Classification of motions admitting linearization of boundary conditions for the relativistic string with massive ends

For the relativistic string with masses at its ends, a classification of motions (world surfaces) admitting a parametrization under which the equations of motion and the boundary conditions are linear (due to the fact that the parameter for the trajectories of the string ends is proportional to the natural parameter) is carried out. These motions can be represented as Fourier series with respect to the eigenfunctions of a generalized Sturm-Liouville problem. The completeness of the family of these eigenfunctions in class C is proved. It is shown that in Minkowski spaces of dimensions2+1 and3+1, the motions in question reduce to a uniform rotation of one or several (spatially coincident) rectilinear strings (the world surface is a helicoid). In spaces of higher dimensions, some other nontrivial motions of this type are also possible.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 187–201, November, 1996.     

On the Reconstruction of a String from Spectral Data

We consider the problem to reconstruct the mass distribution of a string where the mass is concentrated in a finite number of points, or, equivalently, the problem to reconstruct a simply connected mass spring system with unknown masses and stiffness parameters if the following data are given. Problem 1: The spectra of the string and of a modification of the string, or. Problem 2: The spectra of two different modifications of the string. Here a modification of the string is a string which appears if we link the unknown string with another string of known mass distribution. The paper contains a necessary condition for the existence of a solution of Problem 1, and explicit formulas and an algorithm for the solutions of the Problems 1 and 2 under the condition that there exists a solution. For the case that the mass distribution of the unknown string is not discrete we consider the problem to find discrete approximations of this distribution from the respective spectral data. The methods are based on the spectral theory of generalized second order differential operators as developed by M. G. Krein     

Universal and nonperturbative behavior in the one-plaquette model of two-dimensional string theory

The one-plaquette Hamiltonian of large N lattice gauge theory offers a constructive model of a 1+1-dimensional string theory with a stable ground state. The free energy is found to be equivalent to the partition function of a string where the world sheet is discretized by even polygons with signature and the link factor is given by a non-Gaussian propagator. At large, but finite, N we derive the nonperturbative density of states from the WKB wave function and the dispersion relations. This is expressible as an infinite, but convergent, series with the inverse of the hypergeometric function replacing the harmonic oscillator spectrum of the 1+1-dimensional string. In the scaling limit, the series is shown to be finite, containing both the perturbative (asymptotic) expansion of the inverted harmonic oscillator model, and a nonperturbative piece that survives the scaling limit.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 414–429, March, 1994.     

Determination of the world surface of a relativistic string from the trajectory of a massive end

It is shown that the world surface of a relativistic string can be uniquely recovered from the trajectory of a massive end of it if the trajectory is specified in Minkowski space in the form of an arbitrary curve of constant curvature with timelike tangent vector. Conditions are determined for the existence on the world surface of a line that can be identified with the trajectory of the other end of the string and also for the possibility of physical realization of the model in the case when there is no such line.State University, Tver. Translated from Teoreticheskayi i Matematicheskaya Fizika, Vol. 102, No. 1, pp. 150–159, January, 1995.     

Initial—boundary-value problem for the relativistic string with massive ends

It is shown that the initial-boundary-value problem for a relativistic string with masses at the ends can be solved for the most general form of specification of the initial position and initial velocities of the points of the string. An investigation is made of the connection between the freedom in the parametrization of the initial curve and the reparametrization invariance preserving linearity of the equations of motion of the string. The posed problem is solved by extending the solution determined by the initial conditions from the restricted initial region to the entire world surface of the string by means of boundary conditions of various types: the mass at a given end is equal to zero, is infinitely large, or is finite. In the last case it is shown that the problem of the extension reduces to the solution of a normal system of ordinary differential equations. Specific examples are considered.Joint Institute for Nuclear Research, Dubna; State University, Tver. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 253–271, November, 1994.     

Classification of rotational motions for the baryon model “triangle”

A baryon model with three particles (quarks) pairwise connected by relativistic strings forming a curvilinear triangle is considered. The classical analytic solutions corresponding to a uniform plane rotation of the system with arbitrary quark masses m_i are found for this model. The sections of the related world surfaces by the plane t=const are curvilinear triangles composed of segments of a hypocycloid. A complete classification of the types of motion is suggested, based on differences in topological configuration and the presence and number of interior massless points on the string that move at the velocity of light. The classification results from investigating the limiting states as m_i»0. The calculated energy dependence of the angular momentum of the motions creates possibilities to model baryon states on Regge trajectories using these motions.     

Existence and differential geometric properties of continuous families of periodic three-body motions with non-uniform mass distributions

Using the method of analytic continuation in an equivariant differential geometric setting, we exhibit two interesting families of vanishing angular momentum periodic orbits for the Newtonian three-body problem with non-uniform mass distributions having two equal masses which connect at the celebrated figure-8 orbit, exhibited by A. Chenciner and R. Montgomery (2000) in the case of equal masses, and yield a continuous family of periodic three-body motions in the plane.At one end of the family, when the two equal masses are infinitesimal and the third one reaches the value of +1, we arrive at a solution of a double Kepler problem; at the other end of the family, when the third mass is infinitesimal, we have a special case of periodic solution of a restricted three-body problem.     

Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.     

Instanton Effects and Quantum Spectral Curves

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.     

