Chapter 3
Sufficient Conditions of Extrema
of Functionals
The variational problem with fixed boundary conditions has been discussed in
Chap. 2. From the necessary condition δ J = 0 of extremum of a functional, the
Euler equation or the Ostrogradsky equation satisfied by the extremal function are
derived. As was pointed out in Chap. 2, on the extremal function or extremal curve,
the functional doesn’t necessarily obtain extremum. On the basis of some new necessary conditions to deduce extremum of the functional, the sufficient conditions of
extrema of functionals are discussed in this chapter.
3.1 Extremal Curve Fields
Let y = y(x, c) be a single parameter family of curves on the plane domain D, if for
each point in the domain D, one and only one curve in the family of curves passes
through it, then the family of curves is called in the domain D to form an inherent
curve field. Or more accurately, it forms a proper field. The tangent line p(x, y) of
the family of curves y = y(x, c) at any point A(x, y) in the domain D is called the
slope of inherent curve field at point A(x, y). Because point A(x, y) in the inherent
curve field is arbitrary, the slope can be regarded as the function of point A(x, y), so
the slope p(x, y) can be called the slope function of inherent curve field.
For instance, the parallel lines y = 2x + c in the circle x
2
+ y
2
≤ 1 form an
inherent curve field, as shown in Fig. 3.1, the slope of the field is p = 2. For instance
again, a family of the parabolas y = (x − a)
2
− 1 in the preceding circle will not be
able to form an inherent curve field, because in the circle the family of the parabolas
intercross, as shown in Fig. 3.2.
If all of the curves of a family of curves y = y(x, c) pass through a point M(x c , y c )
in the plane domain D, then the point is called the center or centre of the family of
curves y = y(x, c). Let M(x c , y c ) be a point in the plane domain D, y = y(x, c) is
a family of curves with center M(x c , y c ), the family of curves pervades the whole
domain D, and except the center M(x c , y c ), the curves in the family of curves have not
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_3
201
Sufficient Conditions of Extrema
of Functionals
The variational problem with fixed boundary conditions has been discussed in
Chap. 2. From the necessary condition δ J = 0 of extremum of a functional, the
Euler equation or the Ostrogradsky equation satisfied by the extremal function are
derived. As was pointed out in Chap. 2, on the extremal function or extremal curve,
the functional doesn’t necessarily obtain extremum. On the basis of some new necessary conditions to deduce extremum of the functional, the sufficient conditions of
extrema of functionals are discussed in this chapter.
3.1 Extremal Curve Fields
Let y = y(x, c) be a single parameter family of curves on the plane domain D, if for
each point in the domain D, one and only one curve in the family of curves passes
through it, then the family of curves is called in the domain D to form an inherent
curve field. Or more accurately, it forms a proper field. The tangent line p(x, y) of
the family of curves y = y(x, c) at any point A(x, y) in the domain D is called the
slope of inherent curve field at point A(x, y). Because point A(x, y) in the inherent
curve field is arbitrary, the slope can be regarded as the function of point A(x, y), so
the slope p(x, y) can be called the slope function of inherent curve field.
For instance, the parallel lines y = 2x + c in the circle x
2
+ y
2
≤ 1 form an
inherent curve field, as shown in Fig. 3.1, the slope of the field is p = 2. For instance
again, a family of the parabolas y = (x − a)
2
− 1 in the preceding circle will not be
able to form an inherent curve field, because in the circle the family of the parabolas
intercross, as shown in Fig. 3.2.
If all of the curves of a family of curves y = y(x, c) pass through a point M(x c , y c )
in the plane domain D, then the point is called the center or centre of the family of
curves y = y(x, c). Let M(x c , y c ) be a point in the plane domain D, y = y(x, c) is
a family of curves with center M(x c , y c ), the family of curves pervades the whole
domain D, and except the center M(x c , y c ), the curves in the family of curves have not
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_3
201