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

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

Phase-Space Metric for Non-Hamiltonian Systems

We consider an invariant skew-symmetric phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant if the metric tensor field is an integral of motion. We derive the time-dependent skew-symmetric phase-space metric that satisfies the Jacobi identity. The example of non-Hamiltonian systems with linear friction term is considered.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

The Inverse Variational Problem for Autoparallels

We study the problem of the existence of a local quantum scalar field theory in a general affine metric space that in the semiclassical approximation would lead to the autoparallel motion of wave packets, thus providing a deviation of the spinless particle trajectory from the geodesics in the presence of torsion. The problem is shown to be equivalent to the inverse problem of the calculus of variations for the autoparallel motion with additional conditions that the action (if it exists) has to be invariant under time reparametrizations and general coordinate transformations, while depending analytically on the torsion tensor. The problem is proved to have no solution for a generic torsion in four-dimensional spacetime. A solution exists only if the contracted torsion tensor is a gradient of a scalar field. The corresponding field theory describes coupling of matter to the dilaton field.Comment: 13 pages, plain Latex, no figure

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