Lett. A 218 (1996) 235-239.

6. H.Gümral, Lagrangian description, symplectic structure, and invariants
of 3D fluid flow, Phys. Lett. A 232 (1997) 417-424.

7. H.Gümral, A time-extended Hamiltonian formalism, Phys. Lett. A
257 (1999) 43-52.

8. H.Güumral, Kinematical symmetries of 3D incompressible
ows, Physica D 135 (2000) 117-136.

9. H.Gümral, Helicity invariants in 3D: Kinematical aspects, Physica D
139 (2000) 335-359.

10. E. Abadoğlu, H. Güumral, Bi-Hamiltonian structure in Frenet-Serret
frame, Physica D 238 (2009) 526-530.

11. H. Gümral, Existence of Hamiltonian Structure in 3D, Adv. in Dyn.
Sys. and Appl., 5 (2010) 159-171.

12. H. Gümral, Geometry of Plasma Dynamics I: Group of Canonical Diffeomorphisms,
J. Math. Phys. 51 (2010) 083501 (23pp).

13. O. Esen, H. Gümral, Lifts, Jets and Reduced Dynamics, Int. J. of
Geom. Meth. in Mod. Phys. 8 (2011) 331-344.

14. E. Abadoğlu, H. Güumral, Poisson Structures for the Aristotelian Model
of Three-Body Motion, J. Phys. A: Math. Theor. 44 (2011) 325204
(15pp).

15. O. Esen, H. Gümral, Geometry of Plasma Dynamics II: Lie Algebra of
Hamiltonian Vector Fields, to be published in J. Geom. Mech. (2012).

(Articles in Turkish)

1. Kontakt Parçacıkların Kinetik Denklemleri, XVII. Ulusal Mekanik
Kongresi-bildiriler-\ (Yayına hazırlayanlar: ) () (O. Esen ile)

2. Kanonik Dönüşümler Grubu ve Plazma Dinamiği, \XVI. Ulusal Mekanik
Kongresi-bildiriler-\ (Yayına hazırlayanlar: A.Y. Aköz, H. Engin,
Ü. Gülçat, A. Hacınlıyan) (Nisan 2010) 631-649.

3. Yavuz Nutku, \Anılarla/With Memories Yavuz Nutku", (Yayına hazırlayan:
Yılmaz Akyıldız.)

Works in Progress:

O. Esen, H. Gümral, Geometry of Plasma Dynamics III: Orbits of Canonical Diffeomorphisms,

E. Abadoğlu, Z. Demireli, H. Gümral, Gradient Systems, Poisson Structures and Surfaces in 3D,

H. Gümral, Lagrangian Description, Symplectisation and Eulerian Dynamics of Incompressible Fluids,

H. Gümral, On Poisson bi-vectors in 4D,

H. Gümral, Reductions of TMG: Degenerate Second Order Lagrangians,

F. Çağatay Uçgun, H. Gümral, Contraint Analysis of Clement Reduction of TMG,

H. Gümral, Kinetic Theories from Local Lie Algebras,

H.Gümral, Geometry of Plasma Dynamics IV: Space of Displacement Mappings,