scholarly journals An Analysis of Dynamic Vibrations: The Invariant Normal Curvature Wave Equation

2017 ◽  
Author(s):  
Keith C. Afas ◽  
Terry Moschandreou
Keyword(s):  
Wave Equation ◽  
Mean Curvature ◽  
Beltrami Operator ◽  
Actin Dynamics ◽  
Coordinate Space ◽  
Dynamic Surface ◽  
Metric Tensor ◽  
Normal Vibrations ◽  
Biological Entities ◽  
Calculus Of Moving Surfaces

This paper attempts to address the phenomenon of normal vibrations, (referred to as dynamic vibrations) occurring on a surface which is already in vibrational motion due to other kinematic phenomena. Such a surface will have a metric tensor, normal, and ambient velocity which diverges from the surface’s original various dynamic tensorial descriptors. This paper formulates the wave equation defined in a coordinate space, and extends the equation to observe vibrations on a surface with the use of the Laplace-Beltrami operator in a tensorial fashion drawing on conventions from the newly established Calculus of Moving Surfaces (CMS). The Paper then identifies the way which these normal vibrations will manifest within ambient space. Finally, a counter-intuitive relation between the magnitude of such dynamic vibrations and dynamic surface’s time-dependent mean curvature presents itself, for which dynamic vibrations superimposed on original dynamic motion will eliminate the other, and the surface remains static under an arbitrary initial motion. From this condition, resubstitution within the wave equation yields a novel coupled PDE system which within contains the time-evolution of the surface’s mean curvature and its dynamic vibrations. This analysis can have application for algorithms designed for mechanical stabilization during seismic activity, as well as analyzing stabilization algorithms for other various applications and can also have application with respect to biotechnological innovations for analyzing properties defined on the surface of cell-Like biological entities such as metabolism, lipid content, and actin dynamics.

SOME COMMENTS ON THE DYNAMICS OF EXTENDED OBJECTS

1998 ◽  
Vol 13 (17) ◽  
pp. 2979-2990 ◽  
Author(s):  
U. KHANAL
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Pure Shear ◽  
Equation Of Motion ◽  
Conducting Medium ◽  
Constant Density ◽  
Metric Tensor ◽  
World Volume ◽  
The World ◽  
Hilbert Action

A variational method is used to investigate the dynamics of extended objects. The stationary world volume requires the internal coordinates to propagate as free waves. Stationarity of the action which is the integral of a variable energy density over the world volume leads to the wave equation in a medium, with conductivity given by the gradient of the logarithm of reciprocal energy density, constant density corresponding to free space. The Einstein–Hilbert action for the world curvature gives an equation of motion which, in world space with the Einstein tensor proportional to the metric tensor, reduces to the free wave equation. A similar method applied to the action consisting of the surface area enclosing an incompressible world volume undergoing pure shear again yields the wave equation in a conducting medium. Simultaneous stationarity of the volume can be imposed with a stationary area only in the case of pure shear; stationary Einstein–Hilbert action can also be included and lead to an equation of motion which has a similar interpretation of the wave in the conducting medium. Some Green functions applicable to the medium with constant conductivity are also presented.

scholarly journals On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

2014 ◽  
Author(s):  
Christopher D. Sogge ◽  
Steve Zelditch
Keyword(s):  
Wave Equation ◽  
Riemannian Manifold ◽  
Wave Equations ◽  
Nonpositive Curvature ◽  
Universal Cover ◽  
Beltrami Operator ◽  
Main Step ◽  
Two Dimensional ◽  
Restriction Theorem ◽  
Wavefront Analysis

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a wave equation associated with an assigned Laplace–Beltrami operator and a bit of microlocal (wavefront) analysis remains valid as well if a certain variable is part of a periodic orbit under a set of curvature assumptions. This can be done by lifting the wave equation for a compact two-dimensional Riemannian manifold without boundary up to the corresponding one for its universal cover. By identifying solutions of wave equations for this Riemannian manifold with “periodic” ones, this chapter is able to obtain the necessary bounds using a bit of wavefront analysis and the Hadamard parametrix for the universal cover.

scholarly journals Propagating Degrees of Freedom in f(R) Gravity

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Yun Soo Myung
Keyword(s):  
Wave Equation ◽  
Gravitational Waves ◽  
Degrees Of Freedom ◽  
Ricci Scalar ◽  
Metric Tensor ◽  
Breathing Mode

We have computed the number of polarization modes of gravitational waves propagating in the Minkowski background in f(R) gravity. These are three of two from transverse-traceless tensor modes and one from a massive trace mode, which confirms the results found in the literature. There is no massless breathing mode and the massive trace mode corresponds to the Ricci scalar. A newly defined metric tensor in f(R) gravity satisfies the transverse-traceless (TT) condition as well as the TT wave equation.

