Nosso objetivo € consideraruma ampla classe de equaçöes diferenciais ordinarias da qual (*) faz parte, e que aparecem via a equação de Euler– Lagrange no. Palavras-chave: Cálculo Variacional; Lagrangeano; Hamiltoniano; Ação; Equações de Euler-Lagrange e Hamilton-Jacobi; análise complexa (min, +); Equações. Propriedades de transformação da função de Lagrange de covariância das equações do movimento no nível adequado para o ensino de wide class of transformations which maintain the Euler-Lagrange structure of the.
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We apply this complex variational calculus to Born-Infeld theory of electromagnetism and show why it does not exhibit nonlinear effects. As it was said by famous Isaac Newton, Nature likes simplicity.
Physicists and mathematicians have tried to express this metaphysic statement through equations: This principle leads to the so-called Euler-Lagrange equations set 18which is the kernel of past and future laws in physics.
What is the optimal shape of a house with a fixed volume in order to get a minimal surface which optimizes the heat loss? Eskimos have known the solution of this problem since a long time: Those cases are examples of the economy of means, responding to a metaphysical ideal of simplicity.
Soon, practical concerns underlying the calculus of variations, created polemical philosophical and metaphysical discussions. InLeibniz developed the idea that the world had been created as the best of all thinkable world in Essays on the Goodness of God, the Freedom of Man and the Origin of Evil.
Certainly, the creation has been flawed in so far as evil exists in the world, but this does not prevent anybody from considering that there is an almighty Creator, omniscient and infinitely good. God is to be the smartest and most powerful, it follows that our world is the best of all possible worlds: Voltaire used all the derision he mastered in Candidepublished in to attack another great scholar of the time, Pierre-Louis Moreau de Maupertuiswith whom he had few complaints.
Appointed President of the new Prussian Academy of Sciences by Frederick II Voltaire refused before this responsibilityto celebrate his starting in Berlin ddMaupertuis had published the laws of motion and rest, deduced from a metaphysical principle, which was derivated from the least action principle.
According to Maupertuis, this universal axiom is able to describe and explain all physical phenomena. Accordingly, he established that the Nature proceeded always with the maximal possible economy. Maupertuis stated his metaphysical principle in the following way: There remained, euler-largange, with mean of variational calculus to consolidate mathematically the principle of least action.
Maupertuis did not have the competences to build a stronger mathematical theory for that, but the presence in the Berlin Academy of Leonhard Euler foreshadowed fruitful cooperation. Euler was a master in the calculus of variations and he wrote the first treaty, where he showed that the least action principle eu,er-lagrange able to describe the motion of a point mass in a central field, for example trajectory of euler-pagrange planet around the sun.
In the early s, Maupertuis was involved in a violent controversy: We know now that Leibnitz was the first to have formulated and explained this principle in several letters. However, Euler created the corresponding mathematical structure, which served as a model for all the principles of variation subsequently incurred.
The least action principles, and those of virtual work and powers, are nowadays the most important mathematical tools to formulate elegantly and under invariant form the fundamental equations in physical and engineering sciences.
This helped to build other field theories, such as electroweak theory which unifies weak and electromagnetic interactionor quantum chromodynamics strong interactions between quarksand to find new particles through the system symmetries.
Euler–Lagrange equation – Wikidata
Since the 17 th century, theoretical physics paradigm has been based on this approach and from philosophical and metaphysical point of view, it has needed a mathematical approach which has been based on variational calculus. This gives a complex analytical mechanics with complex Euler-Lagrange and Hamilton-Jacobi equations. In section 3 the minimum of a complex valued function is euler-lagraneg and one explores the variational calculus for functionals of such functions, yielding thus to complex Hamilton-Jacobi equations.
Ekler-lagrange results are developed in section 4 to Lagrangian densities in order to derive a generalization of the Euler-Lagrange euler-lagrrange. We end with application of the complex variational calculation to Born-Infeld nonlinear theory of electromagnetism in section 5. When one tries to find the shortest path in a continuous space, optimality equation given by the the classical variational calculus is the well-known Hamilton-Jacobi equation which expresses mathematically the Least Action Principle LAP.
The action S xt has to verify. Distributivity is obtained in two steps. One has first to prove this equality with mean of two inequalities. We start first with eulerr-lagrange simple relations.
This permits to define a distribution theory which is continuously nonlinear in the field of real numbers, but which is linear in the dioid structure. Therefore it is interesting to study anlog results developed in Hilbert spaces functional analysis such as Riesz theorems, Fourier transforms, spectral analysis, measure theory 24 It is defined as.
This transform is very important in physics since it permits to pass from Lagrangian to Hamiltonian and conversely, from microcospic scales to macroscopic ones in statistical physics, and is the keystone mathematical tool for fractal and multifractal analysis 23 Main property is the possibility of passage from macroscopic scales to microscopic ones, and this can be expressed through the Euler-lagrangd Phase Approximation which uses the min operator While the Euler-Lagrange case entails an unknown initial velocity, the Hamilton-Jacobi case implies an unknown initial position.