Optimization of the boundary control induced by the third boundary condition

We study the boundary control by the third boundary condition on the left end of a string, the right end being fixed. An optimality criterion based on the minimization of an integral of a linear combination of the control itself and its antiderivative raised to an arbitrary power p ≥ 1 is established. A method is developed permitting one to find a control satisfying this optimality criterion and write it out in closed form. The uniqueness of the optimal control for p > 1 is proved.     

A terminal-boundary value problem that describes the process of damping the vibrations of a rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times

An explicit analytic expression is obtained for optimal boundary controls exercised on one end of a string by a displacement or by an elastic force under a model nonlocal boundary condition of one of four types.     

Optimization of the boundary control by a displacement or by an elastic force on one end of a string under a model nonlocal boundary condition

An explicit analytic expression is obtained for optimal boundary controls exercised on one end of a string by a displacement or by an elastic force under a model nonlocal boundary condition of one of four types.     

The limiting steady rotations of a body on a string with a suspension point on the axis of symmetry

The steady motions of an axisymmetrical rigid body suspended from a fixed base by a weightless undeformable rod or a non-twisting inextensible string are investigated. The case when the rod is fastened to the body at a point situated on its axis of dynamic symmetry is considered. All types of limiting equilibrium configurations which are possible when there is an unlimited increase in the angular velocity of rotation of the system about the vertical are analysed. Domains in which each type of limiting regular precession and permanent rotation can exist are constructed in the space of dimensionless parameters, and the nature of their asymptotic behaviour when the angular velocity increases is determined. The limiting motions which are possible in the case of suspension on a rod and impossible in the case of suspension on a string are investigated.     

Reparameterization-Invariant Reduction in the Hamiltonian Description of a Relativistic String

We study the time-reparameterization-invariant dynamics of an open relativistic string using the generalized Dirac–Hamilton theory and resolving the constraints of the first kind. The reparameterization-invariant evolution variable is the time coordinate of the string center of mass. Using a transformation that preserves the diffeomorphism group of the generalized Hamiltonian and the Poincaré covariance of the local constraints, we segregate the center-of-mass coordinates from the local degrees of freedom of the string. We identify the time coordinate of the string center of mass and the proper time measured in the string frame of reference using the Levi-Civita–Shanmugadhasan canonical transformation, which transforms the global constraint (the mass shell) in the new momentum such that the Hamiltonian reduction does not require the corresponding gauge condition. Resolving the local constraints, we obtain an equivalent reduced system whose Hamiltonian describes the evolution w.r.t. the proper time of the string center of mass. The Röhrlich quantum relativistic string theory, which includes the Virasoro operators L _n only with n > 0, is used to quantize this system. In our approach, the standard problems that appear in the traditional quantization scheme, including the space–time dimension D = 26 and the tachyon emergence, arise only in the case of a massless string, M ² = 0.     

Representations of fractional Brownian motion using vibrating strings

In this paper, we show that the moving average and series representations of fractional Brownian motion can be obtained using the spectral theory of vibrating strings. The representations are shown to be consequences of general theorems valid for a large class of second-order processes with stationary increments. Specifically, we use the 1–1 relation discovered by M.G. Krein between spectral measures of continuous second-order processes with stationary increments and differential equations describing the vibrations of a string with a certain length and mass distribution.     

A string model of “exotic” particles

The bosonic string is investigated using the approach to 4D string dynamics previously proposed by the author. The physical states and the mass spectrum are constructed. The scale invariance of the theory results in the linear dependence of the squared mass μ² on the “spin” . Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 223–244 May. 1998.     

Small Motions of a Partially Dissipative Hydrodynamic System

A new problem is discussed: small motions of the hydrodynamic system {viscous fluid + set of ideal fluids}. The conditions are determined for the existence of a strong solution of the initial/boundary-value problem corresponding to small motions of the investigated hydrodynamic system, subject to the condition that the system is statically stable in the linear approximation.     

Asymptotic motions and the inversion of the lagrange-dirichlet theorem

The motions of natural mechanical systems which tend to an equilibrium position as time increases without limit are studied. The degenerate case when several frequencies of small oscillations vanish is explained. An existence theorem is proved for asymptotic trajectories on the assumption that the Maclaurin series for the potential energy has the form V₂ + V_m + V_{m + 1} + …(V₈ is a homogeneous form of degree s) and the function V₂ + V_m does not have a local minimum at the equilibrium position. We proved earlier a claim /1, 2/ about the asymptotic motions for the special case when V₂ 0. This theorem is used to solve the question of the existence of asymptotic trajectories in the case of simple and unimodal singularities of the potential energy, for which “canonical” normal forms are known. Similar assertions also hold for the equilibrium positions of gradient dynamic systems. The existence of a trajectory, asymptotic to the equilibrium position, naturally implies that this position is unstable in Lyapunov's sense.     

Inverse problem for a damped Stieltjes string from parts of spectra

We solve the following inverse problem for boundary value problems generated by the difference equations describing the motion of a Stieltjes string (a thread with beads). Given are certain parts of the spectra of two boundary value problems with two different Robin conditions at the left end and the same damping condition at the right end. From these two partial spectra, the difference of the Robin parameters, the damping constant, and the total length of the string, find the values of the point masses, and of the lengths of the intervals between them. We establish necessary and sufficient conditions for two sets of complex numbers to be the eigenvalues of two such boundary value problems and give a constructive solution of the inverse problem.