Chaplygin problem
Encyclopedia
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. The airplane is assumed to be constrained to move in a plane, moving at a constant airspeed
v, 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 eccentricity is w/v.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly in the fields of nonlinear dynamics and the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, the Chaplygin problem is an isoperimetric problem
Isoperimetric problem
Isoperimetric problem may refer to:* Isoperimetric inequality* Any problem in calculus of variations...
with a differential
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
constraint
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
. 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. The airplane is assumed to be constrained to move in a plane, moving at a constant airspeed
Airspeed
Airspeed is the speed of an aircraft relative to the air. Among the common conventions for qualifying airspeed are: indicated airspeed , calibrated airspeed , true airspeed , equivalent airspeed and density airspeed....
v, 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
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
whose eccentricity is w/v.