Physics
Calculus of Variations
The calculus of variations is a mathematical field that deals with finding the path, curve, or function for which a certain integral expression has an extremum. In physics, it is used to find the path taken by a particle that minimizes the action, which is a fundamental principle in classical mechanics. This mathematical tool is essential for understanding the behavior of physical systems.
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eBook - ePubEngineering Optimization
Theory and Practice
- Singiresu S. Rao(Author)
- 2019(Publication Date)
- Wiley(Publisher)
1 The subject of Calculus of Variations is almost as old as the calculus itself. The foundations of this subject were laid down by Bernoulli brothers and later important contributions were made by Euler, Lagrange, Weirstrass, Hamilton, and Bolzane. The Calculus of Variations is a powerful method for the solution of problems in several fields, such as statics and dynamics of rigid bodies, general elasticity, vibrations, optics, and optimization of orbits and controls. We shall see some of the fundamental concepts of Calculus of Variations in this section.12.2.2 Problem of Calculus of Variations
A simple problem in the theory of the Calculus of Variations with no constraints can be stated as follows:Find a function u(x) that minimizes the functional (integral)(12.1)where A and F can be called functionals (functions of other functions). Here x is the independent variable,In mechanics, the functional usually possesses a clear physical meaning. For example, in the mechanics of deformable solids, the potential energy (π) plays the role of the functional (π is a function of the displacement components u, v, and w, which, in turn, are functions of the coordinates x, y, and z).The integral in Eq. (12.1 ) is defined in the region or domain [x 1 , x 2 ]. Let the values of u be prescribed on the boundaries as u(x 1 ) = u 1 and u(x 2 ) = u 2 . These are called the boundary conditions of the problem. One of the procedures that can be used to solve the problem in Eq. (12.1
eBook - ePubLiquid Crystal Displays
Fundamental Physics and Technology
- Robert H. Chen(Author)
- 2011(Publication Date)
- Wiley(Publisher)
7 The Calculus of VariationsThe physical basis for the calculation of energy states in the various liquid crystal theories is that a physical system will always tend to its lowest potential energy state; this is manifested by changes to the system’s free energy, which will seek a minimum value. The states of the minimum free energy are calculated by the methods of the variational calculus, an extremely useful mathematical discipline, introduced below with some of its rather interesting history.Maxima and minima (together extrema ) calculations are commonly used in the natural and the engineering sciences. For example, in using Newton’s fluid dynamics to determine the very practical problem of the best shape of a ship’s hull to offer the lowest resistance to the water, it turns out unsurprisingly that it is just the minimum surface between two points, which result is easy to accept, if not so easy to prove. The extremum problem was also encountered in everyday life, as for instance in feudal Europe, land was ceded from father to sons according to how much land each son could mark off in one day given ropes of equal length, the objective of course being to encompass the maximum area possible.Historically, this type of maximization has roots going back three thousand years (900 B.C.) to the era of the Phoenicians and their Princess Dido. In escaping from her tyrannical brother, the Princess sought asylum in what is today Tunisia on the northwestern shore of Africa. The king there granted asylum but dismissively bequeathed her “all the land that could be contained in a bull’s skin” as her kingdom. The analytical Princess then proceeded to cut the bull’s skin into thin strips, tying them together to form a very long rope, and placing one end at a point on the shoreline, played out the line in semicircle of considerable radius to form a substantial area, which years later was to become the mythical great city of Carthage, over which the Princess reigned to become Queen Dido [1].
eBook - ePubMechanical Vibration
Analysis, Uncertainties, and Control, Fourth Edition
- Haym Benaroya, Mark Nagurka, Seon Han(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
δy ,δ y ≡f ε( x )- f( x )= ε ϕ( x ).There are two fundamental conditions for the variation: (a) it is arbitrary or virtual and (b) it is an infinitesimal change since ɛ can be made arbitrarily small. Note that while both δy and dy represent infinitesimal changes in the function f (x ), dy refers to a change in f (x ) caused by an infinitesimal change of the independent variable dx , whereas δy is an infinitesimal change of y that is due to a change in the function and results in a new function y + δy = f (x ) + ɛφ (x ) .This process of variation applies for each value of x . The value of x is not varied, meaning that δx = 0. If the two end points of this function are prescribed, they also do not vary. The variation is between definite limits. When time is the independent variable, the beginning and ending times are prescribed and therefore not varied. More advanced aspects of variational problems allow variations of the end points.As we will discover, in applying the variational procedures to a particular system, we find the governing equation(s) of motion and the necessary number of boundary conditions. The stationary value conditions imposed by the variational principles result in both the differential equation (s ) and the boundary conditions .The key topics to be examined in this chapter are- the principle of virtual work and its relation to the equilibrium of a body,
- the principle of virtual work, in conjunction with d ’Alembert ’s principle , and
- Lagrange’s equation, and Hamilton’s and Jourdain’s variational principles.
This chapter presents many examples showing the application of these principles. Further applications will be demonstrated for multi degree‐of‐freedom systems in Chapter 6 and continuous systems in Chapters 7 and 8
- Matthew N.O. Sadiku(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
self-adjoint. Solution 〈 L u, v 〉 = − ∫ S v ∇ 2 u d S Taking u and v to be real functions (for convenience) and applying Green’s. identity ∮ ℓ v ∂ u ∂ n d l = ∫ S ∇ u ⋅ ∇ v d S + ∫ S v ∇ 2 u d S yields 〈 L u, v 〉 = ∫ S ∇ u ⋅ ∇ v d S − ∮ ℓ v ∂ u ∂ n d l (4.10) where S is bounded by ℓ and n is the outward normal. Similarly, 〈 u, L v 〉 = ∫ S ∇ u ⋅ ∇ v d S − ∮ ℓ u ∂ v ∂ n d l (4.11) The line integrals in Equations 4.10 and 4.11 vanish under either the homogeneous Dirichlet or Neumann boundary conditions. Under the homogeneous mixed boundary conditions, they become equal. Thus, L is self-adjoint under any one of these boundary conditions. L is also positive definite. 4.3 Calculus of Variations The Calculus of Variations, an extension of ordinary calculus, is a discipline that is concerned primarily with the theory of maxima and minima. Here we are concerned with seeking the extrema (minima or maxima) of an integral expression involving a function of functions or functionals. Whereas a function produces a number as a result of giving values to one or more independent variables, a functional produces a number that depends on the entire form of one or more functions between prescribed limits. In a sense, a functional is a measure of the function. A simple example is the inner product 〈 u, v 〉. Variational formulation refers to the construction of a functional that is equivalent to the governing equation of the given problem. In the calculus of variation, we are interested in the necessary condition for a functional to achieve a stationary value. This necessary condition on the functional is generally in the form of a differential equation with boundary conditions on the required function. Consider the problem of finding a function y (x) such that the function I (y) = ∫ a b F (x, y, y ′) d x, (4.12a) subject to the boundary conditions y (a) = A, y (b) = B, (4.12b) is rendered stationary
