arXiv subjects

Hasan Gumral

Publications and source records attributed to Hasan Gumral.

4 recordsLinked to original sources

arXiv1999

Helicity current as a symplectic dilation

A formal symplectic structure on RxM is constructed for the unsteady flow of an incompressible viscous fluid on a three dimensional domain M. The evolution equation for the helicity density is expressed via the divergence of the Liouville vector field that generates symplectic dilation. For an inviscid fluid this equation reduces to a conservation law. As an application the symplectic dilation is used to generate Hamiltonian automorphisms of the symplectic structure which are then related to the symmetries of the velocity field.

math-ph↗

arXiv1998

Kinematical symmetries of 3D incompressible flows

The motion of an incompressible fluid in Lagrangian coordinates involves infinitely many symmetries generated by the left Lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid. Utilizing a 1+3-dimensional Hamiltonian setting an explicit realization of this symmetry algebra is constructed recursively. A dynamical connection is used to split the symmetries into reparametrization of trajectories and one-parameter family of volume preserving diffeomorphisms of fluid domain. Algebraic structures of symmetries and Hamiltonian structures of their generators are inherited from the same construction. A comparison with the properties of 2D flows is included.

math-ph↗

arXiv1998

A time-extended Hamiltonian formalism

A Poisson structure on the time-extended space R x M is shown to be appropriate for a Hamiltonian formalism in which time is no more a privileged variable and no a priori geometry is assumed on the space M of motions. Possible geometries induced on the spatial domain M are investigated. An abstract representation space for sl(2,R) algebra with a concrete physical realization by the Darboux-Halphen system is considered for demonstration. The Poisson bi-vector on R x M is shown to possess two intrinsic infinitesimal automorphisms one of which is known as the modular or curl vector field. Anchored to these two, an infinite hierarchy of automorphisms can be generated. Implications on the symmetry structure of Hamiltonian dynamical systems are discussed. As a generalization of the isomorphism between contact flows and their symplectifications, the relation between Hamiltonian flows on R x M and infinitesimal motions on M preserving a geometric structure therein is demonstrated for volume preserving diffeomorphisms in connection with three-dimensional motion of an incompressible fluid.

math-ph↗

arXiv1998

General vorticity conservation

The motion of an incompressible fluid in Lagrangian coordinates involves infinitely many symmetries generated by the left Lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid. Utilizing a 1+3-dimensional Hamiltonian setting an explicit realization of this symmetry algebra and the related Lagrangian and Eulerian conservation laws are constructed recursively. Their Lie algebraic structures are inherited from the same construction. The laws of general vorticity and helicity conservations are formulated globally in terms of invariant differential forms of the velocity field.

solv-int↗