Formalization of Euler–Lagrange Equation Set Based on Variational Calculus in HOL Light

2020 ◽  
Author(s):  
Yong Guan ◽  
Jingzhi Zhang ◽  
Guohui Wang ◽  
Ximeng Li ◽  
Zhiping Shi ◽  
...  
Keyword(s):  
Lagrange Equation ◽  
Variational Calculus ◽  
Hol Light

scholarly journals Optimizing Regenerative Braking: A Variational Calculus Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
L. Q. English ◽  
A. Mareno ◽  
Xuan-Lin Chen
Keyword(s):  
Optimization Problem ◽  
Lagrange Equation ◽  
Variational Calculus ◽  
Regenerative Braking ◽  
Battery Charging ◽  
Air Drag ◽  
Battery Charge ◽  
Basic Physics ◽  
Integral Quantity ◽  
Charging Efficiency

We begin by analyzing, using basic physics considerations, under what conditions it becomes energetically favorable to use aggressive regenerative braking to reach a lower speed over “coasting” where one relies solely on air drag to slow down. We then proceed to reformulate the question as an optimization problem to find the velocity profile that maximizes battery charge. Making a simplifying assumption on battery-charging efficiency, we express the recovered energy as an integral quantity, and we solve the associated Euler–Lagrange equation to find the optimal braking curves that maximize this quantity in the framework of variational calculus. Using Lagrange multipliers, we also explore the effect of adding a fixed-displacement constraint.

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.

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 Second Euler-Lagrange Equation of Variational Calculus on Time Scales

2011 ◽  
Vol 17 (1) ◽  
pp. 9-18 ◽  
Author(s):  
Zbigniew Bartosiewicz ◽  
Natália Martins ◽  
Delfim F.M. Torres
Keyword(s):  
Time Scales ◽  
Lagrange Equation ◽  
Variational Calculus

On an Euler-Lagrange equation for color to grayscale conversion

2019 ◽  
Vol 2019 (1) ◽  
pp. 95-98
Author(s):  
Hans Jakob Rivertz
Keyword(s):  
Variational Inequality ◽  
Color Image ◽  
Lagrange Equation ◽  
New Method ◽  
Grayscale Image ◽  
Structure Tensors

In this paper we give a new method to find a grayscale image from a color image. The idea is that the structure tensors of the grayscale image and the color image should be as equal as possible. This is measured by the energy of the tensor differences. We deduce an Euler-Lagrange equation and a second variational inequality. The second variational inequality is remarkably simple in its form. Our equation does not involve several steps, such as finding a gradient first and then integrating it. We show that if a color image is at least two times continuous differentiable, the resulting grayscale image is not necessarily two times continuous differentiable.

The build of nonlinear quantum mechancs and variations of feature of microscopic particles as well as their experimental affirmation

2014 ◽  
Vol 5 (3) ◽  
pp. 871-981 ◽  
Author(s):  
Pang Xiao Feng
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Lagrange Equation ◽  
Minimum Principle ◽  
Superposition Principle ◽  
Type Equation ◽  
Experimental Results ◽  
Traditional Concept ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

We establish the nonlinear quantum mechanics due to difficulties and problems of original quantum mechanics, in which microscopic particles have only a wave feature, not corpuscle feature, which are completely not consistent with experimental results and traditional concept of particle. In this theory the microscopic particles are no longer a wave, but localized and have a wave-corpuscle duality, which are represented by the following facts, the solutions of dynamic equation describing the particles have a wave-corpuscle duality, namely it consists of a mass center with constant size and carrier wave, is localized and stable and has a determinant mass, momentum and energy, which obey also generally conservation laws of motion, their motions meet both the Hamilton equation, Euler-Lagrange equation and Newton-type equation, their collision satisfies also the classical rule of collision of macroscopic particles, the uncertainty of their position and momentum is denoted by the minimum principle of uncertainty. Meanwhile the microscopic particles in this theory can both propagate in solitary wave with certain frequency and amplitude and generate reflection and transmission at the interfaces, thus they have also a wave feature, which but are different from linear and KdV solitary wave’s. Therefore the nonlinear quantum mechanics changes thoroughly the natures of microscopic particles due to the nonlinear interactions. In this investigation we gave systematically and completely the distinctions and variations between linear and nonlinear quantum mechanics, including the significances and representations of wave function and mechanical quantities, superposition principle of wave function, property of microscopic particle, eigenvalue problem, uncertainty relation and the methods solving the dynamic equations, from which we found nonlinear quantum mechanics is fully new and different from linear quantum mechanics. Finally, we verify further the correctness of properties of microscopic particles described by nonlinear quantum mechanics using the experimental results of light soliton in fiber and water soliton, which are described by same nonlinear Schrödinger equation. Thus we affirm that nonlinear quantum mechanics is correct and useful, it can be used to study the real properties of microscopic particles in physical systems.

Virtual Work & d’Alembert’s Principle

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Euler Equations ◽  
Virtual Work ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Rigid Bodies ◽  
Gibbs Function ◽  
Constraint Force ◽  
Appell Equations ◽  
Jourdain's Principle ◽  
D'alembert's Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

Kinematic characteristics of longitudinal double folding wings

2021 ◽  
pp. 1-25
Author(s):  
L. Tiegang ◽  
C. Guoguang ◽  
L. Shuai
Keyword(s):  
Angular Velocity ◽  
Structural Model ◽  
Initial Conditions ◽  
Aerodynamic Force ◽  
Model Simulation ◽  
Lagrange Equation ◽  
Grid Method ◽  
Weapon Systems ◽  
Folding Wing ◽  
Double Folding

ABSTRACT A folding wing is a tactical missile launching device that needs to be miniaturised to facilitate storage, transportation, and launching; save missile and transportation space; and improve the combat capability of weapon systems. This study investigates the aeroelastic characteristics of the secondary longitudinal folding wing during the unfolding process. First, the Lagrange equation is used to establish the structural dynamics model of the folding wing, the kinematics characteristics during the deformation process are analysed, and the unfolding movement of the folding wing is obtained using the dynamic equations in the process. Then, the generalised unsteady aerodynamic force is calculated using the dipole grid method, and the multi-body dynamics equation of the folding wing is obtained. The initial angular velocity required for the deployment of the folding wing is analysed through structural model simulation, and the influence of the initial angular velocity on the opening process is studied. Finally, aeroelastic flutter analysis is performed on the folding wing, and the physical model of the folding wing verified experimentally. Results show that the type of aeroelastic response is sensitive to the initial conditions and the way the folding wing opens.

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