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The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital , but Leonhard Euler first elaborated the subject, beginning in 1733.
The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis, 1698 – 1759) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). [1] It is a special case of the more generally stated principle of least action.
The shortest path or geodesic entails finding that function φ(λ) which minimizes s 12. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity, ′ ′ = Substituting for L and using Eqs.
Download as PDF; Printable version; ... Direct method in the calculus of variations; Dirichlet energy; ... Path of least resistance;
A locally shortest path between two given points in a curved space, assumed [b] to be a Riemannian manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations. This has some minor technical ...
Download as PDF; Printable version; In other projects ... is a shortest path(i.e., calculus of variations) problem with = () as the cost function. It turns out that ...
Download as PDF; Printable version; ... provide the shortest path between two points on the Earth's surface. ... a co-developer of the calculus of variations, ...