Extremal

A smooth solution of the Euler equation, which is a necessary extremum condition in the problem of variational calculus.

In the case of the simplest problem of variational calculus, which requires one to find the extremum of a functional

$$ \tag{1 } J ( y) = \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x , y , y ^ \prime ) d x $$

among all curves $ y ( x) $ satisfying the boundary conditions

$$ \tag{2 } y ( x _ {1} ) = y _ {1} ,\ \ y ( x _ {2} ) = y _ {2} , $$

Euler's equation has the form

$$ F _ {y} - \frac{d}{dx} F _ {y ^ \prime } = 0 , $$

that is, it is an ordinary differential equation of the second order. Its explicit form is

$$ \tag{3 } F _ {y ^ \prime y ^ \prime } y ^ {\prime\prime} + F _ {y ^ \prime y } y ^ \prime + F _ {y ^ \prime x } - F _ {y} = 0 . $$

If the extremum in the problem (1), (2) is a attained for a smooth curve $ y ( x) $, $ x _ {1} \leq x \leq x _ {2} $, then $ y ( x) $ is an extremal, that is, a solution of Euler's equation (3) with the initial condition $ y ( x _ {1} ) = y _ {1} $.

For $ F _ {y ^ \prime y ^ \prime } \neq 0 $, $ x _ {1} \leq x \leq x _ {2} $, Euler's equation has only smooth solutions (provided that $ F ( x , y , y ^ \prime ) $ is twice continuously differentiable). If $ F _ {y ^ \prime y ^ \prime } $ can vanish, then the solutions of Euler's equation may also include piecewise-smooth curves. Suppose that a piecewise-smooth curve $ y ( x) $, $ x _ {1} \leq x \leq x _ {2} $, yields the minimum in the problem (1), (2). Then every one of its smooth parts is an extremal, and at the corner points $ ( c , y ( c) ) $ the necessary Weierstrass–Erdmann conditions (cf. Weierstrass–Erdmann corner conditions) must be satisfied:

$$ \left . F _ {y ^ \prime } \right | _ {y ^ \prime ( c - 0 ) } = \left . F _ {y ^ \prime } \right | _ {y ^ \prime ( c + 0 ) } , $$

$$ \left . \left ( F - y ^ \prime F _ {y ^ \prime } \right ) \right | _ {y ^ \prime ( c - 0) } = \left . \left ( F - y ^ \prime F _ {y ^ \prime } \right ) \right | _ {y ^ \prime ( c + 0) } . $$

A piecewise-smooth curve that consists of pieces of extremals and satisfies the Weierstrass–Erdmann corner conditions is called a polygonal extremal. If the extremum in the problem (1), (2) is attained for a piecewise-smooth curve, then this curve is a polygonal extremal. However, the term "polygonal" is often omitted and one speaks of an extremal of the functional (1), for short, meaning a polygonal extremal.

In the case of a functional $ J ( y) $ depending on several functions, that is, when $ y $ in (1) is an $ n $- dimensional vector $ y = ( y _ {1} \dots y _ {n} ) $, Euler's equation becomes a system of $ n $ ordinary differential equations of the second order:

$$ \tag{4 } F _ {y _ {i} } - \frac{d}{dx} F _ {y _ {i} ^ \prime } = 0 ,\ \ i = 1 \dots n , $$

and the definition of an extremal (polygonal extremal) is similar.

In the more general case of a problem on a conditional extremum (see Isoperimetric problem; Bolza problem; Lagrange problem; and Mayer problem) an extremal is defined by means of the rule of multipliers.

Suppose, for example, that a piecewise-smooth curve $ y ( x) = ( y _ {1} ( x) \dots y _ {n} ( x) ) $ realizes the extremum in Lagrange's problem

$$ \tag{5 } \left . \begin{array}{c} J ( y) = \int\limits _ { x _ {1} } ^ { {x _ 2 } } f ( x , y , y ^ \prime ) d x , \\ f : \mathbf R ^ {1} \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R ^ {1} , \\ \end{array} \right \} $$

$$ \tag{6 } \left . \begin{array}{c} \phi _ \beta ( x , y , y ^ \prime ) = 0 ,\ \beta = 1 \dots m < n , \\ \phi _ \beta : \mathbf R ^ {1} \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R ^ {1} , \\ \end{array} \right \} $$

$$ \tag{7 } \psi _ {k} ( x _ {1} , y ( x _ {1} ) , x _ {2} , y ( x _ {2} ) ) = 0 ,\ k = 1 \dots p \leq 2 n + 1 . $$

Then, by the rule of multipliers, there is a constant (generally speaking, non-zero) $ \lambda _ {0} $ and multipliers $ \lambda _ {i} ( x) $, $ i = 1 \dots n $, such the vector function $ y ( x) $ is an ordinary (unconditional) extremal for the functional

$$ \tag{8 } I ( y , x ) = \int\limits _ {x _ {1} } ^ { {x } _ {2} } F ( x , y , y ^ \prime , \lambda ) d x , $$

where

$$ F ( x , y , y ^ \prime , \lambda ) = \lambda _ {0} f + \lambda _ {1} \phi _ {1} + \dots + \lambda _ {m} \phi _ {m} . $$

The system of Euler equations for the problem on an unconditional extremum for the functional (8),

$$ \tag{9 } F _ {\lambda _ \beta } - \frac{d}{dx} F _ {\lambda _ \beta ^ \prime } = \phi _ \beta ( x , y , y ^ \prime ) = 0 ,\ \ \beta = 1 \dots m , $$

$$ \tag{10 } F _ {y _ {1} } - \frac{d}{dx} F _ {y _ {i} ^ \prime } = 0 ,\ i = 1 \dots n , $$

consists of the $ m $ equations (9), which coincide with the constraints (6), and the $ n $ additional equations (10), which together with (9) yield the unknown functions $ y _ {1} ( x) \dots y _ {n} ( x) $, $ \lambda _ {1} ( x) \dots \lambda _ {m} ( x) $( for given initial conditions).

A smooth (piecewise-smooth) solution of the system (9), (10) of Euler equations rewritten for the problem on an unconditional extremum for the functional (8) is called an extremal (polygonal extremal) of the problem (5), (6) on a conditional extremum.

Being an extremal is not the only necessary condition for a curve to realize the extremum of a functional. This is explained by the fact that Euler's equation is derived as a necessary condition for the first variation of the functional to vanish, so that there still remains the problem of investigating the sign of the second variation of the functional. The extremal is subsequently studied by means of the necessary conditions of Lagrange, Weierstrass and Jacobi, as well as by means of sufficient conditions based on the construction of an extremal field.

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A polygonal extremal is also called a broken extremal. The Euler equation is also called the Euler–Lagrange equation.

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