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1.
The generalized variational principle of Herglotz type provides an effective way to study the problems of conservative and non-conservative systems in a unified way. According to the differential variational principle of Herglotz type, we study the adiabatic invariants for a disturbed Birkhoffian system in this paper. Firstly, the differential equations of motion of the Birkhoffian system based upon this variational principle are given, and the exact invariant of Herglotz type of the system is introduced. Secondly, a new type of adiabatic invariants for the system under the action of small perturbation is obtained. Thirdly, the inverse theorem of adiabatic invariant for the disturbed Birkhoffian system of Herglotz type is obtained. Finally, an example is given. 相似文献
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We give a brief review of the theory of quantum universal invariants and their counterparts in the physics of light and particle beams. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, or the transverse phase-space coordinates of the paraxial beams of light or particles. They are conserved in time (or along the beam axis) independently of the concrete form of the coefficients of the Schrödinger-like equations governing the evolution of the systems, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation of the quantum (optical) invariants to the classical universal integral invariants of Poincaré and Cartan. The specific features of Gaussian beams are discussed as examples. The concept of the universal quantum integrals of motion is introduced, and examples of the “universal invariant solutions” to the Schrödinger equation, i.e., self-consistent eigenstates of the universal integrals of motion, are given. 相似文献
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Bhimsen K. Shivamoggi 《Physics letters. A》2009,373(7):708-710
Impulse formulations of Hall magnetohydrodynamic (MHD) equations are developed. The Lagrange invariance of a generalized ion magnetic helicity is established for Hall MHD. The physical implications of this Lagrange invariant are discussed. The discussion is then extended to compressible Hall MHD and a generalized ion magnetic potential helicity Lagrange invariant is established. The physical implications of the generalized ion magnetic potential helicity Lagrange invariant are shown to be the same, as to be expected, as those of the generalized ion magnetic helicity Lagrange invariant. 相似文献
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We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections
of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions
whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations,
characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that
generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for
a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state
which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian
potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions
generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the
sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written
on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process
generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal
rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes
equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant
diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore,
the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian
motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and
Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate
the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form
of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field
equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that
the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between
an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when
we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss
the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted
to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to
include the electromagnetic potential terms of Q 相似文献
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《Physica A》1995,214(1):82-94
New terms describing dissipation are added to currents in the spin hydrodynamic equations for superfluid phases A and B of 3He. The terms contain kinetic coefficients connected with external magnetic field and nonconventional fields conjugated to the hydrodynamic variables describing suitable broken symmetries. Due to the Onsager type relations only two new coefficients in each phase (both connected with magnetic field) remain in the equations and can be expressed in terms of the “old” kinetic coefficients and some generalized susceptibilities. Experimental data for superfluid 3He may be able to provide information about “viscosity” of magnetic fields. 相似文献
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E.V. Ferapontov C. A. P. Galvão O. I. Mokhov Y. Nutku 《Communications in Mathematical Physics》1997,186(3):649-669
We exhibit the bi-Hamiltonian structure of the equations of associativity (Witten-Dijkgraaf-Verlinde-Verlinde-Dubrovin equations)
in 2-d topological field theory, which reduce to a single equation of Monge-Ampère type in the case of three primary fields. The first Hamiltonian structure of this equation is based on its representation as a
3-component system of hydrodynamic type and the second Hamiltonian structure follows from its formulation in terms of a variational
principle with a degenerate Lagrangian.
