On normal modes in classical Hamiltonian systems

Normal modes of Hamittonian systems that are even and of classical type are characterized as the critical points of a normalized kinetic energy functional on level sets of the potential energy functional. With the aid of this constrained variational formulation the existence of at least one family of normal modes is proved and, for a restricted class of potentials, bifurcation of modes is investigated. Furthermore, a conjecture about a lower bound for the number of normal modes in case the potential is homogeneous, is proved

Bifurcating electric wires

The stationary states of a string through which an electric current is sent and which is placed in an axial magnetic field, are investigated. Using methods of constrained variational principles, it is shown that, in case the string is inextensible, only those stationary states which have least total potential energy are stable

Dual and inverse formulations of constrained extremum problems

AbstractIn this paper we consider constrained extremum problems of th form Pp:infu∈t−1(p)f(u) where f and t are continuously differentiable functionals on a reflexive Banach space V and where t-1(p) denotes the level set of the functional t with value p ϵ R.Related to problems Pp we investigate inverse extremum problems, which are extremum problems for the functional t on level sets of the functional f. Under conditions that guarantee the existence of solutions of Pp, let h(p) denote the value of f at such a solution. If h is a (locally) convex function at some p̄ ϵ R, we show that it is possible to define a dual problem of Pp̄. This dual problem is a saddle-point formulation for the functional V × RЭ(u,μ)→f(u)−μ[t(u)−p̄]: for some extreme value μ̄ (which is the Lagrange multiplier of a solution of Pp̄) the solutions of Pp̄ are precisely the (local) minimal points of the functional f − μ̄t on V.It is shown how these results can be used to describe solution branches of nonlinear eigenvalue problems (of semilinear elliptic type) with a global parameter, such as p ϵ R, instead of with the eigenvalue as a parameter

Simple wave interaction of an elastic string

The equations for the two-dimensional motion of a completely flexible elastic string can be derived from a Lagrangian. The equations of motion possess four characteristic velocities, to which the following four simple wave solutions correspond: leftward and rightward propagating longitudinal and transverse waves. The latter are exceptional (constant shape). By expanding the solution about a steady solution the interaction of simple waves may be studied. A typical result is the following: As a consequence of their interaction two transverse waves running into opposite directions emit a longitudinal wave and undergo themselves a translation over a finite distance but remain otherwise unchanged. The results are also valuable for a full comprehension of the interaction process of simple waves on inextensible strings

Stable model equations for long water waves

In this paper, a sequel to two others [1, 2], some extensions and improvements of this earlier work are presented. Among these are: A more precise version of the proof of the basic canonical theorem, some considerations on conservation laws and their relation, a more complete treatment of the stability of the models, especially with respect to the wave amplitude, a short treatment of the Lagrangian version of the theory, a stable discrete model which might be useful for numerical experiments and an extension of the method to the case of slowly varying water depth

Iterative methods for efficient generation of wave fields in hydrodynamic laboratories

Motivated to be applied in the Indonesian Hydrodynamic Laboratory (LHI), we design an iteration scheme for the motion of a wavemaker on one side of a large wave tank in such a way that a desired energy spectrum for the wave field results. Due to imperfections of the mechanical operation of the wavemaker, the mapping from the wavemaker control to the generated wave field in the tank is not known. Taking some realistic qualitative assumptions for this mapping, we design and study possible steering strategies. From a class of iteration schemes, one preferred iteration scheme can be selected based on the global conditions for convergence; the iteration will enable to generate any realisable wave field. The use of the proposed iteration led to a satisfying result for an actual experiment, details of which will be described