Hamilton's principle as substationarity principle

Summary The present paper deals with Hamilton's principle for nonconvex, generally non-differentiable functions. It is shown that in this case Hamilton's principle can be formulated as a substationarity principle which is a generalization of a stationarity principle. To this end the recently defined notion of Clarke's generalized gradient is used. Finally a generalization of modified Hamilton's principle is discussed.With 1 Figure     

Hamilton's principle and the calculus of variations

Summary In the recent literature of the Calculus of variations, mathematical proofs have been presented for what the writers claim to be a more precise statement of Hamilton's Principle for conservative systems. Nothing is said about Hamilton's Principle for nonconservative systems. According to these writers, the action integral, the variation of which is Hamilton's Principle for conservative systems, is a minimum for discrete systems for small time intervals only and is never a minimum for continuous systems. The proof of this more precise statement is based in the sufficiency theorems of the Calculus of Variations. In this paper, two contradictions to the statement are demonstrated — one for a discrete system and one for a continuous system.     

Hamilton's principle for systems of changing mass

Summary An extended form of Hamilton's principle is developed for systems whose constituent particles change with time. By a suitable choice of system boundary it is demonstrated that in some cases stationary forms of the principle are possible. A simple example is considered.     

Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle

This work deals with a general energetic approach to establish an accurate electromechanical model of a piezoelectric transformer (PT). Hamilton's principle is used to obtain the equations of motion for free vibrations. The modal characteristics (mass, stiffness, primary and secondary electromechanical conversion factors) are also deduced. Then, to illustrate this general electromechanical method, the variational principle is applied to both homogeneous and nonhomogeneous Rosen-type PT models. A comparison of modal parameters, mechanical displacements, and electrical potentials are presented for both models. Finally, the validity of the electrodynamical model of nonhomogeneous Rosen-type PT is confirmed by a numerical comparison based on a finite elements method and an experimental identification.     

Numerical solution of dynamical systems by direct application of Hamilton's principle

In this paper, a method is described which allows a direct derivation of a set of first-order finite difference equations to numerically compute the motion of any conservative or non-conservative dynamic system with a finite number of degrees of freedom. The derivation of the method is based on an application of Lagrangian multipliers to a functional form of Hamilton's equations, and reduces the work required to obtain the most desirable form for numerical integration from the standpoint of computational efficiency and accuracy. For systems with many degrees of freedom, the required matrix inversions produce first derivatives of the co-ordinates, instead of second derivatives, thus eliminating a potential source of error in numerical integration. Two examples are given to illustrate the method.     

A variational principle in optics

We derive a new variational principle in optics. We first formulate the principle for paraxial waves and then generalize it to arbitrary waves. The new principle, unlike the Fermat principle, concerns both the phase and the intensity of the wave. In particular, the principle provides a method for finding the ray mapping between two surfaces in space from information on the wave's intensity there. We show how to apply the new principle to the problem of phase reconstruction from intensity measurements.     

Reissner's variational principle and elastodynamics

Summary There are two variational principles in elastostatics:Green's principle andCastigliano's principle. These are special cases ofReissner's variational principle.Green's principle andCastigliano's principle have been generalized in a symmetric manner to include elastodynamic problems. In the present paper,Reissner's principle is generalised to include these problems.

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Zusammenfassung In der Elastostatik gibt es zwei Variationsprinzipe: das Prinzip vonGreen und das Prinzip vonCastigliano. Diese sind Spezialfälle desReissnerschen Variationsprinzips. DasGreensche Prinzip und dasCastiglianosche Prinzip wurden in einer symmetrischen Weise verallgemeinert, um elastodynamische Probleme umfassen zu können. In der vorliegenden Arbeit wird dasReissnersche Prinzip so verallgemeinert, daß es die genannten Probleme beinhaltet.

     

A variational principle for magneto-elastic buckling

A variational principle that can serve as the basis for a magneto-elastic stability (or buckling) problem is constructed. For the two cases of soft ferromagnetic media and superconductors, respectively, it is shown how the variational principle directly yields an explicit expression for the buckling value. The formulation starts from a specific choice for a magneto-elastic Lagrangian L (associated with the so-called Maxwell-Minkowski model for magneto-elastic interactions). For the evaluation of the principle the first and second variations of L are calculated both inside and outside the solid magneto-elastic body. Thus, a general buckling criterion, consisting of an expression for the critical field value, together with a set of constraints for the field variables occurring in the right-hand side of this expression, is constructed. Finally, more detailed formulations are given for, successively, soft ferromagnetic bodies and superconductors. Applications to specific structures, yielding explicit numerical values for the magneto-elastic buckling fields, will be given in a forthcoming paper.     

Physical variational principle in dissipative media

The physical variational principle (PVP) is a physical principle which is implied in the thermodynamiclaws. For a conservative system, the PVP is implied in the first thermodynamic law and gives the motionequation. But for a dissipative system, PVP is implied in the extended Gibbs equation, which is the result ofthe first and second thermodynamic laws. The precision of the PVP in a dissipative system is in the same orderof the Gibbs equation. The dissipative work and its converted internal irreversible heat are simultaneouslyincluded in the PVP to get the governing equation and the boundary condition of the dissipated variables.The “generalized motion equations” including governing equations of the mechanical momentum and thermoelastic,thermal viscoelastic, thermal elastoplastic, linear thermoelastic diffusive and linear electromagneticthermoelastic materials etc. can be derived by the PVP of dissipative media in this paper. The conservativesystem is the special case of the dissipative system. Other than the mathematical variational principle, whichis obtained by a known governing equation, the PVP is used to deduce the governing equation. The PVPsincluding the hyperbolic temperature wave equation with a finite phase speed are also discussed shortly.     

