Extremal field

A domain in the $( n + 1 )$- dimensional space of the variables $x , y _ {1} \dots y _ {n}$, covered without intersections by an $n$- parameter family of extremals of the functional

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

where $A$ and $B$ are the initial and final points through which the extremals of the family pass.

One must distinguish between proper (or general) and central extremal fields. A proper extremal field corresponds to the case when the extremals of the family are transversal to some surface

$$\tag{2 } \phi ( x , y _ {1} \dots y _ {n} ) = 0 ,$$

that is, on this surface the transversality conditions

$$\tag{3 } \frac{F - \sum _ {i=} 1 ^ {n} y _ {i} ^ \prime F _ {y _ {i} ^ \prime } }{\phi _ {x} } = \frac{F _ {y _ {1} ^ \prime } }{\phi _ {y _ {1} } } = \dots = \frac{F _ {y _ {n} ^ \prime } }{\phi _ {y _ {n} } }$$

hold. For a proper extremal field the point $A$( or $B$) in (1) belongs to the surface (2) and the condition (3) is satisfied at it.

A central extremal field corresponds to the case when the extremals of the family emanate from one point lying outside the field, for example, from a common initial point $A$.

The slope of an extremal field is the vector-function $u ( x , y ) = ( u _ {1} ( x , y ) \dots u _ {n} ( x , y ) )$ associating with every point $( x , y ) = ( x , y _ {1} \dots y _ {n} )$ of the field the vector $y ( x) = ( y _ {1} ^ \prime ( x) \dots y _ {n} ^ \prime ( x) )$.

For problems with moving end points, when $y ( x)$ is an extremal, the differential of the integral (1) has the form

$$\tag{4 } d J = \left [ \left ( F - \sum _ { i= } 1 ^ { n } y _ {i} ^ \prime F _ {y _ {i} ^ \prime } \right ) d x + \sum _ { i= } 1 ^ { n } F _ {y _ {i} ^ \prime } d y _ {i} \right ] _ {x _ {1} } ^ {x _ {2} } ,$$

where the differentials $dx$ and $dy$ are computed along the lines of displacement of the moving end points $A ( x _ {1} , y ( x _ {1} ) )$ and $B ( x _ {2} , y ( x _ {2} ) )$, and $y ^ \prime$ is the angular coefficient of the tangent to the extremal $y ( x)$.

The expression between square brackets in (4) can be rewritten in the form

$$\tag{5 } - H d x + \sum _ { i= } 1 ^ { n } p _ {i} d y _ {i} ,$$

where

$$H = - F ( x , y , u ( x , y ) ) + \sum _ { i= } 1 ^ { n } u _ {i} ( x , y ) F _ {y _ {i} ^ \prime } ( x , y , u ( x , y ) ) ,$$

$$p _ {i} = F _ {y _ {i} ^ \prime } ( x , y , u ( x , y ) ) .$$

In an extremal field the expression (5) is the total differential of some function of $x , y _ {1} \dots y _ {n}$, since

$$- \frac{\partial H }{\partial y _ {i} } = \ \frac{\partial p _ {i} }{\partial x } ,\ \ \frac{\partial p _ {i} }{\partial y _ {k} } = \ \frac{\partial p _ {k} }{\partial y _ {i} } ,\ \ i , k = 1 \dots n .$$

This function is, up to a constant term, equal to the curvilinear integral

$$\tag{6 } \int\limits _ { C } - H ( x , y , p ) d x + \sum _ { i= } 1 ^ { n } p _ {i} d y _ {i} ,$$

and is called the invariant Hilbert integral. In (6) $C$ denotes an arbitrary curve $y ( x)$ lying in the extremal field and joining the points $A$ and $B$. The term "invariant" emphasizes the fact that the integral (6) does not depend on the choice of $C$ and is determined only by the given end points.

The Hilbert integral (6) can be rewritten in the equivalent form

$$\tag{7 } \int\limits _ { C } \left [ F ( x , y , u ) - \sum _ { i= } 1 ^ { n } u _ {i} F _ {y _ {i} ^ \prime } ( x , y , u ( x , y )) \right ] d x +$$

$$+ \sum _ { i= } 1 ^ { n } F _ {y _ {i} ^ \prime } ( x , y , u ) d y _ {i\ } =$$

$$= \ \int\limits _ { C } \left ( F ( x , y , u ) + \sum _ { i= } 1 ^ { n } \left ( \frac{d y _ {i} }{dx} - u _ {i} \right ) F _ {y _ {i} ^ \prime } ( x , y , u ) \right ) d x .$$

If an extremal $E$ is taken as the comparison curve $C$, then $d y _ {i} / d x = u _ {i}$, and the Hilbert integral (7) becomes

$$\tag{8 } \int\limits _ { E } F ( x , y , u ( x , y ) ) d x ,$$

which coincides with the geodesic distance between the points $A$ and $B$, defined as the value of the functional (1) on the extremal joining $A$ and $B$.

The above property of an extremal field and of the invariant Hilbert integral lies at the basis of the theory of sufficient conditions for an extremum, as developed by K. Weierstrass. It makes it possible, in the study of the sign of the increment of the functional

$$\tag{9 } \Delta J = J ( C) - J ( E) ,$$

to express the value of $J ( E)$ along an extremal (under the assumption that the latter is enclosed by the extremal field) in terms of the invariant Hilbert integral over the comparison curve $C$ joining those two points. Thus, the increment of the functional (9) is represented as a curvilinear integral over $C$:

$$\tag{10 } \Delta J =$$

$$\int\limits _ { C } \left ( F ( x , y , y ^ \prime ) - F ( x , y , u ) - \sum _ { i= } 1 ^ { n } ( y _ {i} ^ \prime - u _ {i} ) F _ {y _ {i} ^ \prime } ( x , y , u ) \right ) d x =$$

$$= \ \int\limits _ { C } {\mathcal E} ( x , y , u , y ^ \prime ) d x .$$

The integrand ${\mathcal E} ( x , y , u , y ^ \prime )$ in (10) is called the Weierstrass ${\mathcal E}$- function. If this function is non-negative (non-positive) at any point of the extremal field for arbitrary finite values of $y ^ \prime$, then the extremal $E$ yields a strong minimum (maximum) of the functional (1) on the set of all comparison curves joining the points $A$ and $B$.

References

 [1] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947) [2] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian) [3] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)

For Weierstrass' approach see Weierstrass conditions (for a variational extremum); Weierstrass ${\mathcal E}$- function.