scholarly journals A tale of two actions: A variational principle for two-dimensional causal sets

2021 ◽  
Author(s):  
Luca Bombelli ◽  
Benjamin Pilgrim
Keyword(s):  
Variational Principle ◽  
Two Dimensional ◽  
Causal Sets

scholarly journals A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

2019 ◽  
Vol 475 (2224) ◽  
pp. 20180642 ◽  
Author(s):  
H. Alemi Ardakani ◽  
T. J. Bridges ◽  
F. Gay-Balmaz ◽  
Y. H. Huang ◽  
C. Tronci
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Variational Principle ◽  
Stream Function ◽  
Fluid Motion ◽  
Two Dimensional ◽  
Euclidean Group ◽  
Fluid Sloshing ◽  
Pressure Boundary Condition ◽  
Vessel Motion

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler–Poincaré variations, the derivation of free surface variations and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.

On the existence and ‘blow-up’ of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling

2003 ◽  
Vol 133 (1) ◽  
pp. 119-149 ◽  
Author(s):  
Otared Kavian ◽  
Michael Vogelius
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Finite Number ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Boundary Value ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Limiting Behaviour ◽  
Corrosion Modelling

Let Ω be a bounded C2,α domain in R2. We prove that the boundary-value problem Δυ = 0 in Ω, ∂υ/∂n = λsinh(υ) on ∂Ω, has infinitely many (classical) solutions for any given λ > 0. These solutions are constructed by means of a variational principle. We also investigate the limiting behaviour as λ → 0+; indeed, we prove that each of our solutions, as λ → 0+, after passing to a subsequence, develops a finite number of singularities located on ∂Ω.

The unbounded two-dimensional guiding-centre plasma

1995 ◽  
Vol 54 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Michael K. -H. Kiessling
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Mean Field ◽  
Analytical Techniques ◽  
Larmor Frequency ◽  
Two Dimensional ◽  
Algebraic Relation ◽  
Density Profiles ◽  
Translation Invariant ◽  
Computer Based

The thermal mean-field equilibrium of a translation-invariant, unbounded one- component guiding-centre plasma is studied by analytical techniques. A variational principle is constructed. It is shown that only radial symmetric, decreasing density profiles occur. Prescribing the total number of gyro centres N ∈ (0, ∞), the energy E ∈ (E0, ∞) and the canonical angular momentum M ∈ (0, ∞]) uniquely determines a profile. Metastable or unstable profiles do not exist. A simple algebraic relation between N, M, the guiding-centre temperature β−1 and the characteristic Larmor frequency ω is derived. This explains Williamson's computer-based observations.

scholarly journals DETERMINATION OF JOIN REGIONS BETWEEN CARBON NANOSTRUCTURES USING VARIATIONAL CALCULUS

2013 ◽  
Vol 54 (4) ◽  
pp. 221-247 ◽  
Author(s):  
D. BAOWAN ◽  
B. J. COX ◽  
J. M. HILL
Keyword(s):  
Carbon Nanotubes ◽  
Variational Principle ◽  
Carbon Nanostructures ◽  
Lagrange Equation ◽  
Variational Calculus ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Dimensional Plane ◽  
Continuous Models

AbstractWe review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two-dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other.

scholarly journals Geometric electrostatic particle-in-cell algorithm on unstructured meshes

2021 ◽  
Vol 87 (4) ◽  
Author(s):  
Zhenyu Wang ◽  
Hong Qin ◽  
Benjamin Sturdevant ◽  
C.S. Chang
Keyword(s):  
Variational Principle ◽  
Electric Field ◽  
Energy Conservation ◽  
Unstructured Meshes ◽  
Magnetized Plasmas ◽  
Two Dimensional ◽  
Particle In Cell ◽  
Conservation Property ◽  
Electron Gyrofrequency ◽  
Pic Method

We present a geometric particle-in-cell (PIC) algorithm on unstructured meshes for studying electrostatic perturbations with frequency lower than electron gyrofrequency in magnetized plasmas. In this method, ions are treated as fully kinetic particles and electrons are described by the adiabatic response. The PIC method is derived from a discrete variational principle on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that the electric field is interpolated using Whitney 1-forms, the charge is deposited using Whitney 0-forms and the electric field is computed by discrete exterior calculus. The algorithm has been applied to study the ion Bernstein wave (IBW) in two-dimensional magnetized plasmas. The simulated dispersion relations of the IBW in a rectangular region agree well with theoretical results. In a two-dimensional circular region with fixed boundary condition, the spectrum and eigenmode structures of the IBW are obtained from simulations. We compare the energy conservation property of the geometric PIC algorithm derived from the discrete variational principle with that of previous PIC methods on unstructured meshes. The comparison shows that the new PIC algorithm significantly improves the energy conservation property.

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