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We discuss the exact nonlinear equations for the dynamics of fluid films, modeled as a two dimensional manifold. Our main goal is to illustrate the differences and similarities between the fluid film equations and Euler's equations, their classical three dimensional counterpart. Since the geometry of fluid films is fundamentally different -- three dimensional velocity field on a two dimensional support with a time varying Riemannian metric -- all classical theorems must be properly modified. We offer adaptations of the following theorems: conservation of mass and energy, pointwise conservation of vorticity and Kelvin's circulation theorem. We present proofs of these theorems by employing the calculus of moving surfaces. It is of great interest to develop a simplified model that captures normal deformations of fluid films by assuming that tangential velocities vanish while preserving the exact nonlinear nature of the full system. This cannot be accomplished simply by neglecting the tangential components, for such an attempt leads to internal contradictions. Instead, we modify the initial formulation and present a modified variational approach that leads to a simplified system of equations capable of capturing a broad range of deeply nonlinear effects.
The aim of the present study is to characterize and compute closed geodesics on toroïdal surfaces. We show that a closed geodesic must make a number of rotations about the equatorial part ($k$ rotations) and the axis of revolution ($k'$ rotations) of the surface. We give the relation that exists between the numbers $k$ and $k'$, and the Clairaut's constant $C$ corresponding to the geodesic. Moreover, we prove that the numbers $k$ and $k'$ are relatively prime. We validate our findings by constructing closed geodesics on some examples of toroïdal surfaces using MAPLE. Finally, using experimental data on cardiac fiber direction, we show that fibers run as geodesics in the left ventricle whose geometrical shape looks like a toroïdal surface.
The presence of defects in material continua is known to produce internal permanent strained states. Extending the theory of defects to four dimensions and allowing for the appropriate signature, it is possible to apply these concepts to space-time. In this case a defect would induce a non-trivial metric tensor, which can be interpreted as a gravitational field. The image of a defect in space-time can be applied to the description of the Big Bang. A review of the four-dimensional generalisation of defects and an application to the expansion of the universe will be presented.
It is shown that the Bohm equations for the phase $S$ and squared modulus $\rho$ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum $p_s$ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat grad$S$ and $p_s$ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schrödinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.