Specify the endpoints of the curve: (0,1) and (1, 1/2) ODE1 := remove(has, EL, diff(y(x),x,x)) Ĭompute y(x) as a parametric curve (x(s), y(s)) using the dsolve command with the parametric option. This returns a set of ODEs.ĮL := EulerLagrange( fallTime, x, y(x) ) We then use the EulerLagrange function to compute the Euler-Lagrange equations for this functional in terms of y(x) and its derivatives. This is found in standard textbooks on classical mechanics.įallTime := sqrt( (1+diff(y(x),x)^2)/(2*(yInit-y(x))) ) The Brachistochrone problem can be stated as follows: Given two endpoints in the plane, find the curve y(x) between them such that a ball of unit mass rolls along the curve under the influence of gravity in minimum time.įirst we write down the falling time over an infintesimal distance dx in terms of y(x) and yInit, assuming the gravitational constant is 1. ![]() The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem for minimizing functions of real variables subject to constraints. Find the shape of a soap film having minimum surface area spanning a given wire frame.Shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time (the Brachistochrone problem). ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |