Further, it is seen that elastica gives a natural solution for the variational problem which deals with the minimizing of bending energy of the elastic curve. Later, the equivalence between the motion of the simple pendulum and elastica's fundamental differential equation was investigated.
We firstly determine differential equations satisfied by non-rigid deformable curves in order to model the behaviorof elastic curves in 4-dimensional Minkowski space [E.sup.4.sub.1].
At present we are only concerned with the eigenanalysis of a particular physical entity, the thin elastic beam, and its mathematical analog, the interpolating elastic curve
, but the method described bears upon vibration analyses of all kinds.
An interest elastic curve
is one for which a small change in the interest rate leads to a large change in quantity, so that when plotted as in the figure, an elastic curve
will be close to horizontal.
Thus, [Theta] is the angle that the tangent vector to the elastic curve
makes with a parallel direction to the curve.
In Section 4, we establish a connection between invariant Willmore-like tori with [xi]-invariant potentials in the total space N of Killing submersions and elastic curves
with related potentials in the base surface M.
3, it is easy to recognise similar character of our elastic curves
to those of elastic wave  dissipation mechanisms.
Modern mathematicians recognize this eighteenth-century wonder worker with 21 papers on such diverse topics as Euler's work from 1750 to 1760, his fourteen most significant problems, his archives, his proof, with Bernoulli, of the fundamental theory of algebra, and his work on the quadrature of lunes, the Basel problem, elliptic integrals, harmonic progressions, power series expansions of the logarithmic and exponential functions, the summation formula, combinatories, the partition function, parallels with Clausen, the motion of the lunar apsides, elastic curves
, the advection equation, the propulsion of ships and the creation of accurate maps.
In Flow Motion, 2001, for example, sections of tan, green, and red are corralled by elastic curves
of lavender and orange, each hue somehow balancing the next.
Jurdjevic covers Cartan decomposition and the generalized elastic problems, the maximum principle and the Hamiltonians, the left-invariant symplectic form, symmetries and the conservation laws, complex Lie groups and complex Hamiltons, complexified elastic problems, complex elasticae of Euler and its n-dimensional extensions, Cartan algebras, root spaces and extra integrals of motion, and elastic curves
in the cases of Lagrange and Kowalewski.