|XXXIX Workshop on Geometric Methods in Physics||27.06-3.07.2021|
|IX School on Geometry and Physics||21-25.06.2021|
Generalized Relativistic Helicity, A new Topological Invariant.
In the isentropic and compressible relativistic and non-relativistic flows, fluid trajectories are considered as orbits of a family of one parameter, smooth, orientation preserving, and nonsingular diffeomorphisms on a compact and smooth-boundary domain in the Euclidian 3-space which necessarily preserve a finite measure, later interpreted as the fluid mass in non-relativistic flows and as the particles number in relativistic flows. Under such diffeomorphisms the Biot-Savart helicity and the well-known Physical helicity are conserved. The difference between these two helicities reflects some topological features of the domain and is shown for simply connected domains the two helicities coincide. For fluid domains consisting of several disjoint solid tori, at each time, the harmonic knot subspace of smooth vector fields on the fluid domain is found to have two independent base sets with a special type of orthogonality between these two bases by which a topological description of the generalized momentum and its curl depending on the helicity difference is achieved since this difference is shown to depend only on the harmonic knot parts of velocity, vorticity, and its Biot-Savart vector field.
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