Classical Variational Principle Research Articles

On t-entropy and variational principle for the spectral radius of weighted shift operators

AbstractIn this paper we introduce a new functional invariant of discrete time dynamical systems—the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on L1(X,m) generated by the dynamical system. This result is called the variational principle and is similar to the classical variational principle for the topological pressure.

Oscillation criteria for forced second-order nonlinear differential equations with damping

In this paper, we are concerned with oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using a classical variational principle and an averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. An example is also given to illustrate the results.

Lateral buckling of web-tapered I-beams: A new theory

This paper presents a new theory for the lateral buckling of web-tapered I-beams. Linear analysis is first conducted by taking account into the tapering effects of web-tapered I-beams, where the deformation compatibilities of the two flanges and web are considered in terms of the basic assumptions of thin-walled members. Subsequently, the total potential for the lateral buckling analysis of web-tapered I-beams is developed, based on the classical variational principle for buckling analysis. The lateral buckling loads of web-tapered cantilevers and simply supported beams of I-sections from the proposed theory are compared with those from the finite element (FE) analyses using two shell element models and two widely used beam element models. The two beam element models respectively represent the equivalent method using prismatic beam elements and the typical tapered beam theory in existing literature. These comparisons show that the results based on the total potential proposed in this paper are more accurate in predicting the lateral buckling loads of web-tapered I-beams than those in existing theories, indicating that the theory proposed in this paper is superior to existing theories. It is also found that the equivalent method using prismatic beam elements may yield unreliable buckling loads of tapered beams.

Elastic flexural-torsional buckling of thin-walled cantilevers

Previous studies by the authors revealed that the two representative theories with slight differences between, widely used in investigating the flexural-torsional buckling of thin-walled beams, have led to two different solutions in well-known literature for assessing critical loads of simply supported beams of monosymmetric cross section. With these two solutions, significant differences in critical loads may be found for these monosymmetric beams. Based on the classical variational principle for buckling analyses, a new theory on the flexural-torsional buckling of thin-walled members was proposed by the authors. In this paper, this new theory as well as the other two typical theories is employed to investigate the flexural-torsional buckling of cantilevers. This paper first gives a brief review and a careful comparative study on the flexural-torsional buckling of thin-walled cantilevers employing three different buckling theories. Differences between these theories are demonstrated with investigations on buckling of cantilevers under pure bending and two typical transverse loads. Explicit solutions, capable of considering variations of beam length and loading position along the vertical axis of cross section, are presented for predicting the critical loads of doubly symmetric cantilevers under two typical transverse loads. Advantages of presented solutions, such as good accuracy and ease of use, are exploited through the comparisons of critical results with those from existing solutions and finite element analyses.

New oscillation criteria for a class of second-order nonlinear forced differential equations

By using classical variational principle and averaging technique, new oscillation criteria are established for second-order nonlinear forced differential equations [ r ( t ) ψ ( x ( t ) ) ϕ ( x ′ ( t ) ) ] ′ + q ( t ) f ( x ( t ) ) = e ( t ) , t ⩾ t 0 , where the functions r , q , e ∈ C ( [ t 0 , ∞ ) , R ) , and ϕ , ψ , f ∈ C ( R , R ) improve and extend some recent results.

Measure theoretical entropy of covers

In this paper we introduce three notions of measure theoretical entropy of a measurable cover \(\mathcal{U}\) in a measure theoretical dynamical system. Two of them were already introduced in [R] and the new one is defined only in the ergodic case. We then prove that these three notions coincide, thus answering a question posed in [R], and recover a variational inequality (proved in [GW]) and a proof of the classical variational principle based on a comparison between the entropies of covers and partitions.

