arXiv subjects

H. Gumral

Publications and source records attributed to H. Gumral.

4 recordsLinked to original sources

arXiv2011

On geometry of the Lagrangian description of ideal fluids

The Euler equation for an inviscid, incompressible fluid in a three-dimensional domain M implies that the vorticity is a frozen-in field. This can be used to construct a symplectic structure on RxM. The normalized vorticity and the suspended velocity fields are Hamiltonian with the function t and the Bernoulli function, respectively. The symplectic structure incorporates the helicity conservation law as an identity. The infinitesimal dilation for the symplectic two-form can be interpreted as a current vector for the helicity. The symplectic dilation implies the existence of contact hypersurfaces. In particular, these include contact structures on the space of streamlines and the Bernoulli surfaces. The symplectic structure on RxM can be realized as symplectisations of these through the Euler equation.

math-ph↗

arXiv2010

Existence of Hamiltonian Structure in 3D

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. Explicit integration of conserved quantities are connected with the coefficients of Riccati equation which are elements of the third cohomology class. All explicitly constructed examples of bi-Hamiltonian systems are exhausted when this class along with the first one vanishes. The latter condition provides integrating factor for explicit integration of Hamiltonian functions. For the Darboux-Halphen system, the Godbillon-Vey invariant is shown to arise as obstruction to integrability of integrating factor.

math.DS↗

arXiv2007

Bi-Hamiltonian Structure in Serret-Frenet Frame

We reduced the problem of constructing bi-Hamiltonian structure in three dimensions to the solution of a Riccati equation in moving coordinates of Serret-Frenet frame. We then show that either the linearly independent solutions of the corresponding second order equation or the normal vectors of the moving frame imply two compatible Poisson structures.

math-ph↗

arXiv1997

Lagrangian Description, Symplectic Structure, and Invariants of 3D Fluid Flow

Three dimensional unsteady flow of fluids in the Lagrangian description is considered as an autonomous dynamical system in four dimensions. The condition for the existence of a symplectic structure on the extended space is the frozen field equations of the Eulerian description of motion. Integral invariants of symplectic flow are related to conservation laws of the dynamical equation. A scheme generating infinite families of symmetries and invariants is presented. For the Euler equations these invariants are shown to have a geometric origin in the description of flow as geodesic motion; they are also interpreted in connection with the particle relabelling symmetry.

solv-int↗