Motion (geometry)

Abstract: A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Abstract: In this paper we are concerned with the manifold structure of certain groups of diffeomorphisms, and with the use of this structure to obtain sharp existence and uniqueness theorems for the classical equations for an incompressible fluid (both viscous and non-viscous) on a compact C^∞ riemannian, oriented n-manifold M, possibly with boundary.

Abstract: There exist intricate geometric relations between multiple views of a 3D scene. These relations are related to the camera motion and calibration as well as to the scene structure. In this chapter w ...

Abstract: The fact that our present laws of physics admit of a formal extension to spaces of an arbitrary number of dimensions suggests that there must be some principle (or principles) operative which in conjunction with these laws entails the observed specificity of spatial dimensionality,n=3. Generalizing from an approach suggested by the work ofEhrenfest (and independently byG. J. Whitrow) on the Newtonian keplerian problem inn dimensions, it is proposed that this principle may be tentatively summarized in the postulate that there shall be stable bound orbits or «states» for the equations of motion governing the interaction of bodies (considered as «material points’). This postulate is applied to the geodesic equations of motion obtained from a generalization of the Schwarzschild field to static systems with hyper-spherical symmetry, and it is shown that the bound state postulate uniquely entails the spatial dimensionality. This result is not entirely peculiar, to general relativity because it also holds for Newtonian theory (Ehrenfest-Whitrow) if one also introduces an asymptotic condition to exclude casesn 3, and in conjunction with the asymptotic conditionn<3. An attempt is made to understand the logical origin of this postulate and it is argued that if one assumes the basic representatives of a dynamics with a metric to be material points, one needs such a postulate to construct Einstein’s «practically rigid rods», since point bodies in themselves do not provide us with a measure of distance. Some brief qualitative applications of these ideas are, made to quantum electrodynamics.