
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
April 9th, 2015, 05:05 AM  #1 
Newbie Joined: Apr 2015 From: Moscow, Russia Posts: 1 Thanks: 0  Variational problem: quadratic functional
Let me bring to your attention the following problem. Suppose we have the functional $\displaystyle F = \int\limits_{a}^{b} f(y(x))\cdot(\frac{dy}{dx})^2 dx $ with differential constraint $\displaystyle \frac{dy}{dx} / n(y(x))  1 = 0 $  DE. We write the Lagrangian for this problem $\displaystyle L = f(y(x))\cdot(\frac{dy}{dx})^2 + \lambda(x) \cdot ((\frac{dy}{dx} / n(y(x)))  1 ) $ Value at the end of the interval y(b) is not fixed (free end point) EL equation is: $\displaystyle \frac{d( f(y(x))\cdot (\dot{y})^2)}{dx} + \dot{\lambda} =0 $ or $\displaystyle f(y(x))\cdot (\dot{y})^2 + {\lambda(x)} =const $. Very strange EL equation. From my point of view, it means the following. Lagrange multiplier is a function only of the argument x, then it (and therefore the Lagrangian, as can be seen from equation) can be represented as a derivative with respect to x of a function p. And this is in accordance with the wellknown theorem sufficient criterion of "zero Lagrangian" (EL equation is an identity  every function which satisfies the boundary conditions is the extremum). Boundary condition for the non fixed end point: $\displaystyle (\frac{dL}{d\dot{y}})_b = 2(f(y)\cdot (\dot{y}))_b + ({\lambda(x)}/ n(y(x))_b = 0 $ The question is how to determine whether the extremum is minimum or maximum? Thank you for your help. 

Tags 
functional, problem, quadratic, variational 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Energy norm from Variational formulation (FEM)  ChidoriPOWAA  Differential Equations  0  December 16th, 2014 10:00 PM 
constrained functional minimization  functional linear in the derivatives  el keke  Calculus  0  October 31st, 2014 01:07 PM 
variational principle  sm61  Differential Equations  0  May 14th, 2014 03:44 AM 
Solving differential equation from variational principle  JulieK  Differential Equations  1  March 26th, 2013 04:42 PM 
A puzzel about variational method  doctorF  Applied Math  0  April 10th, 2012 09:31 PM 