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PeroK said:I think it should be -2, but that's a minor point.
Orodruin said:This would depend on the order of the integration limits (note that ##a## was the lower integration limit in ##x## but ##A## is the upper integration limit in ##u## - of course I am just making the arbitrary inference that ##a## corresponds to ##A## here ...).
bobred said:but if we are trying to find stationary points of a functional constant multiples can be ignored?
bobred said:Thanks guys.
I may have missed this in my notes PeroK but if we are trying to find stationary points of a functional constant multiples can be ignored?
James
Ray Vickson said:Positive multiples can be ignored, but omitting negative multiples changes the direction of optimization.
Orodruin said:But it does not change the fact that the point is stationary. Just exchanges minima for maxima.
Calculus of variations is a branch of mathematics that deals with finding the optimal solution to a functional, which is a mathematical expression involving a function. It involves finding the function that minimizes or maximizes the value of the functional, often subject to certain constraints.
In calculus of variations, changing variables refers to the process of transforming the original functional into an equivalent one using a different set of independent variables. This is often done to simplify the problem or to make it easier to solve.
Changing variables can make the problem easier to solve by reducing the complexity of the functional or by making it more amenable to existing methods and techniques. It can also reveal hidden relationships between different functions and provide a deeper understanding of the problem.
Some commonly used techniques for changing variables in calculus of variations include the substitution method, the integration by parts method, and the Legendre transformation method. These methods can help transform the functional into a simpler form or reveal new relationships between the variables.
While changing variables can be a powerful tool in solving problems in calculus of variations, it does have its limitations. In some cases, the transformation may not result in a simpler functional, or it may introduce new constraints that make the problem more difficult to solve. Additionally, the choice of variables can greatly affect the outcome of the problem, so it is important to carefully consider which variables to use for the transformation.