N-vortex equilibrium theory

The problem of finding and classifying all relative equilibrium configurations of $N$-point vortices in the plane is first described as a classical variational principle and then formulated as a problem in linear algebra. Given a configuration of $N$ points in the plane, one must understand the structure of the $N(N-1)/2 \times N$ configuration matrix $A$ obtained by requiring that all interparticle distances remain fixed in time. If the determinant of the square, symmetric $N \times N$ covariance matrix $A^T A$ is zero, there is a non-trivial nullspace of $A$ and a basis set for this nullspace can be used to determine all vortex strengths $\vec{\Gamma} \in R^N$ for which the configuration remains rigid. Optimal basis sets are obtained by using the singular value decomposition of $A$ which allows one to categorize exact equilibria, approximate equilibria, and the distance between different equilibria in the appropriate vector space, as characterized by the Frobenius norm.

Electromagnetic Field with Constraints and Papapetrou Equation

It is shown that a geometric optical description of the electromagnetic wave with respect to its polarization in a curved space-time can be obtained straightforwardly from the classical variational principle for the electromagnetic field. For this purpose the entire functional space of electromagnetic fields must be reduced to its subspace of locally plane monochromatic waves. We have formulated the constraints under which this can be achieved. These constraints introduce variables of another kind which specify a field of local frames associated with the wave. They contain some congruence with null-curves. The Lagrangian for constrained electromagnetic fields contains variables of two kinds, namely a congruence of null-curves and the field itself. This in turn yields two kinds of Euler- Lagrange equations. The equations of the first kind are trivial due to the constraints imposed. The variation of the curves yields the Papapetrou equations for a classical massless particle with helicity 1

Second Noether-type theorem for the generalized variational principle of Herglotz

The generalized variational principle of Herglotz defines the functional, whose extrema are sought, by a differential equation rather than by an integral. For such functionals the classical Noether theorems are not applicable. First and second Noether-type theorems which do apply to the generalized variational principle of Herglotz were formulated and proved. These theorems contain the classical first and second Noether theorems as special cases. We published the first Noether-type theorem previously in this journal. Here we prove the second Noether-type theorem and show that it reduces to the classical second Noether theorem when the Herglotz variational principle reduces to the classical variational principle.

Flexural–torsional buckling of thin-walled beam members based on shell buckling theory

This paper summarized currently available techniques of setting up flexural–torsional buckling theory of thin-walled members. It is found that all the existing methods introduced a nonlinear load potential in their total potentials, while based on the classical variational principle for stability of a solid structure, no such load potential should be included. This situation has led to an inconsistency between some widely referenced monographs in buckling theories of beams with mono-symmetrical cross-sections. This paper provides a new theory for flexural–torsional buckling of thin-walled members based on the classical variational principle and the theory for thin-walled shells. No nonlinear load potential is included, but a new term: nonlinear strain energy from transverse stresses, which has been neglected in previous theories of thin-walled members, is introduced in. It is found that the nonlinear load potential is not equivalent to the contribution of transverse stresses for beams with mono-symmetrical cross-sections, which causes the inconsistency mentioned above. The comparison shows that the proposed theory and the traditional theory are the same for most cases encountered in practice.

Variational methods of solving dynamic problems for fluid-containing bodies

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.

Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem

This paper extends the generalized variational principle of Herglotz to one with several independent variables and derives the corresponding generalized Euler–Lagrange equations. The extended principle contains the classical variational principle with several independent variables and the variational principle of Herglotz as special cases. A first Noether-type theorem is proven for the new variational principle, which gives the conserved quantities corresponding to symmetries of the associated functional. This theorem contains the classical first Noether theorem as a special case. As examples for applications we calculate a conserved quantity for the damped nonlinear Klein–Gordon equation and we show that the equations which describe the propagation of electromagnetic fields in a conductive medium can be derived from the generalized variational principle of Herglotz (but not from a classical variational principle).

The nearly isotropic behaviour of high-rank nonlinear sequentially laminated composites

The effective transport behaviour, in the context of electrical conductivity, of sequentially laminated composites with nonlinear behaviour of the constituents is considered. An expression for the effective electric potential of two-dimensional composites is developed from the classical variational principle. This provides the means for determining the effective potential of a rank– N sequentially laminated composite with arbitrary volume fractions and lamination directions of the core laminates, in terms of an N –dimensional optimization problem. The effective potentials of sequentially laminated composites with power–law behaviour of the constituent phases are determined. It is demonstrated that the behaviour of sufficiently high–rank sequentially laminated composites tends to be isotropic. Two families of composites with both resistive and conductive inclusions are considered. The overall behaviour of these composites is compared with corresponding available bounds and estimates. It is found that the behaviours of composites belonging to these two families are, respectively, more conductive and more resistive than the corresponding second–order estimates of Ponte Castan∼eda and Kailasam.

