130
2 Variational Problems with Fixed Boundaries
y =
2c 1
a
ax + b − c
2
1 + c 2
Example 2.5.9 Find the extremal curve of the functional J [y] =
x 1
x 0
x
m
x n +y dx, where,
m > 0, n > 0.
Solution Because the integrand F =
x
m
x n +y does not contain y, so the Euler equation
of the functional has the first integral
x
m
(x n + y ) 2 = c
or
y
= −x
n
+ c 0 x
m
2
Integration gives
y = −
x
n+1
n + 1
+ c 1 x
m
2 +1
+ c 2
Example 2.5.10 The determination of the optimal shape of a sewing machine
needlepoint. In the process of sewing of a sewing machine, the needlepoint is one
of the main parts pierced a fabric. Determine the optimal shape of the needlepoint,
the purpose is to make it in the process of piercing the fabric consume the minimal
energy. When the fabric thickness is more than length of the needlepoint, using the
variational method to solve the above problem. In general, the needlepoint shape can
be expressed as a surface that a curve y = y(x) rotates around the y axis, as shown
in Fig. 2.6. In the process of the needlepoint piercing the fabric, they are interacting,
obviously in this case, the fundamental forces acting on the surface element dF are
the axial resistance dP along the needlepoint direction, the elastic force dQ of the
fabric perpendicular to the needlepoint direction, the normal pressure dN and the
frictional force f dN produced by the needlepoint contacting the fabric, This force is
along the tangent direction of the vertical section of surface element, with needlepoint
direction is α angle degree, where, f is the coefficient of friction in the process of
the pinpoint piercing fabric.
Solution The various forces are projected on the coordinate axes, the equilibrium
equations are
f dN cos α − dP + dN sin α = 0
f dN sin α + dQ − dN cos α = 0
(1)
According to Eq. (1), the relation between dP and dQ is
dP =
f + tan α
1 − f tan α
dQ
(2)
2 Variational Problems with Fixed Boundaries
y =
2c 1
a
ax + b − c
2
1 + c 2
Example 2.5.9 Find the extremal curve of the functional J [y] =
x 1
x 0
x
m
x n +y dx, where,
m > 0, n > 0.
Solution Because the integrand F =
x
m
x n +y does not contain y, so the Euler equation
of the functional has the first integral
x
m
(x n + y ) 2 = c
or
y
= −x
n
+ c 0 x
m
2
Integration gives
y = −
x
n+1
n + 1
+ c 1 x
m
2 +1
+ c 2
Example 2.5.10 The determination of the optimal shape of a sewing machine
needlepoint. In the process of sewing of a sewing machine, the needlepoint is one
of the main parts pierced a fabric. Determine the optimal shape of the needlepoint,
the purpose is to make it in the process of piercing the fabric consume the minimal
energy. When the fabric thickness is more than length of the needlepoint, using the
variational method to solve the above problem. In general, the needlepoint shape can
be expressed as a surface that a curve y = y(x) rotates around the y axis, as shown
in Fig. 2.6. In the process of the needlepoint piercing the fabric, they are interacting,
obviously in this case, the fundamental forces acting on the surface element dF are
the axial resistance dP along the needlepoint direction, the elastic force dQ of the
fabric perpendicular to the needlepoint direction, the normal pressure dN and the
frictional force f dN produced by the needlepoint contacting the fabric, This force is
along the tangent direction of the vertical section of surface element, with needlepoint
direction is α angle degree, where, f is the coefficient of friction in the process of
the pinpoint piercing fabric.
Solution The various forces are projected on the coordinate axes, the equilibrium
equations are
f dN cos α − dP + dN sin α = 0
f dN sin α + dQ − dN cos α = 0
(1)
According to Eq. (1), the relation between dP and dQ is
dP =
f + tan α
1 − f tan α
dQ
(2)