![]() The time it takes for the bead to go through a segment of the trajectory depends only on the speed of the bead on a particular segment (that can be inferred from its height) and the length of the segment which again only depends on the derivative of the state variable.The state variable is characterized by the coordinates of the successive segments of the trajectory (also known as a path).The trajectory duration is obtained by summing up the time that it takes the bead to go through successive segments of the trajectory.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. The Lagrangian and Hamiltonian variational approaches to mechanics are the only approaches that can handle the Theory of Relativity, statistical mechanics, and the dichotomy of philosophical approaches to quantum physics.The calculus of variations is a powerful technique to solve some dynamic problems that are not intuitive to solve otherwise. In fact, not only is it an exceedingly powerful alternative approach to the intuitive Newtonian approach in classical mechanics, but Hamilton’s variational principle now is recognized to be more fundamental than Newton’s Laws of Motion. This variational approach is both elegant and beautiful, and has withstood the rigors of experimental confirmation. The calculus of variations provides the mathematics required to determine the path that minimizes the action integral. \) follows a path that minimizes the scalar action integral \(S\) defined as the time integral of the Lagrangian.
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