On the action principle for a system of differential equations

We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Euler-Lagrange equations for an action principle. Two ways of action principle construction are presented. From simple consideration, we derive necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order differential equations with the structure of Euler-Lagrange equations. An explicit form of the action is constructed in case if such a multiplier exists. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples.Comment: 10 page

Lie conformal algebra cohomology and the variational complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a g-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.Comment: 56 page

The Unique Determination of Neuronal Currents in the Brain via Magnetoencephalography

The problem of determining the neuronal current inside the brain from measurements of the induced magnetic field outside the head is discussed under the assumption that the space occupied by the brain is approximately spherical. By inverting the Geselowitz equation, the part of the current which can be reconstructed from the measurements is precisely determined. This actually consists of only certain moments of one of the two functions specifying the tangential part of the current. The other function specifying the tangential part of the current as well as the radial part of the current are completely arbitrary. However, it is also shown that with the assumption of energy minimization, the current can be reconstructed uniquely. A numerical implementation of this unique reconstruction is also presented

Lagrange formalism of memory circuit elements: classical and quantum formulations

The general Lagrange-Euler formalism for the three memory circuit elements, namely, memristive, memcapacitive, and meminductive systems, is introduced. In addition, {\it mutual meminductance}, i.e. mutual inductance with a state depending on the past evolution of the system, is defined. The Lagrange-Euler formalism for a general circuit network, the related work-energy theorem, and the generalized Joule's first law are also obtained. Examples of this formalism applied to specific circuits are provided, and the corresponding Hamiltonian and its quantization for the case of non-dissipative elements are discussed. The notion of {\it memory quanta}, the quantum excitations of the memory degrees of freedom, is presented. Specific examples are used to show that the coupling between these quanta and the well-known charge quanta can lead to a splitting of degenerate levels and to other experimentally observable quantum effects

Derivation of Boltzmann Principle

We present a derivation of Boltzmann principle $S_{B}=k_{B}\ln \mathcal{W}$ based on classical mechanical models of thermodynamics. The argument is based on the heat theorem and can be traced back to the second half of the nineteenth century with the works of Helmholtz and Boltzmann. Despite its simplicity, this argument has remained almost unknown. We present it in a modern, self-contained and accessible form. The approach constitutes an important link between classical mechanics and statistical mechanics

Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy

The lattice Gel'fand-Dikii hierarchy was introduced by Nijhoff, Papageorgiou, Capel and Quispel in 1992 as the family of partial difference equations generalizing to higher rank the lattice Korteweg-de Vries systems, and includes in particular the lattice Boussinesq system. We present a Lagrangian for the generic member of the lattice Gel'fand-Dikii hierarchy, and show that it can be considered as a Lagrangian 2-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus the multiform structure proposed in arXiv:0903.4086v2 [nlin.SI] is extended to a multi-component system.Comment: 12 page

The spike train statistics for consonant and dissonant musical accords

The simple system composed of three neural-like noisy elements is considered. Two of them (sensory neurons or sensors) are stimulated by noise and periodic signals with different ratio of frequencies, and the third one (interneuron) receives the output of these two sensors and noise. We propose the analytical approach to analysis of Interspike Intervals (ISI) statistics of the spike train generated by the interneuron. The ISI distributions of the sensory neurons are considered to be known. The frequencies of the input sinusoidal signals are in ratios, which are usual for music. We show that in the case of small integer ratios (musical consonance) the input pair of sinusoids results in the ISI distribution appropriate for more regular output spike train than in a case of large integer ratios (musical dissonance) of input frequencies. These effects are explained from the viewpoint of the proposed theory.Comment: 22 pages, 6 figure

Helmholtz's inverse problem of the discrete calculus of variations

International audienceWe derive the discrete version of the classical Helmholtz's condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations

Point vortices on the sphere: a case with opposite vorticities

We study systems formed of 2N point vortices on a sphere with N vortices of strength +1 and N vortices of strength -1. In this case, the Hamiltonian is conserved by the symmetry which exchanges the positive vortices with the negative vortices. We prove the existence of some fixed and relative equilibria, and then study their stability with the ``Energy Momentum Method''. Most of the results obtained are nonlinear stability results. To end, some bifurcations are described.Comment: 35 pages, 9 figure

Historical roots of gauge invariance

Gauge invariance is the basis of the modern theory of electroweak and strong interactions (the so called Standard Model). The roots of gauge invariance go back to the year 1820 when electromagnetism was discovered and the first electrodynamic theory was proposed. Subsequent developments led to the discovery that different forms of the vector potential result in the same observable forces. The partial arbitrariness of the vector potential A brought forth various restrictions on it. div A = 0 was proposed by J. C. Maxwell; 4-div A = 0 was proposed L. V. Lorenz in the middle of 1860's . In most of the modern texts the latter condition is attributed to H. A. Lorentz, who half a century later was one of the key figures in the final formulation of classical electrodynamics. In 1926 a relativistic quantum-mechanical equation for charged spinless particles was formulated by E. Schrodinger, O. Klein, and V. Fock. The latter discovered that this equation is invariant with respect to multiplication of the wave function by a phase factor exp(ieX/hc) with the accompanying additions to the scalar potential of -dX/cdt and to the vector potential of grad X. In 1929 H. Weyl proclaimed this invariance as a general principle and called it Eichinvarianz in German and gauge invariance in English. The present era of non-abelian gauge theories started in 1954 with the paper by C. N. Yang and R. L. Mills.Comment: final-final, 34 pages, 1 figure, 106 references (one added with footnote since v.2); to appear in July 2001 Rev. Mod. Phy