Research
General interests
Research Summary
References to Mathematical Reviews
Publication List
Works in Progress
General interests:
Geometric mechanics: symplectic, multisymplectic, Poisson, contact, Jacobi
and multiHamiltonian structures with applications to nonlinear dierential
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 nonautonomous dynamical systemswith applications to (magneto
)hydrodynamic motions in Lagrangian description, 3) local and global aspects
of Poisson structures in three dimensions and, 4) multiHamiltonian
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 dieomorphisms gives the momentumVlasov 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 onedimensional plasma and
twodimensional 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) 331344, to kinetic theories
of particles whose congiguration space is the group of contact
diffeomorphisms.

A Hamiltonian formulation for nonautonomous dynamical systems in
the framework of contravariant geometry is given in Phys. Lett. A
218 (1996) 235239 (MR 97e: 58080). This is further elaborated to
show that any Poisson bivector in timeextended space possesses two
innitesimal automorphism and applied to the investigations of the
phase space geometry and of the invariants of hydrodynamical motions
in Phys. Lett. A 257 (1999) 4352 (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) 417424 (MR 98i: 76003), Physica D 135
(2000) 117136, Physica D 139 (2000) 335359 (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. Timeextended formalism is also
applied for Aristotelian model of threebody motion in J. Phys. A:
Math. Theor. 44 (2011) 325204 (15pp) where it is also shown to be an
autonomous biHamiltonian 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)
56915723 (MR 94i: 58066). The problem in Lichnerowicz complex is
converted into the one on de Rham complex. Frobenius theorem is
used to characterize integrable biHamiltonian systems. GodbillonVey
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) 526530, it is shown that in three dimensions, the construction
of biHamiltonian structure can be reduced to the solutions of a Riccati
equation with the arclength coordinate of a FrenetSerret 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
biHamiltonian. Based on this, the work Adv. in Dyn. Sys. and Appl.,
5 (2010) 159171 shows that explicit integration of conserved quantities
are connected with the coecients of Riccati equation which are
elements of the third cohomology class. The DarbouxHalphen system,
as the only nontrivial example of locally biHamiltonian system in the
literature, is revisited and it is concluded that the GodbillonVey 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 threebody motion on the line is shown
to be biHamiltonian for all physical parameters involved.
 Integrability of equations of hydrodynamic type in one space dimension
is studied in the framework of multiHamiltonian 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, 26062611 (MR 91j: 35229). In J. Phys. A: Math. Gen. 25
(1992) 51415149 (MR 93h: 35164) a combinatorial method is developed
for the construction of biHamiltonian structures and is applied to
Ncomponent Kodama equations. Some transformations are used in J.
Phys. A: Math. Gen. 27 (1994) 193200 (MR 95e: 58085) for the same
purpose and the integrability of dispersionlessBoussinesq, BenneyLax
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 CharlesMichel 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 HansPeter Kruse.
Publication List:
(Articles in International Refereed Journals)
1. H. Gümral, Y. Nutku, MultiHamiltonian structure of equations of hydrodynamic
type, J. Math. Phys. 31(11) 1990, 26062611.
2. H. Gümral, BiHamiltonian structure of Ncomponent Kodama equations,
J. Phys. A: Math. Gen. 25 (1992) 51415149.
3. H.Gümral, Y. Nutku, Poisson structure of dynamical systems with
three degrees of freedom, J. Math. Phys. 34 (1993) 56915723.
4. H.Gümral, Y. Nutku, BiHamiltonian structures of dispersionlessBoussinesq
and Benney equations, J. Phys. A: Math. Gen. 27 (1994) 193200.
5. H.Gümral, Contravariant geometry of timedependent dynamical systems,
Phys. Lett. A 218 (1996) 235239.
6. H.Gümral, Lagrangian description, symplectic structure, and invariants
of 3D fluid flow, Phys. Lett. A 232 (1997) 417424.
7. H.Gümral, A timeextended Hamiltonian formalism, Phys. Lett. A
257 (1999) 4352.
8. H.Güumral, Kinematical symmetries of 3D incompressible
ows, Physica D 135 (2000) 117136.
9. H.Gümral, Helicity invariants in 3D: Kinematical aspects, Physica D
139 (2000) 335359.
10. E. Abadoğlu, H. Güumral, BiHamiltonian structure in FrenetSerret
frame, Physica D 238 (2009) 526530.
11. H. Gümral, Existence of Hamiltonian Structure in 3D, Adv. in Dyn.
Sys. and Appl., 5 (2010) 159171.
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) 331344.
14. E. Abadoğlu, H. Güumral, Poisson Structures for the Aristotelian Model
of ThreeBody 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
Kongresibildiriler\ (Yayına hazırlayanlar: ) () (O. Esen ile)
2. Kanonik Dönüşümler Grubu ve Plazma Dinamiği, \XVI. Ulusal Mekanik
Kongresibildiriler\ (Yayına hazırlayanlar: A.Y. Aköz, H. Engin,
Ü. Gülçat, A. Hacınlıyan) (Nisan 2010) 631649.
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 bivectors 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,