This is the principle of least action defined by Euler 8 in and Lagrange 18 in In equxo, the principle euler-lagrangs least action used in this equation does not choose the velocity at each time s between 0 and tbut only when the particle arrives at x at time t.
The resolution of this problem implies that the ejler-lagrange solves the Euler-Lagrange equations 2. This is an a posteriori point of view.
The Hamiton-Jacobi action will overcome this a priori lack of knowledge of the initial velocity in the Euler-Lagrange action. Indeed, at the initial time, the Hamilton-Jacobi action S 0 x is known. The Hamilton-Jacobi euler-oagrange S xt at x and time t is then the function defined by.
This Hamilton-Jacobi action with its initial solution S 0 x is well known in the mathematical textbooks 9 for optimal control problems, but is ignored in physical ones 151920 euler-lagrrange, where there is no mention of the initial condition S 0 x.
It is often confused in the textbooks with the so-called principal function of Hamilton. Squao introduction of the Hamilton-Jacobi action highlights the importance of the initial action S 0 xwhile texbooks do not well differentiate these two actions.
The initial condition S 0 x is mathematically necessary to obtain the general solution to the Hamilton-Jacobi equations 2. Physically, it is the condition that describes the preparation of the particles. We will see that this initial condition is the key to the least action principle understanding. Nothing eeuler-lagrange because S 0 x 0 does not play a role in 2. It is an equation that generalizes the Hopf-Lax and Lax-Oleinik formula 9.
From the solution S xt of the Hamilton-Jacobi equations, one deduces the particle trajectories with 2. One can understand with equation 2. This problem is solved sequentially by the local evolution equation 2. This is an a equaoo point of view.
Equilibrium and Euler-Lagrange equation for hyperelastic materials
It is the problem solved by Nature with the principle of least action. The classical Euler-Lagrange action S cl xt ; x 0 is the elementary solution to the Hamilton-Jacobi equations 2.
One introduces for complex valued functions, the definition of a minimum in a first step and develops variational calculus for functionals applied to such kind of functions in euldr-lagrange second step. For a complex function f: Therefore x 0y 0 has to be a saddle-point of P xy.
For such a function, its real part P xy strictly convex in x is equivalent to P xy strictly concave in y Cauchy-Riemann conditions. For all convex function f: Theorem 2 Complex Euler-Lagrange equation. One defines the complex action S zt as the complex minimum of the integral. The complex action S zt verifies the complex Jamilton-Jacobi equations.
Analogous to the classical variational calculus with the assumption that the action S zt euler-lzgrange holomorphic in z. If a 0 x is strictly convex, one can show that the solution S xt of this system of equations 3.
In order to get the solution of those equations, it has been necessary to use complex variables. This is a general method. One can generalize the resolution of Hamilton-Jacobi equations for the complex ones. The following theorem gives a generalisation of Hopf-Lax formula. The solution S zt of complex Hamilton-Jacobi equations.
One develops in this sections euler-layrange complex analytical mechanics from complex valued Lagrangian density. One would like to generalize the Least Action Principle to this complex field in order to derive Euler-Lagrange-like equations.
We present in this section an application of our previous development about complex variational calculus to the Born-Infeld theory of electromagnetism. In almost all texbooks. Euler-lgrange two electromagnetic tensors are not combined into only one as for other fields in physics? From a fundamental point of view, one can not define the Lagrangian density 5. Let us note that it is important to obtain the right electromagnetic tensor if one wants to combine it with another one such as the metric tensor.
In this paper, we show that a well-suited candidate for the electromagnetic tensor is the complex Faraday tensor. This vector F has a long history since its introduction in by L.
Silberstein 32 It is the following Eqhao density 221 This density describes a non-interacting gauge theory but has not been validated by experiments in order to demonstrate nonlinear classical effects However, it still remains a relevant and useful theory for membranes and superstrings theories 430 Using the expression of F in 5. This shows first that the right complex Born-Infeld Lagrangian has to be the complex Faraday one.
It means that there are no nonlinear effects, excepted in the quantum treatment of electrodynamics. Indeed, the real part of the square root of a complex number is not equal to the square root of its real part.
This permits to develop a well-defined complex variational calculus, to generalize Hamilton-Jacobi and Euler-Lagrange equations to the complex case.
The analysis of the Born-Infeld theory through the complex Faraday tensor explains why experiments have never demonstrated nonlinear Born-Infeld effects and then confirms the Faraday complex tensor as a better candidate to represent the electromagnetic field.
It is a pragmatic way to consider the existence of this wave function, as stated by P. The complex Lagrangian euler-lagrangr proposed here is therefore an explicit functional of the wave function. In order to unify General Relativity and Electrodynamics Theories, Einstein also defined a complex tensor which linked metric and electromagnetic tensors 7.
We have proved with equation 5.