The Stationary Action Principle

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Boundary Conditions ◽  
Lagrange Equation ◽  
Variational Calculus ◽  
Action Principle ◽  
Dirichlet Boundary Conditions ◽  
Lagrange Formulation ◽  
Helmholtz Conditions ◽  
Stationary Action ◽  
The Difference ◽  
Corner Conditions

This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse problem is described. The chapter then considers the Ostrogradsky equation and discusses the properties of the extrema using the second-order variation to the action. It then discusses the difference between action functions (of Dirichlet boundary conditions) and action functionals of the extremal path. The different types of boundary conditions (Dirichlet vs Neumann) are elucidated. Topics discussed include Hessian conditions, Douglas’s theorem, the Jacobi last multiplier, Helmholtz conditions, Noether-type variation and Frenet–Serret frames, as well as concepts such as on shell and off shell. Actions of non-continuous extremals are examined using Weierstrass–Erdmann corner conditions, and the action principle is written in the most general form as the Hamilton–Suslov principle. Important applications of the Euler–Lagrange formulation are highlighted, including protein folding.

scholarly journals The Extrema of an Action Principle for Dissipative Mechanical Systems

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Tongling Lin ◽  
Qiuping A. Wang
Keyword(s):  
Numerical Simulation ◽  
Small Particle ◽  
Variational Calculus ◽  
Action Principle ◽  
Least Action Principle ◽  
Friction Forces ◽  
Intermediate Case ◽  
Least Action ◽  
Stationary Action ◽  
Energy Dissipated

A least action principle for damping motion has been previously proposed with a Hamiltonian and a Lagrangian containing the energy dissipated by friction. Due to the space-time nonlocality of the Lagrangian, mathematical uncertainties persist about the appropriate variational calculus and the nature (maxima, minima, and inflection) of the stationary action. The aim of this work is to make a numerical simulation of the damped motion and to compare the actions of different paths in order to obtain evidence of the existence and the nature of stationary action. The model is a small particle subject to conservative and friction forces. Two conservative forces and three friction forces are considered. The comparison of the actions of the perturbed paths with that of the Newtonian path reveals the existence of extrema of action which are minima for zero or very weak friction and shift to maxima when the motion is overdamped. In the intermediate case, the action of the Newtonian path is neither least nor most, meaning that the extreme feature of the Newtonian path is lost. In this situation, however, no reliable evidence of stationary action can be found from the simulation result.

scholarly journals Evaluation of the Capability of the Multigrid Method in Speeding Up the Convergence of Iterative Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Iman Harimi ◽  
Mohsen Saghafian
Keyword(s):  
Convergence Rate ◽  
Boundary Conditions ◽  
Laplace Equation ◽  
Multigrid Method ◽  
Low Frequency ◽  
Dirichlet Boundary Conditions ◽  
Frequency Function ◽  
Frequency Error ◽  
Multigrid Algorithm ◽  
The Difference

The performance of the multigrid method and the effect of different grid levels on the convergence rate are evaluated. The two-, three-, and four-level V-cycle multigrid methods with the Gauss-Seidel iterative solver are employed for this purpose. The numerical solution of the one-dimensional Laplace equation with the Dirichlet boundary conditions is obtained using these methods. For the Laplace equation, a two-frequency function involving high- and low-frequency components is defined. It is observed that, however, the GS method can smooth out the high-frequency error components properly, but because the difference scheme for Laplace equation is remarkably concise, in the fine grids, a very large number of iterations are needed for extending the boundary conditions into the domain. Furthermore, the obtained results reveal that the number of necessary iterations for convergence is reduced considerably by employing the two-level multigrid algorithm. But increasing the number of levels of algorithm does not have any significant effect on the convergence rate in this study.

Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions

2019 ◽  
Vol 12 (4) ◽  
pp. 333-361 ◽  
Author(s):  
Sascha Eichmann ◽  
Hans-Christoph Grunau
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Lagrange Equation ◽  
Order Reduction ◽  
Dirichlet Boundary Conditions ◽  
Surfaces Of Revolution ◽  
Geometric Information ◽  
Willmore Surfaces ◽  
Willmore Surfaces Of Revolution ◽  
Willmore Energy

AbstractIn this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible class is strictly below {4\pi}. Under a more restrictive but still explicit geometric smallness condition we obtain a quite interesting additional geometric information: The profile curve of this solution can be parameterised as a graph over the x-axis. By working below the energy threshold of {4\pi} and reformulating the problem in the Poincaré half plane, compactness of a minimising sequence is guaranteed, of which the limit is indeed smooth. The last step consists of two main ingredients: We analyse the Euler–Lagrange equation by an order reduction argument by Langer and Singer and modify, when necessary, our solution with the help of suitable parts of catenoids and circles.

