mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly in the fields of

nonlinear dynamics In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathem ...

and the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, the Chaplygin problem is an isoperimetric problem with a differential constraint. Specifically, the problem is to determine what flight path an airplane in a constant wind field should take in order to encircle the maximum possible area in a given amount of time. The airplane is assumed to be constrained to move in a plane, moving at a constant

airspeed In aviation, airspeed is the speed of an aircraft relative to the air it is flying through (which itself is usually moving relative to the ground due to wind). In contrast, the ground speed is the speed of an aircraft with respect to the sur ...

''v'', for time ''T'', and the wind is assumed to move in a constant direction with speed ''w''. The solution of the problem is that the airplane should travel in an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

whose major axis is perpendicular to ''w'', with eccentricity ''w''/''v''.

References

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References

See also