Calculus of Variations in Higher Dimensions
نویسنده
چکیده
Many problems in Calculus of Variations require the use functionals containing several independent and dependent variables. The first part of this report covers the derivation of the Euler-Lagrange equation for the general functional involving multi-variables vectorvalued functions with fixed boundary. Some of its corollaries and applications will be explored. In the second part we focus on an abstract yet elegant generalisation of the classic geodesic problem on surfaces to n-dimensional manifolds and explore it’s critical applications in physics, especially in Einstein’s Theory of Relativity. Part I The general functional
منابع مشابه
An analytic study on the Euler-Lagrange equation arising in calculus of variations
The Euler-Lagrange equation plays an important role in the minimization problems of the calculus of variations. This paper employs the differential transformation method (DTM) for finding the solution of the Euler-Lagrange equation which arise from problems of calculus of variations. DTM provides an analytical solution in the form of an infinite power series with easily computable components. S...
متن کاملNON-POLYNOMIAL SPLINE FOR THE NUMERICAL SOLUTION OF PROBLEMS IN CALCULUS OF VARIATIONS
A Class of new methods based on a septic non-polynomial spline function for the numerical solution of problems in calculus of variations is presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Convergence anal...
متن کاملPower Allocation Strategies in Block-Fading Two-Way Relay Networks
This paper aims at investigating the superiority of power allocation strategies, based on calculus of variations in a point-to-point two-way relay-assisted channel incorporating the amplify and forward strategy. Single and multilayer coding strategies for two cases of having and not having the channel state information (CSI) at the transmitters are studied, respectively. Using the notion of cal...
متن کاملA Brief Introduction to Calculus of Variations
To find a stationary point (i.e., a local maximum or minimum) the derivative ḟ(x) needs to vanish. In higher dimensions, the gradient ∇f needs to be identical to zero. The latter may also be expressed as fu = 0 ∀u. In a sense these simple derivatives are the most basic form of the calculus of variations. Usually, however, one talks about calculus of variations in the context of determining func...
متن کاملNumerical solution of variational problems via Haar wavelet quasilinearization technique
In this paper, a numerical solution based on Haar wavelet quasilinearization (HWQ) is used for finding the solution of nonlinear Euler-Lagrange equations which arise from the problems in calculus of variations. Some examples of variational problems are given and outcomes compared with exact solutions to demonstrate the accuracy and efficiency of the method.
متن کامل