![]() ![]() ![]() We are trying to minimize the duration of the rolling bead trajectory.Let’s check if each of these conditions applies: The calculus of variations provides a methodology to find out which path is the shortest in such a case. In the continuous space, we are trying to compare an infinite number of possible trajectories, each of them varying infinitesimally from another one. The optimal trajectory does not depend on few metro line changes but on every single steering decision. Now consider the case where you are driving and need to rally point A to point B with full control on the steering wheel. It is pretty easy to compare all possible combinations of metro stops and find out the optimal trajectory: in discretized space, the set of trajectories is a fixed number. Now why is the calculus of variations especially good at solving these problems? Consider the case where we have to find the fastest transportation time riding metro. The simulation allows us to compare multiple investment paths and choose the one leading to the highest revenue. If the company has an investment simulation, it can try multiple investment strategies and determine what is the optimal sequence of daily investment through time (the path) that leads to the highest revenue. ![]() The company wants to earn as much possible over the year, and to do so sums up every day’s revenue into a yearly income. The money that is invested each day is generating a revenue on the same day. Take an investment company that can decide on its daily investments with a limited capital to use over one year. The problem we want to solve is finding a path, a set of continuous values, that leads to the minimum overall cost (any maximization problem can be turned into a minimization problem by adding a negative sign to the cost). Each function is mapped to a single value by the mean of adding up a cumulative cost. Instead, it enables us to find the ~minimum~ of a set of functions (which we call a functional). Intuition on the theoryĬontrary to function optimization theory, the calculus of variations does not try to find the minimum of a function. Understanding the calculus of variations framework will then allow you to put a steady foot in the optimal control theory framework as well as discovering key mathematical concepts. Moreover, it is the basis for Lagrangian mechanics, less famous than its counterpart Newtonian mechanics yet just as powerful. ![]() It is the precursor to optimal control theory as it allows us to solve non-complex control systems. Some would say it's the main difficulty.īy the way this is the reason why some people think there are "different versions" of stochastic calculus/stochastic integrals: you're just using different vector spaces and bases to do your calculations.The calculus of variations is a powerful technique to solve some dynamic problems that are not intuitive to solve otherwise. Half the difficulty in functional analysis/PDEs is finding the right space and topology to work in. These two structures will have different sets of continuous and differentiable functions. You can have the same vector space but with two different and inequivalent norms. This essentially means there is only one way to do calculus on a given finite dimensional vector space, because if a function is differentiable or continuous with respect to one norm it will be differentiable/continuous with respect to every norm you can think of for that space. You may remember the result that "All norms on a finite dimensional vector space are equivalent." meaning notions of limits/convergence are the same. The difficult part of calculus of variations is that the vector spaces you're working with are infinite dimensional. The point is that you can do calculus in any situation where you have a vector space and a usable notion of length/topology. Or rather analysis if you're being pedantic. As others have said, calculus of variations is an antiquated term. ![]()
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