Received: 1 March 1996 / Accepted: 25 October 1996 相似文献
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The paper presents an improved derivation of the dynamic equations for the conformally invariant generalization of Einstein's equations. The consistency of the variational procedure with the causality principle is studied. The well-posedness of the Cauchy problem in the synchronous coordinate system is proved as applied to the generalized equations. The possibility of generalized equations at finding quantitative relations between observed values is noted. 相似文献
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H. Roos 《Communications in Mathematical Physics》1975,42(1):83-100
The well-known results concerning the equilibrium of a translation invariant quantum lattice system — existence of the pressure and of the time automorphisms, variational principle for the pressure — are generalized to a large class of quantum lattice systems with potentials not exhibiting covariance under the group of lattice translations. 相似文献
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We consider (classical and generalized) Massey products on the Chekanov homology of a Legendrian knot, and we prove that they are invariant under Legendrian isotopies. We also construct a minimal A∞-algebra structure on the Chekanov algebra of a Legendrian knot, we prove that this structure is invariant under Legendrian isotopy, and we observe that its higher multiplications allow us to find representatives for classical Massey products. Finally, we consider differential equations: we remark that the Massey product Legendrian invariants admit a “dynamical interpretation”, in the sense that they provide solutions for a Maurer-Cartan equation posed on an infinite-dimensional bigraded Lie algebra, and we show how to set up and solve a (twisted) Kadomtsev-Petviashvili hierarchy of equations starting from the Chekanov algebra of a Legendrian knot. 相似文献
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Jan Makarewicz 《Physics letters. A》1985,111(5):217-219
A new form of the Hartree equations containing variational potentials is presented. These equations are linear and not coupled. Additive “one-particle” energies are defined. Advantages of the new formulation are discussed. 相似文献
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Shaofan Li 《哲学杂志》2013,93(7):1059-1084
In this work, we study invariant properties of defect potentials that are capable of describing defect motions in a continuum. By formulating two canonical defect theories, a generalized Nye theory and the Kröner–de Wit theory, we have found three defect potentials that are variational, i.e. their associated Euler–Lagrange equations are differential compatibility conditions of the continuum and defects. Consequently, symmetry properties of these variational functionals render several classes of new conservation laws and invariant integrals that are related with continuum compatibility conditions, which are independent of the constitutive relations of the continuum. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by such an integral is specifically defect-free. The invariant integral is applied to study macroscopically brittle fracture, and a multiscale Griffith criterion is proposed, which leads to a rigorous justification of the well-known Griffith–Irwin theory. 相似文献
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The hydrodynamics of mixtures of liquids of nonspherical molecules is considered by using the method of “generalized hydrodynamic matrix” (Felderhof-Selwyn-Oppenheim theory). The resulting hydrodynamic equations are applied to calculate the low frequency depolarized light scattering spectra based on Gershon-Oppenheim theory. Two possible cases in the VH geometry are considered. 相似文献
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The “nonlinear superposition law” for systems of the Ermakov-type is generalized to systems of n + 1 second-order differential equations having n invariants, quadratic in the velocities. 相似文献
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Adiabatic invariants are specific physical quantities which do not change appreciably even after a very long time when the Hamiltonian of a mechanical system undergoes a slow change in time. Existing proofs of this nice feature range from sophistication, and typically resort to a sort of averaging principle using Hamilton’s equations of motion. We show that a much simpler argument based directly on Hamilton’s principle per se is possible. Furthermore, this approach readily reveals an interesting local recurrent property of the adiabatic invariants that is rarely emphasized in the existing literature. We also show how our simpler approach can be easily generalized to derive the time dependence of the adiabatic invariant. 相似文献
19.
The Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems
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This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results. 相似文献
20.
Fruitful analogies, partially first established by C. M. Newman,(1) between the variables, functions, and equations which describe the equilibrium properties of classical ferro- and antiferromagnets in the Mean Field Approximation (MFA) and those which describe the space-time evolution of compressible Burgers' liquids are developed here for one-dimensional systems. It is shown that the natural analogies are: magnetic field and position coordinate; ferro-/antiferromagnetic coupling constants and negative/positive times; free energy per spin and velocity potential; magnetization and velocity field; magnetic susceptibility and mass density. An unexpected consequence of these analogies is a derivation of the Morette–Van Hove relation. Another novelty is that they necessitate the investigation of weak solutions of Burgers' equation for negative times, corresponding to the Curie–Weiss transition in ferromagnets. This is achieved by solving the “final-value” problem of the homogenous Hamilton–Jacobi equation. Unification of the final- and initial-value problems results in an extended Hopf–Lax variational principle. It is shown that its applicability implies that the velocity potentials at time zero be Lipschitz continuous for the velocity field to be bounded. This is a rather mild condition for the class of physically interesting and functionally compatible velocity potentials, compatible in the sense of satisfying the Morette–Van Hove relation. 相似文献