![]() The result is surprising because the curve actually dips below the right end point. The path along which a ball would roll in minimum time under the influence of gravity. The finite-length curve (x(s), y(s)) is parametrized over, so Find the curve between two given points in the plane that yields a surface of revolutionof minimum area when revolved around a given axis. Use the endpoints to solve for the unknown constants K and _C1 The calculus of variations gives us precise analytical techniques to answer questions of thefollowing type: Find the shortest path (i.e.,geodesic) between two given points on a surface. He gives a detailed discussion of the Hamilton-Jacobi theory. The parameter s is not equivalent to time t, so we must solve for the value of s at which They are the problem of geodesics, the brachistochrone, and the minimal surface of revolution. ![]() Specify the endpoints of the curve: (0,1) and (1, 1/2) The calculus of variations gives us precise analytical techniques to answer questions of thefollowing type: Find the shortest path (i.e. 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.The geodesic equations are formulated by means of the theory of differential geometry. Shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time (the Brachistochrone problem). Abstract The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically in geodetic and Cartesian coordinates.Find the shortest path between two points on a 3-D surface, such as a cone (a geodesic problem).Classical problems from the calculus of variations include: The new VariationalCalculus package provides routines for solving problems in the calculus of variations, which studies nature's most "efficient" curves and surfaces.
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