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

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.

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 Necessary Condition for an Euler-Lagrange Equation on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Monika Dryl ◽  
Delfim F. M. Torres
Keyword(s):  
Time Scales ◽  
Integrodifferential Equation ◽  
Dynamic Equation ◽  
Time Scale ◽  
Lagrange Equation ◽  
Second Order ◽  
Necessary Condition ◽  
Quantum Calculus ◽  
Arbitrary Time

We prove a necessary condition for a dynamic integrodifferential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. An example of a second order dynamic equation, which is not an Euler-Lagrange equation on an arbitrary time scale, is given.

LVSEM: Past Present and Future

1990 ◽  
Vol 48 (1) ◽  
pp. 364-365
Author(s):  
James B. Pawley
Keyword(s):  
Field Emission ◽  
Line Width ◽  
Time Scales ◽  
Silicon Oxide ◽  
Beam Current ◽  
High Brightness ◽  
High Beam ◽  
Fixed Charges ◽  
Beam Voltage ◽  
Deposited Metal

Past: In 1960 Thornley published the first description of SEM studies carried out at low beam voltage (LVSEM, 1-5 kV). The aim was to reduce charging on insulators but increased contrast and difficulties with low beam current and frozen biological specimens were also noted. These disadvantages prevented widespread use of LVSEM except by a few enthusiasts such as Boyde. An exception was its use in connection with studies in which biological specimens were dissected in the SEM as this process destroyed the conducting films and produced charging unless LVSEM was used.In the 1980’s field emission (FE) SEM’s came into more common use. The high brightness and smaller energy spread characteristic of the FE-SEM’s greatly reduced the practical resolution penalty associated with LVSEM and the number of investigators taking advantage of the technique rapidly expanded; led by those studying semiconductors. In semiconductor research, the SEM is used to measure the line-width of the deposited metal conductors and of the features of the photo-resist used to form them. In addition, the SEM is used to measure the surface potentials of operating circuits with sub-micrometer resolution and on pico-second time scales. Because high beam voltages destroy semiconductors by injecting fixed charges into silicon oxide insulators, these studies must be performed using LVSEM where the beam does not penetrate so far.

A Two Time Scales Turbulence Model of Turbulent Bubbly Flows

2013 ◽  
Vol 40 (3) ◽  
pp. 185-203
Author(s):  
Dhahri Maher ◽  
Bellakhel Ghazi ◽  
Chahed Jamel
Keyword(s):  
Time Scales ◽  
Turbulence Model ◽  
Bubbly Flows
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