In
mathematics, particularly in the fields of
nonlinear dynamics
In mathematics and science, a nonlinear 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, mathematicians, and many other ...
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 functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, the Chaplygin problem is an isoperimetric problem with a
differential constraint
Constraint may refer to:
* Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies
* Constraint (mathematics), a condition of an optimization problem that the solution ...
. 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. Among the common conventions for qualifying airspeed are:
* Indicated airspeed ("IAS"), what is read on an airspeed gauge connected to a Pitot-static system;
* Calibrate ...
''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 whose major axis is perpendicular to ''w'', with eccentricity ''w''/''v''.
References
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*
Calculus of variations
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