MA209 Variational principles
Exercise sheet II
1. Elastic rope fixed at the two end-points: Then the change of the elastic energy can be approximated by
E [ y ] = k
Z
( y
0
)
2 dx, with k being the material constant. Determine the shape of a rope y = y ( x ) by finding critical points of the functional
I [ y ] = E [ y ] + g
Z y ( x ) dx, where g is the gravitational constant. Assume that the rope is fixed at x = ± 1, therefore y ( ± 1) = 0.
2. Elastic beam: The change of the elastic energy can be approximated by
E [ y ] = H
Z
( y
00
)
2 dx, where H is a material constant. Determine the shape of the beam y = y ( x ) by finding the critical points of the functional
I [ y ] = E [ y ] + g
Z y ( x ) dx, where g denotes the gravitational constant. Assume that the beam is fixed at its endpoint x = ± 1, that is y ( ± 1) = 0 and y
0
( ± 1) = 0.
3. Find critical points of the functional
I [ y ] = c
Z
L y ( y
0
)
3 dx,
0 with y (0) = 0 and y ( L ) = R .
4. Show that there is no solution y = y ( x ) to the minimisation problem
I [ y ] =
Z
1
( xy + y
2
0
− 2 yy
0
) dx, with y (0) = 1 and y (1) = 2.
5. Determine the equation of a geodesic on a circular cylinder of radius R
(you may use the expression of the arc length stated on Exercise Sheet I).
Show that its solution is a circular helix.
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