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1.
The generalized variational principle of Herglotz type provides a variational method for describing nonconservative or dissipative processes. The purpose of this letter is to extend this variational principle to a first order linear nonholonomic system and study the conservation laws of the nonconservative nonholonomic system based on Herglotz variational problem. A new differential variational principle of the nonconservative nonholonomic system is proposed, which is based on Herglotz variational problem. And the differential equations of motion of the system are also obtained. Then, according to the condition for the invariance of the differential variational principle, the conservation theorem based on Herglotz variational problem for the nonconservative nonholonomic system are obtained. The theorem contains the conservation theorem of the nonconservative holonomic system as its special case, which can be reduced to the first Noether's theorem based on Herglotz variational problem under proper conditions. The inverse theorem of the conservation theorem is also provided and proved. An example is given to illustrate the application at the end of this letter.  相似文献   

2.
It is proposed that the universal thermodynamic energy variational principle is included in the first law of thermodynamics. Some variational principles in the electroelastic media under finite deformation are derived from this universal thermodynamic variational principle. It is suggested that in the general electroelastic analysis the environment should be considered together with the discussed electroelastic medium. For the variational principle of nonlinear electroelastic media the variation of the electric potential is coupled with the virtual displacement, and the variation of the initial volume should be considered. The Maxwell stress in the initial configuration is naturally derived from this variational principle and it is unique in the second order precision. Supported by the National Natural Science Foundation of China (Grant No. 10472069)  相似文献   

3.
According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the un-conventional Hamilton-type variational principles of holonomic conservative system in analytical mechanics can be established systematically. This unconventional Hamilton-type variational principle can fully characterize the initial-value problem of analytical mechanics, so that it is an important innovation for the Hamilton-type variational principle. In this paper, an important integral relation is given, which can be considered as the expression of the generalized principle of virtual work for analytical mechanics in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work of holonomic conservative system in analytical mechanics, but also to derive systematically the complementary functionals for three-field and two-field unconventional variational principles, and the functional for the one-field one by the generalized Legendre transformation given in this paper. Further, with this new approach, the intrinsic relationship among various principles can be explained clearly. Meanwhile, the unconventional Hamilton-type variational principles of nonholonomic conservative system in analytical mechanics can also be established systematically in this paper.  相似文献   

4.
The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.  相似文献   

5.
广义Birkhoff系统的时间积分定理   总被引:1,自引:0,他引:1       下载免费PDF全文
葛伟宽  梅凤翔 《物理学报》2009,58(2):699-702
研究了广义Birkhoff系统的时间积分定理.给出系统的时间积分等式,并由此等式导出类能量方程、类维里定理、一个积分变分原理和一个微分变分原理. 关键词: 广义Birkhoff系统 时间积分定理 类能量方程 变分原理  相似文献   

6.
In many metrics of physical interest, the gravitational field can be represented as an optical medium with an effective index of refraction. We show that, in such a metric, the orbits of both massive and massless particles are governed by a variational principle which involves the index of refraction and which assumes the form of Fermat's principle or of Maupertuis's principle. From this variational principle we derive exact equations of motion of Newtonian form which govern both massless and massive particles. These equations of motion are applied to some problems of physical interest.  相似文献   

7.
A Schwinger variational principle has been derived for use in quantum, manybody systems at finite temperatures. The variational principle is a stationary expression for the density matrix which may be iterated to improve an approximate density matrix. It also can be used to find stationary expressions for observables. If an approximate, parametrized density matrix is used, the parameters are varied to find the regions where the variational principle is stationary. The variational density matrix obtained with the optimal parameters can be regarded as optimal for that observable. The method has been applied to two model problems, a particle in a box and two hard spheres at finite temperatures. The advantages and shortcomings of the method are discussed.  相似文献   

8.
Taking into account the circumstance that the decay of an unstable microscopic system into two fragments is established by the counting of one of the decay products in a detector, the observed exponential decay law then asserts only knowledge of the spatiotemporal behaviour of the probability density (and therewith knowledge of the decaying state) at a large finite distance from the site of decay. We therefore formulate a variational principle, of which stationary functions show this decay behaviour. In addition to the resonant wave functions there are also solutions of the variational principle, which decrease exponentially with increasing distance, i.e., functions which could be used to describe the bound states. As the time-dependent treatment shows, the decaying states cannot occur in isolation in a scattering process. The mathematical characterisation of the decaying states via a variational principle is incorporated in a theory of open physical systems. In contradiction to the variational principle of Schrödinger our principle does not provide complete knowledge of the quantum states, but this is not needed in order to describe the decay.  相似文献   

9.
The methods of Ritz, Galerkin, and complementary energy are applied to a nonconservative problem in the theory of elastic stability. The numerical calculations are based upon (i) a variational expression, for which no functional can be determined, and (ii) an adjoint variational principle, for which a functional is established in terms of the variables of the original non-self-adjoint eigenvalue problem and the adjoint problem. The adjoint variational principle yields somewhat more accurate values for the critical load parameter than does the variational expression. In addition, the results obtained by means of the complementary energy method are more precise than the corresponding results obtained from the Ritz and Galerkin methods.  相似文献   

10.
We propose a new approach to construct structure-preserving algorithms for Birkhoffian systems. First, the Pfaff–Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Then, taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhoffian systems. Simulation results of the given example indicate that structure-preserving algorithms obtained by this method have great advantage in conserving conserved quantities.  相似文献   

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