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Variational Problems with

Moving Boundaries

1. INTRODUCTION

We know that a functional associates a unique number to each function belonging to a certain class of functions. In the preceding chapter, we studied the methods of finding stationary function of the functional of the form

z

x2

x1

F(x, y(x), y′(x)) dx, where the boundary points (x1, y1)

and (x2, y2) were fixed points. In the present chapter, we shall allow one or both boundary points to move. In this case the class of admissible functions is extended because, in addition to the comparison curves with fixed boundary points, we have to admit functions with variable boundary points.

2. FUNCTIONALS DEPENDENT ON ONE FUNCTION

Theorem. Let J[y(x)] =

z

x2

x1

F(x, y(x), y′(x)) dx be a functional defined on the set of functions

y = y(x) which admits of continuous first order derivative and the end points of which (x1, y1) and (x2, y2) lie on the given curves y = φ(x) and y = ψ(x) respectively so that y1 = φ(x1) and y2 = ψ(x2), where y = φ(x) and y = ψ(x) are functions from the class C1[x1, x2]. Also, F is differentiable three times with respect to all its arguments.

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DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

.

.

.

.

.

.

(Fx) x + (Fy) y + (Fz) z = 0

(Gx) x + (Gy) y + (Gz) z = 0

and

.

.

.

Solving these equations for x, y and z , we get

.

.

.

x y z

=

=

Fy G z − Fz G y Fz G x − Fx G z Fx G y − Fy G x

FyGz – FzGy , FzGx – FxGz , FxGy – FyGx are also d.r.’s of the tangent to the curve given by the equations F(x, y, z) = 0, G(x, y, z) = 0 at the point t.

Example 7. Show that the equation of the tangent to the curve of intersection of the ellipsoid

x2 a2

+

y2

+

b2

z2 c2

x( X − x )

2

x2

= 1 and the confocal

2

2

a (b − c )(a − λ )

y2 b2 − λ

y(Y − y)

=

2

a2 − λ

+

2

2

2

2

b (c − a ) (b − λ )

+

=

z2 c2 − λ

= 1 is

z(Z − z)

2

2

c (a − b 2 ) (c 2 − λ )

,

where (x, y, z) is an arbitrary point on the curve.

Sol. The given surfaces are x2 a2

x2

and

+

y2

y2

z2

+

b2

–1=0

c2

...(1)

z2

–1=0

...(2) a 2 − λ b2 − λ c 2 − λ

Let the equation of the curve of intersection of given surfaces be r = r(t), where t is an arbitrary parameter.

Differentiating the equations (1) and (2) w.r.t. t, we get

+

2x

2

+

.

x+

2y

2

.

y+

2z

2

.

z=0

a b c

2x .

2y .

2z . y+ 2 z =0 x+ 2 a2 − λ b −λ c −λ

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DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

Remark. It can be proved that a curve is uniquely determined (except for its position in space) if we are given its curvature κ(≠ 0) and torsion τ as continuous functions of arc length s. This result shows the importance of curvature and torsion in the study of differential geometry of space curves.

Example 1. For the helix r = a cos ti + a sin tj + btk, a > 0, b ≠ 0, find the torsion at the point t.

Sol. We have r = a cos ti + a sin tj + btk

r = – a sin ti + a cos tj + bk

a 2 sin 2 t + a 2 cos 2 t + b2 =

|r | =

t=

r

=

|r|

t′ =

dt dt dt dt

=

= ds dt ds dt

2

a + b2

= −

=–

|t′| =

n=

a + b2

1

=

1

2

a +b

a

2

a + b2 a

a 2 + b2

(– a sin ti + a cos tj + bk) ds dt

(– a cos ti – a sin tj) |r|

a

2

a 2 + b2

2

(cos ti + sin tj)

a 2 + b2

(cos ti + sin tj)

(cos2 t + sin2 t)1/2 =

a a 2 + b2

a a 2 + b2 t′

(cos t i

+ sin t j

)

.

=− 2

= – (cos ti + sin tj) a a + b2

|t′| i

b=t×n= −

=–

=–

b′ =

1 a 2 + b2

1

2

a + b2

a a 2 + b2

− cos t

1

2

a + b2

sin t

a

k cos t

a 2 + b2

− sin t

i j k

− a sin t a cos t b cos t sin t 0

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CURVATURE AND TORSION

The centre of the osculating circle at point P is called the centre of curvature of the curve C at the point P.

By the definition of contact between curves, the osculating circle to the curve C at point

P can be considered as the intersection of a sphere with at least 3-point contact with the curve

C at point P and a plane with at least 3-point contact with C at P. If κ ≠ 0 at P, then the osculating plane at P is the unique plane having at least 3-point contact with the curve C at P.

In particular, if τ ≠ 0 in addition to κ ≠ 0 at P, then the osculating plane is the unique plane having exactly 3-point contact with C at P.

Therefore the osculating circle to a curve at a point always lies on the osculating plane to the curve at that point, provided κ ≠ 0 at the point under consideration.

Thus, the osculating circle to a curve at a point can be considered as the intersection of a sphere with at least 3-point contact with the curve at that point and the osculating plane to the curve at the point under consideration, provided κ ≠ 0.

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Curvature and Torsion

1. INTRODUCTION

For curves in space, the concepts of curvature and torsion are of fundamental importance.

We know that line segments are uniquely determined by their lengths, circles by their radii, triangles by side-angle-side etc. In geometry, we look for geometric quantities which distinguish one figure from another. The importance of curvature and torsion can easily be estimated from the fact that it can be proved that a curve is uniquely determined (except for its position in space) if its curvature and torsion are given as continuous functions of arc length ‘s’.

2. CURVATURE OF A CURVE

Let r = r(s) be a regular curve C of class Cm(m ≥ 2), where s is the parameter ‘arc length’.

The vector r″(s) is called the curvature vector on the curve C at the point r(s) and it is denoted by t(s)

κ(s) (or by κ). The magnitude of the curvature vector r(s) is called the curvature of the curve C at the point r(s) and it is denoted by κ(s) (or by κ).

C k(s)

κ(s) = |r″(s)|

Also t(s) = r′(s), so we have

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