Meshchersky–Lagrange equation for mass variable system: theory and application

In this paper, we develop a fractional cyclic integral and a Routh equation for fractional Lagrange system defined in terms of fractional Caputo derivatives. The fractional Hamilton principle and the fractional Lagrange equations of the system are obtained under a combined Caputo derivative. Furthermore, the fractional cyclic integrals based on the Lagrange equations are studied and the associated Routh equations of the system are presented. Finally, two examples are given to show the applications of the results.

A formal approach for the optimization of the final design of reload cores has been devised and verified. The method is based on applying the calculus of variations (Pontryagin's principle) to the normal flux and depletion system equations. The resulting set of coupled system, Euler-Lagrange (E-L), and optimality equations are solved iteratively. This is done by assuming a loading pattern for the old fuel, first solving the system equations, and then the E-L equations. The pattern is then modified by using the optimality (or Pontryagin) condition, and the process is repeated until no further improvements can be made. A computer program, OPMUV, implementing these procedures has been written and verified. The code can handle two-dimensional, quarter-core symmetric configurations with up to 241 assemblies and 4 nodes per assembly with modified one-group theory. It also has the capability of optimizing over the entire depletion cycle as well as just at the beginning of cycle (BOC). The results show that the procedure does work. In all cases tried, the method led to a reduction in nodal peaks of 1 to 3% over the final designer-obtained loading pattern within a couple of iterations. These savings carry over to comparable reductions in pin peaksmore » when the optimized patterns are used in four-group, fine-mesh calculations. Since the changes on each iteration are limited to ensure convergence, the method is thus well suited for the final fine tuning of the normally obtained patterns to gain an extra few percent in power flattening.« less

An explicit generalization of the classical Lagrange's equations (for discrete coordinate dynamical systems) to cover a large family of multibody hybrid discrete/distributed parameter systems is presented. The coupled system of ordinary and partial differential equations follows directly from spatial and time differentiation of various Lagrangian functional, whereas the boundary conditions are directly established from another explicit set of symbolic variational equations. Five illustrative examples are presented. E consider a family of multibody hybrid discrete/dis- tributed parameter systems that can be regarded as consisting of a collection of interconnecte d rigid and elastic bodies. Such models are useful for dynamics and control anal- ysis of flexible spacecraft. The equations of motion are hybrid, in the sense that the rigid-body motions are described by dis- crete time-varying coordinates, and the elastic motions are described by time- and space-varying coordinates; the resulting hybrid system of ordinary and partial integro-differential equations embodies significant coupling between the rigid- body and elastic motions.1 Meirovitch2 extended the classical Lagrange's equations for hybrid systems using the extended Hamilton's principle. Al- though Meirovitch found the correct forms for the hybrid system, his equations embodied a differential operator that must be developed through integration by parts for each specific application. Also, the boundary condition operator in Meirovitch's developments must be found by integration by parts for each specific application. Berbyuk and Demid- yuk3 formulated the dynamic equations and boundary condi- tions for a specific mechanical system (two-link manipulator with one rigid link and one flexible link) by means of the extended Hamilton's principle. In deriving the kinetic energy and boundary conditions, they included the effects of end pay load. Low and Vidyasagar4 presented a procedure for deriving dynamic equations for manipulators containing both rigid and flexible links. They proposed a method for producing a compact symbolic expression for the equation of flexible manipulator systems. As they mentioned, their boundary con- ditions do not make allowance for ends that involve discrete elements, such as lumped masses and springs. Of course, Hamilton's principle can produce the appropriate boundary conditions in such cases, but the procedure is system-specific and tedious, especially when dealing with multiple-connected flexible bodies. We were motivated by Meirovitch's developments to estab- lish, at least for significant classes of systems, explicit La- grange differential equations and boundary conditions that make allowance for lumped masses, springs, and similar forces at the boundaries. In essence, we seek to symbolically carry out the integration by parts once and for all for a large class of systems. In the present paper, explicit Lagrange's equations

Systems of stochastic differential equations, for which the Riemannian manifold generated by a diffusion matrix has zero curvature, are considered in this article. The method for approximate evaluation of characteristics of the solution of the systems of stochastic differential equations is proposed. This method is based on the representation of the probability density function through the functional integral. To compute functional integrals we use the expansion of action with respect to a classical trajectory, for which the action takes an extreme value. The classical trajectory is found as the solution of the multidimensional Euler – Lagrange equation.

Dynamic response of composite plates with cut-outs, part II: Clamped-clamped plates

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton's principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d'Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.