scholarly journals Symmetry in the composite plate problem

2019 ◽  
Vol 21 (02) ◽  
pp. 1850019 ◽  
Author(s):  
Francesca Colasuonno ◽  
Eugenio Vecchi
Keyword(s):  
Boundary Conditions ◽  
Composite Plate ◽  
Dirichlet Boundary ◽  
Lagrange Equation ◽  
Composite Membranes ◽  
Radial Symmetry ◽  
Dirichlet Boundary Conditions ◽  
Qualitative Properties ◽  
Optimal Pair ◽  
Math Phys

In this paper, we deal with the composite plate problem, namely the following optimization eigenvalue problem: [Formula: see text] where [Formula: see text] is a class of admissible densities, [Formula: see text] for Dirichlet boundary conditions and [Formula: see text] for Navier boundary conditions. The associated Euler–Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator [Formula: see text]. In the spirit of [S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys. 214 (2000) 315–337], we study qualitative properties of the optimal pairs [Formula: see text]. In particular, we prove existence and regularity and we find the explicit expression of [Formula: see text]. When [Formula: see text] is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of [Formula: see text] and radial symmetry of both [Formula: see text] and [Formula: see text].

On the Field-Induced Cholesteric-Nematic Transition in Cholesteric Liquid Crystals with Homeotropic Boundary Conditions

1983 ◽  
Vol 38 (1) ◽  
pp. 1-9 ◽  
Author(s):  
J. Brokx ◽  
G. Vertogen ◽  
E. W. C. van Groesen
Keyword(s):  
Magnetic Field ◽  
Liquid Crystals ◽  
Boundary Conditions ◽  
Difference Method ◽  
Lagrange Equation ◽  
Helix Axis ◽  
Cholesteric Liquid Crystals ◽  
Director Field ◽  
Fréedericksz Transition ◽  
The Difference

Abstract The influence of homeotropic boundary conditions on the cholesteric-nematic transition is discussed, starting from the assumption that the director field consists of only components perpendicular to the helix axis, which is directed parallel to the boundary planes. Attention is paid to the solution of Cladis and Kléman for the simpler case without magnetic field. Their solution, consisting of regularly spaced line signularities, is derived in another more transparent way. Besides it is shown that even director fields without singularities are possible. The presence of a magnetic field complicates the solution considerably. Now the relevant Euler-Lagrange equation is solved in an approximate way by partly resorting to the difference method. The usefulness of the method is demonstrated by applying this technique to the nematic-cholesteric transition and the Fréedericksz transition in nematics.

scholarly journals Free energy and defect C-theorem in free scalar theory

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Tatsuma Nishioka ◽  
Yoshiki Sato
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Flat Space ◽  
Neumann Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Scalar Theory ◽  
Free Scalar ◽  
The Difference ◽  
Recent Classification

Abstract We describe conformal defects of p dimensions in a free scalar theory on a d-dimensional flat space as boundary conditions on the conformally flat space ℍp+1× 𝕊d−p−1. We classify two types of boundary conditions, Dirichlet type and Neumann type, on the boundary of the subspace ℍp+1 which correspond to the types of conformal defects in the free scalar theory. We find Dirichlet boundary conditions always exist while Neumann boundary conditions are allowed only for defects of lower codimensions. Our results match with a recent classification of the non-monodromy defects, showing Neumann boundary conditions are associated with non-trivial defects. We check this observation by calculating the difference of the free energies on ℍp+1× 𝕊d−p−1 between Dirichlet and Neumann boundary conditions. We also examine the defect RG flows from Neumann to Dirichlet boundary conditions and provide more support for a conjectured C-theorem in defect CFTs.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

scholarly journals Mode Shape-Based Damage Detection Method (MSDI): Experimental Validation

2021 ◽  
Vol 11 (10) ◽  
pp. 4589
Author(s):  
Ivan Duvnjak ◽  
Domagoj Damjanović ◽  
Marko Bartolac ◽  
Ana Skender
Keyword(s):  
Boundary Conditions ◽  
Damage Detection ◽  
Mode Shape ◽  
Experimental Validation ◽  
Detection Method ◽  
Dynamic Properties ◽  
Damage Index ◽  
Main Principle ◽  
The Difference ◽  
Different Levels

The main principle of vibration-based damage detection in structures is to interpret the changes in dynamic properties of the structure as indicators of damage. In this study, the mode shape damage index (MSDI) method was used to identify discrete damages in plate-like structures. This damage index is based on the difference between modified modal displacements in the undamaged and damaged state of the structure. In order to assess the advantages and limitations of the proposed algorithm, we performed experimental modal analysis on a reinforced concrete (RC) plate under 10 different damage cases. The MSDI values were calculated through considering single and/or multiple damage locations, different levels of damage, and boundary conditions. The experimental results confirmed that the MSDI method can be used to detect the existence of damage, identify single and/or multiple damage locations, and estimate damage severity in the case of single discrete damage.

scholarly journals Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions

2021 ◽  
pp. 104123
Author(s):  
Firdous A. Shah ◽  
Mohd Irfan ◽  
Kottakkaran S. Nisar ◽  
R.T. Matoog ◽  
Emad E. Mahmoud
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Wavelet Method ◽  
Telegraph Equations
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