First-order covariant treatment of the influence of space charge and external fields on finite-emittance relativistic beams of charged particles in generalized curvilinear coordinates

1989 ◽  
Vol 67 (7) ◽  
pp. 678-685
Author(s):  
Giulio Bosi
Keyword(s):  
Distribution Function ◽  
Space Charge ◽  
Charged Particles ◽  
Curvilinear Coordinates ◽  
Coordinate Space ◽  
Transverse Motion ◽  
Metric Tensor ◽  
First Order ◽  
Dimensional Phase Space ◽  
Principal Axes

A first-order covariant treatment of space-charge effects on relativistic beams of charged particles under the influence of applied fields of arbitrary forms is presented. The beams and fields are assumed to fulfill no particular symmetry conditions, and curvilinear orthogonal coordinates are used. Relative to a given distribution function, a suitable choice of both metric tensor and optical axis allows for the selection of appropriate coordinates, reducing the overall transverse motion of a charged particle to a pair of independent motions along the principal axes of a tensor that is related to both geometry and field. Detailed formulas applying to a microcanonical distribution in an eight-dimensional phase space are worked out, and the envelope equations for a monokinetic beam are carried out explicitly. The distribution function is shown to separate into the product of a stationary distribution in transverse-coordinate space and momentum space by using factors carrying the relativistic corrections. As a result, the envelope equations are formally identical to the customary equations describing the envelope of a nonrelativistic beam under the influence of merely transverse forces. A few numerical applications are finally presented; the behavior of beams moving through a magnetic prism and through a cycloidal analyzer is displayed graphically.

scholarly journals A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces

2019 ◽  
Vol 12 (1) ◽  
pp. 70-78
Author(s):  
Sudip Kumar Das ◽  
Mirza Cenanovic ◽  
Junfeng Zhang
Keyword(s):  
Mean Curvature ◽  
Beltrami Operator ◽  
Force Balance ◽  
Normal Vector ◽  
Triangulated Surface ◽  
Balance Principle ◽  
Alternative Expression ◽  
Triangulated Surfaces ◽  
The Mean ◽  
Mesh Structures

In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.

CONVERGENCE OF A SEMI-DISCRETE SCHEME FOR THE CURVE SHORTENING FLOW

1994 ◽  
Vol 04 (04) ◽  
pp. 589-606 ◽  
Author(s):  
GERHARD DZIUK
Keyword(s):  
Linear Systems ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Time Derivative ◽  
Curvature Flow ◽  
Beltrami Operator ◽  
Position Vector ◽  
Asymptotic Convergence ◽  
Curve Shortening Flow ◽  
Curve Shortening

Convergence for a spatial discretization of the curvature flow for curves in possibly higher codimension is proved in L∞((0, T), L2(ℝ/2π)) ∩ L2((0, T) H1(ℝ/2π)). Asymptotic convergence in these norms is achieved for the position vector and its time derivative which is proportional to curvature. The underlying algorithm rests on a formulation of mean curvature flow which uses the Laplace-Beltrami operator and leads to tridiagonal linear systems which can be easily solved.

Local Fourier transform of the Helmholtz wave equation

Geophysics ◽  
1987 ◽  
Vol 52 (9) ◽  
pp. 1303-1305 ◽  
Author(s):  
Steve Hildebrand
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Phase Space ◽  
Wave Field ◽  
Fourier Method ◽  
Coordinate Space ◽  
Spatial Coordinate ◽  
Time Space ◽  
Helmholtz Wave Equation ◽  
Ray Parameter

A local Fourier transform of a wave field is an integral transformation from space‐time coordinate space to phase space, i.e., ray parameter, spatial coordinate, and intercept‐ time space. McMechan (1983) introduced the concept of a local slant‐stack/Fourier transform. Using this integral transform, he was able to define a wave field in phase space. By projecting the resulting wave field to the spatial coordinate space, he obtained a representation of the wave field in the ray‐parameter and the spatial‐coordinate plane. In this paper, the local Fourier method is used to transform the Helmholtz wave equation into a phase‐space coordinate system. The resulting wave equation is then written in a state‐variable form of coupled first‐order differential equations. A propagator solution is then shown.

High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation

2004 ◽  
Vol 47 (4) ◽  
pp. 504-514 ◽  
Author(s):  
Fernando Cardoso ◽  
Georgi Vodev
Keyword(s):  
Wave Equation ◽  
Real Axis ◽  
Riemannian Manifolds ◽  
High Frequency ◽  
Beltrami Operator ◽  
Energy Decay ◽  
Local Energy ◽  
Resolvent Estimates ◽  
The Real ◽  
Decay Of Solutions

AbstractWe prove an uniform Hölder continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation.

scholarly journals The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 398 ◽  
Author(s):  
Erhan Güler ◽  
Hasan Hacısalihoğlu ◽  
Young Kim
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Gauss Map ◽  
Beltrami Operator ◽  
The Third ◽  
The Mean

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

The wave equation, O(2, 2), and separation of variables on hyperboloids

1978 ◽  
Vol 79 (3-4) ◽  
pp. 227-256 ◽  
Author(s):  
E. G. Kalnins ◽  
W. Miller
Keyword(s):  
Wave Equation ◽  
Harmonic Analysis ◽  
Separation Of Variables ◽  
Beltrami Operator ◽  
Eigenvalue Equation ◽  
Coordinate Systems ◽  
Group Manifold ◽  
Orthogonal Systems

SynopsisWe classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

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