On nonlinear oscillations in electromechanical systems

We apply the Galerkin method in order to obtain numerical solution of the Euler - Lagrange equations for the variational problem for the upper bounds on the convective heat transport in a fluid layer under the action of intermediate and strong rotation. The role of the numerical investigation in such kind of variational problems is to obtain the upper bounds for the case of small and intermediate values of the Rayleigh and Taylor numbers in addition to the analytical asymptotic theory which leads to the upper bounds for the case of large values of the above two characteristic dimensionless numbers. The application of the Galerkin method reduces the Euler - Lagrange equations to a system of nonlinear algebraic equations. This system is solved numerically by the Powel hybrid method. We observe that the Powel hybrid method guarantees satisfactory fast rate of convergence from the guess solution to the solution of the system of equations. We present and discuss several results from the numerical computations.KeywordsPrandtl NumberVariational ProblemRayleigh NumberHeat TransportGalerkin MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

THE orthodox solution of Lagrangian frequency equations involves the expansion into polynomial form of the characteristic determinantal equation in the latent roots, but this method becomes exceedingly laborious if a large number of frequencies and their associated modes are required accurately for any system of equations of high order, say above the sixth. We define a system of Lagrangian frequency equations to be of the nth order if it consists of n equations for n homogeneous unknowns, which we call modes. A useful contribution to the problem was made by the iteration solution of Duncan and Collar, which is especially valuable when only the highest one or two latent roots are required. But when an aircraft propeller vibration problem required the first seven frequencies and their associated modes for a 12th‐order equation whose coefficients involved a variable pitch angle, the labour of calculation by this method appeared at that time (1941) to be prohibitive. The ‘Escalator’ method was therefore devised jointly by the author and Captain J. Morris of the Royal Aircraft Establishment as an alternative. In the propeller problem all the latent roots involved were necessarily real. Dr L. Fox, using relaxation methods, has recently solved a similar problem in a remarkably short time. Unfortunately, relaxation methods cannot easily be extended to the case of complex latent roots, which can occur in connexion with flutter, radio circuits and other problems. In this paper it is shown how the Escalator method can be adapted without essential change to cases in which complex quantities occur.

In the present work, the current-voltage (I-V) characteristics in a coupled long Josephson junction based on magnesium diboride are studied by establishing a system of equations of phase differences of various inter- and intra-band channels starting from the microscopic Hamiltonian of the junction system and simplifying it through the phenomenological procedures such as action, partition function, Hubbard-Stratonovich transformation (bosonization), Grassmann integral, saddle-point approximation, Goldstone mode, phase dependent effective Lagrangian and, finally, Euler-Lagrange equation of motion. The system of equations are solved using finite difference approximation for which the solution of unperturbed sine-Gordon equation is taken as the initial condition. Neumann boundary condition is maintained at both the ends so that the fluxon is capable of reflecting from the end of the system. The phase dependent current is calculated for different tunnel voltage and averaged out over space and time. The current-voltage characteristics are almost linear at low voltage and non-linear at higher voltage which indicates that the more complicated physical phenomena at this situation may occur. At some region of the characteristics, there exist a negative resistance which means that the junction system can be used in specific electronic devices such as oscillators, switches, memories etc. The non-linearity is also sensitive to the layer as well as to the junction thicknesses. Non-linearity occurs for lower voltage and for higher junction and layer thicknesses.

In this paper, we define a Donaldson type functional whose Euler–Lagrange equations are a system of differential equations, which corresponds to Hitchin’s self-duality equations for a suitable choice of Higgs bundle on closed Riemann surfaces. The main challenge of this functional is its lack of regularity and lack of compactness when defined in its natural domain of definition. Though a standard variational approach cannot directly be applied, we provide the appropriate analytical tools that make Donaldson functional treatable by a variational viewpoint. We prove that this functional admits a unique critical point corresponding to its global minimum. As an immediate consequence, we find that this system of self-duality equations admits a unique solution. Among the applications in geometry of this fact, we obtain a parametrization of closed constant mean curvature immersions in hyperbolic manifolds (possibly incomplete), and their moduli spaces.

Every underdetermined system of partial differential equations arising from a variational principle admits an infinite hierarchy of higher‐order generalized symmetries. These symmetries are a consequence of the Noether dependencies among the Euler–Lagrange equations that follow from Noether's Second Theorem. This result is a consequence of a more general theorem on the existence of higher‐order generalized symmetries for any system of differential equations that admits an infinitesimal symmetry generator depending on an arbitrary function of the independent variables.

On the Lagrangian form of the variational equations of Lagrangian dynamical systems

Shapes of phospholipid vesicles that involve narrow neck(s) were studied theoretically. It is taken into account that phospholipid molecules are intrinsically anisotropic with respect to the membrane normal and that they exhibit quadrupolar orientational ordering according to the difference between the local principal membrane curvatures. Direct interactions between oriented molecules were considered within a linear approximation of the energy coupling with the deviatoric field. The equilibrium shapes of axisymmetric closed vesicles were studied by minimization of the free energy of the phospholipid bilayer membrane under relevant geometrical constraints. The variational problem was stated by a system of Euler-Lagrange differential equations that revealed a singularity in the derivative of the meridian curvature at points where the effect of the orientational ordering exactly counterbalances the effect of the isotropic bending. The system of Euler-Lagrange differential equations was solved numerically to yield consistently related equilibrium orientational distribution of the phospholipid molecules and vesicle shape. According to our estimation of the model constants the formation of the neck is promoted if direct interactions between the oriented molecules are taken into account. It was shown that the energy of the equilibrium shapes is considerably affected by the quadrupolar ordering of phospholipid molecules.

Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.