First Noether-type theorem for the generalized variational principle of Herglotz

In this paper we formulate and prove a theorem, which provides the conserved quantities of a system described by the generalized variational principle of Herglotz. This new theorem contains as a special case the classical first Noether theorem. It reduces to it when the generalized variational principle of Herglotz reduces to the classical variational principle. Several examples for applications to physics are given.

TWO GENERALIZED HIGHER ORDER THEORIES IN FREE VIBRATION STUDIES OF MULTILAYERED PLATES

This paper presents an extension of two-dimensional models for the analysis of freely vibrating laminated plates. The extension concerns the enlargement of higher order theories, recently introduced by different authors in several forms, to encompass higher order terms over the cubic one usually taken into consideration. Higher order effects such as rotatory inertia and transverse shear stress are naturally included without any shear correction factors. Namely, two different models are introduced by expanding, on different functional bases, displacements (D2D) and transverse shear stresses in conjunction with displacements (M2D). The expansion is considered to be consistent with the traction-type boundary condition on the external surfaces of the plate. The governing equations and associated boundary conditions are consistently obtained by the classical Hamilton's variational principle and Reissner's mixed variational theorem. Both models are equivalent single layer type and, therefore, differ according to the layer-wise descriptions, preserve the independence of the number of unknown variables on the number of layers. However, this feature is presented together with intrinsic physical violations for both models. Model D2D violates the interlaminar stress continuity requirement and model M2D violates in a weaker from the same requirement (derivatives are not piecewise continuous), besides neglecting the transverse normal stress. The importance of completely fulfilling the mentioned continuity is then discussed once the relevant governing equations are tailored for the cylindrical bending condition. The effectiveness of the models is indicated by making numerical comparisons with the exact three-dimensional theory of the elasticity for several lamination schemes, angle/cross-ply lay-ups, and characteristic geometric ratios for low and higher frequencies.

KMS States, Entropy and the Variational Principle¶in Full C*-Dynamical Systems

To any periodic and full C *-dynamical system , an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius type theorem asserts the existence of KMS states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Unital completely positive maps implemented by partitions of unity {x j } of grade 1 are considered, resembling the “canonical endomorphism” of the Cuntz algebras. The relationship between the Voiculescu topological entropy of and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary. The measure-theoretic entropy of , in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure-theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy is obtained for the “canonical endomorphism” of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz–Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done.

Construction of variational factors for quasilinear second order partial differential equations

Using our methodology the next results are delivered. For concrete types of quasi-linear partial differential equations (PDE) for which classical variational principle is fulfilled, are constructed in explicit form some variational factors. It has also been demonstrated the existence of some particular cases where the above mentioned solution does not exist. For certain linear PDE types such solutions are described. Concerning linear PDE hyperbolic and elliptic types, the solutions led to lower bounded functionals.

Deriving generalized variational principles in general mechanics by using Lagrangian multiplier method

By using the involutory transformations, the classical variational principle—Hamiltonian principle— of two kinds of variables in general mechanics is advanced and by using undetermined Lagrangian multiplier method, the generalized variational principles and generalized variational principles with subsidiary conditions are established. The stationary conditions of various kinds of variational principles are derived and the relational problems discussed.

Path integral approach to fluctuations in relativistic transport

In this paper we analyze the fluctuations in an approximation to relativistic heat transport theory from a path integral approach. We start with a telegraphist type equation for the heat transport derived from a Meixner–Prigogine scheme and construct a classical variational principle for it. We use Hamiltonian-like function to obtain the transition probability between two different non-equilibrium states. We show that this probability obeys the Chapman–Kolmogorov equation.

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.