Stability of Equilibria in the Calculus of Variations
Suppose that we have a solution to the Euler-Lagrange equilibrium equations for
some calculus of variations functional.
This computed equilibrium is a critical point of the
functional, but not necessarily a stable local minimum.
In the classic theory of the unconstrained calculus of variations,
the Jacobi/Morse conjugate point theory is used to classify critical
points as local minima or some other type.
The number of conjugate points is called the index, and
should give the number of downward-turning directions on the
(infinite-dimensional) graph of the functional
in the neighborhood of the critical point.
This conjugate point test can be generalized to
isoperimetrically constrained problems. This generalization was
done over a century ago for some particular cases by Bolza and others, e.g. for one or
two variables and a few constraints. In
Manning, Rogers & Maddocks, 1998,
we have treated the general isoperimetric problem,
framing the process in modern functional analytic language.
With the appropriate inclusion of projection operators
(the projection being onto the space orthogonal to the linearized constraints),
the isoperimetric theory can be made to look almost exactly
parallel to the unconstrained theory.
For example, the twisted ring is described by
an isoperimetric calculus of variations problem.
In the bifurcation diagram for the "perfect"
(intrinsically straight with circular cross-section) rod (recreated below),
the index is denoted by color: green = 0 (local minimum),
red = 1 (saddle with one downward-turning direction),
light blue = 2, orange = 3, purple = 4, and dark blue = 5.
Another extension to the conjugate point theory allows its application
to cases where the second variation operator is more complicated than
a second-order differential operator as in the classic theory.
For example, in contact problems, the second variation operator is
often an integrodifferential operator. One simple contact problem
is the buckling of an elastic rod in the presence of an external field
designed to mimic contact with a wall (hence, the field acts as a sort
of "soft" wall).
By virtue of the general functional analytic approach taken to
isoperimetric conjugate points in
Manning, Rogers & Maddocks, 1998,
the theory can be extended relatively easily to this soft wall case. All that is
required are some basic properties of the integrodifferential second
variation operator, which are readily verified. In addition, although in
general one would expect the conjugate point equation to become an integrodifferential
equation, in this case it can be transformed into a differential equation
for easy numerical solution.
We have used this test to determine the stability of various equilibria of
the soft wall problem, for various values of the rod and wall parameters.
(This material is based upon work supported by the National Science Foundation under Grant No. DMS-9973258.
Any opinions, findings and conclusions or recomendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).)
The above theory is designed primarily for numerical implementation, since, in
most cases, it is impossible to solve the conjugate point equation in closed
form. Sometimes, however, an analytic solution is possible, or at least
a semi-analytic one (one which involves solving algebraic but not differential
equations numerically). This is particularly the case when a parameter
appears linearly in the second-variation operator, in which case we have
shown in
Hoffman, Manning, and Paffenroth, 2001 (preprint)
that one can completely determine stability by locating values of the
parameter where there are conjugate points at s=1.
Since conjugate points at s=1 are often easier to determine than conjugate points at general s,
this is a significant simplification.
We have applied this technique to two problems:
* The stability of multi-covered circular
configurations of an elastic rod with constant
planar intrinsic curvature, as a function of the intrinsic curvature kappa and the
rod stiffness parameters rho and gamma (rho is ratio of the out-of-plane bending stiffness
to the in-plane bending stiffness, and gamma is the ratio
of the twisting stiffness to the in-plane bending stiffness).
The index can be determined in closed form; for example, for an intrinsically
straight rod, the 1-covered circle
is stable if and only if rho>=1 (a fact that was previously known), while for n>1, the n-covered
circle is stable if and only if rho>1, gamma>1,
and (n-1)2/n2 < (gamma-1)(rho-1)/(gamma rho).
Below we show the dependence of the index on n, kappa, rho, and gamma.
The subfigure columns are for n=1, 2, and 3, while the rows are for increasing
values of kappa as labeled. In each subfigure, we color the gamma-rho plane by the index
of the n-covered circle for that value of rho, gamma, n, and kappa.
* The stability of an intrinsically straight
elastic strut under twisting and endloading, as a function of the twist,
endloading, and rod anisotropy (we consider rods with elliptical cross-section).
In this case we can determine the index semi-analytically, in that there are no
differential equations to be solved, but an algebraic equation remains for determining
the index.
Below we show the dependence of the index on rho (as in previous example), gamma (as in
previous example), alpha (twist angle), and lambda (endloading).
The subfigure columns are for gamma = 1/2, 1, and 2, while the rows are for increasing
values of rho as labeled. In each subfigure, we color the alpha-lambda plane by the index
of the straight strut for that value of alpha, lambda, gamma, and rho, using the
color scheme shown at the top.
For this problem, we have also computed families of buckled configurations numerically,
and assigned stabilities by the numerical isoperimetric conjugate point test described
above. These results include force-extension curves for lambda varying at fixed values
of rho, gamma, and alpha; and also 2-dimensional bifurcation surfaces for alpha and lambda
varying at fixed rho and gamma, as shown below:
See Hoffman, Manning,
and Paffenroth, 2001 (preprint) for more examples.
(This material is based upon work supported by the National Science Foundation under Grant No. DMS-9973258.
Any opinions, findings and conclusions or recomendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).)
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rmanning@haverford.edu