Chaplygin problem

In mathematics, particularly in the fields of nonlinear dynamics and the calculus of variations, 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 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

• Akhiezer, N. I. (1962). The Calculus of variations. New York: Blasidel. pp. 206–208. ISBN 3-7186-4805-9.
• Klamkin, M. S. (1976). "On Extreme length flight paths". SIAM Review. 18 (2): 486–488. doi:10.1137/1018079.
• Klamkin, M. S.; Newman, D. J. (1969). "Flying in a wind field I, II". Amer. Math. Monthly. 76 (1). Mathematical Association of America: 16–23, pp. 1013–1019. doi:10.2307/2316780. JSTOR 2316780.

See also

• Isoperimetric inequality : zero wind speed case