Constrained Minimisation Problems (Method of Lagrange Multipliers)
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In many applications of the calculus of variations, what we want to determine is not just the minimum of a function f on an open set U but the minimum of f subject to certain restrictions on the points x ∈ U. A well-known example from classical mechanics can be used to illustrate this type of problem. We want to determine the minimum of the action functional subject to the subsidiary condition that the motion be on a given surface. The restriction in this case is, therefore, that the points x ∈ U. satisfy an equation of the form g(x) = 0, i.e. the equation of the surface.
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