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Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?

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In summary, the conversation discusses the use of calculus of variations to determine the least time-taking course between two points in two dimensions. It is then questioned if the same method can be applied in three dimensions. The response states that while the cycloid solution can be applied to a 3D curve, it only works when the two points lie in a plane that also contains the gravitational force vector. Therefore, the trajectory will always be a cycloid in this scenario.

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Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?

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In principle yes, in fact the cycloid solution applies to a 3D curve when the two points lie in a plane that also contains the gravitational force vector. Since such plane always exists for any given pair of points, the trajectory will always be a cycloid.Serenie said:Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?

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The purpose of studying Calculus of Variations is to find the optimal solutions for problems involving maximizing or minimizing a functional. This is achieved by finding the function that minimizes the integral of a given functional over a certain interval.

A functional is a mathematical expression that takes in a function as its input and produces a real number as its output. In Calculus of Variations, functionals are typically integrals of a function and its derivatives over a certain interval.

The Euler-Lagrange equation is a necessary condition for finding the function that minimizes a given functional. It is derived by setting the functional's first derivative equal to zero and solving for the function. This equation is used to find the critical points of a functional, which correspond to the optimal solutions.

Calculus of Variations has many applications in physics, engineering, and economics. It is used to find the optimal paths for objects to follow in order to minimize their energy consumption, as well as the optimal shapes for structures to withstand external forces. It is also used in solving optimization problems in economics, such as finding the most efficient production processes.

Calculus of Variations is an extension of the fundamental theorem of calculus, which states that the derivative of a definite integral is equal to the integrand evaluated at the upper and lower limits of integration. In Calculus of Variations, the derivative of a functional is set equal to zero to find the optimal solutions, similar to how the derivative of a definite integral is set equal to the integrand at the limits of integration to find the area under a curve.

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