A variational principle for nonlinear water waves

Summary This paper is concerned with a variational formulation of non-axisymmetric water waves and of two-dimensional surface waves in a running stream of finite depth. The full set of equations of motions for the non-axisymmetric water wave problem in cylindrical polar coordinates and for the two-dimensional surface waves in the running stream in Cartesian coordinates is obtained from a Lagrangian function which is equal to the pressure.With 1 Figure     

A variational principle for the equations of viscopiezoelectricity

The three-dimensional equations of linear viscopiezoelectricity and an accompanying electromechanical energy theorem are deduced, by the quasielectrostatic approximation, from the equations of viscoelectromagnetism and a generalized Poynting's theorem, respectively. For a viscopiezoelectric solid of volume V and bounding surface S, the internal energy, kinetic energy, and electric enthalpy densities as well as the variation of work done over S and the variation of energy dissipation in V are defined. A variational principle in terms of the defined functions is presented. It is shown that, from the principle, the equations of viscopiezoelectricity in V and the natural boundary conditions on S are obtained.     

Boundary integral equations from Hamilton's principle for surface acoustic waves under periodic metal gratings

The procedure describes the derivation of boundary integral equations for surface acoustic waves propagating under periodic metal strip gratings with piezoelectric films. It takes into account the electrical and mechanical perturbations, including the effects of mass loading caused by the gratings with an arbitrary shape. First, an integral equation is derived with line integrals on the boundaries within one period. This derivation is based on Hamilton's principle and uses Lagrange's method of multipliers to alleviate the continuous conditions of the displacement and the electric potential on the boundaries. Second, boundary integral equations corresponding to each substrate, piezoelectric film, metal strip, and free space region are obtained from the integral equation using the Rayleigh-Ritz method for admissible functions. With this procedure, it is not necessary to make any assumptions for separation of the boundary conditions between two neighboring regions. Consequently, we clarify the theoretical basis for the analytical procedure using boundary integral equations for longitudinal LSAW modes.     

Time-dependent linear systems derivable from a variational principle

This paper presents a study of time-varying linear systems of second order ordinary differential equations, which can be derived from a Lagrangian after multiplication by a suitable matrix. It concerns a generalization of previous studies on systems with constant coefficients. After a simplification of the Helmholtz conditions, it is shown that the problem is reduced to a purely algebraic one, provided one can solve a matrix differential equation which produces the transformation to canonical form of the given system. This further leads to a theoretical characterization of all systems admitting a multiplier. Various algebraic relations are derived, involving constant matrices only, which can help to detect, prior to any integration procedure, whether or not: a multiplier exists. They are referred to as the generalized commutativity conditions. The first of these, which is sufficient for the existence of a Lagrangian, is shown to allow also a simple construction of a quadratic first integral, and to have some other interesting features. The paper ends with an example.     

A mixed variational principle for finite element analysis

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of modified potential and complementary energy principles. Compatibility and equilibrium are satisfied throughout the domain a priori, leaving only the boundary conditions to be satisfied by the variational principle. This leads to a finite element model capable of relaxing troublesome interelement continuity requirements. The nodal concept is also abandoned and, instead, generalized parameters serve as the degrees-of-freedom. This allows for easier construction of higher order elements with the displacements and stresses treated in the same manner. To illustrate these concepts, plane stress and plate bending analyses are presented.     

Dispersive optical solitons by the semi-inverse variational principle

In this paper an analytical expression for an optical soliton is obtained with the aid of He's semi-inverse variational principle in the presence of third- and fourth-order dispersion as well as inter-modal dispersion. Three laws of nonlinear media are considered in this paper: the Kerr law, the power law and the log law.     

A variational principle for a solid-water interaction system

A variational principle for a coupled dynamic system of a solid and a water field with a free surface is derived with the aid of Hamilton and Toupin's dual principles. It is helpful for better understanding solid-water interaction problems.     

A variational principle for ideal flow over a spillway

A variational principle is presented which represents steady ideal flow with a free surface under gravity in terms of the stream function with Dirichlet boundary conditions. A different principle of Ikegawa and Washizu is shown to require restricted variations not consistent with the mass flow specification. The new principle is a special case of the of O'Carroll and Harrison in terms of enthalpy discontinuity between two streams. It is also reciprocal to a velocity potential functional. Finite element procedures and the determination of critical flow are discussed.     

A variational principle motivated by the optimal rod theory

Summary In this work we show that a number of well known nonlinear second order ODE appearing in theoretical physics provide the necessary condition for the minimum of the functional with the Lagrangian . Also we prove that those second-order differential equations may be viewied as conservation laws for the corresponding Euler-Lagrange equations that are the fourth-order ODE. Several special cases that have importance in physics, mechanics and optimal rod theory are studied in detail.