From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.Comment: To appear in: Proceedings of Symmetries in Science XV, Journal of Physics: Conference Series, (2012

Modular Solutions to Equations of Generalized Halphen Type

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have $q$--series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.Comment: PlainTeX 36gs. (Formula for Hecke operator corrected.

From geodesics of the multipole solutions to the perturbed Kepler problem

A static and axisymmetric solution of the Einstein vacuum equations with a finite number of Relativistic Multipole Moments (RMM) is written in MSA coordinates up to certain order of approximation, and the structure of its metric components is explicitly shown. From the equation of equatorial geodesics we obtain the Binet equation for the orbits and it allows us to determine the gravitational potential that leads to the equivalent classical orbital equations of the perturbed Kepler problem. The relativistic corrections to Keplerian motion are provided by the different contributions of the RMM of the source starting from the Monopole (Schwarzschild correction). In particular, the perihelion precession of the orbit is calculated in terms of the quadrupole and 2$^4$-pole moments. Since the MSA coordinates generalize the Schwarzschild coordinates, the result obtained allows measurement of the relevance of the quadrupole moment in the first order correction to the perihelion frequency-shift

Characterization of a Double Mesospheric Bore Over Europe

Observations of a pair of mesospheric bore disturbances that propagated through the nighttime mesosphere over Europe are presented. The observations were made at the Padua Observatory, Asiago (45.9\ub0N, 11.5\ub0E), by the Boston University all-sky imager on 11 March 2013. The bores appeared over the northwest horizon, approximately 30 min apart, and propagated toward the southeast. Using additional satellite and radar data, we present evidence indicating the bores originated in the mesosphere from a single, larger-scale mesospheric disturbance propagating through the mesopause region. Furthermore, the large-scale mesospheric disturbance appeared to be associated with an intense weather disturbance that moved southeastward over the United Kingdom and western Europe during 10 and 11 March

Quasi-doubly periodic solutions to a generalized Lame equation

We consider the algebraic form of a generalized Lame equation with five free parameters. By introducing a generalization of Jacobi's elliptic functions we transform this equation to a 1-dim time-independent Schroedinger equation with (quasi-doubly) periodic potential. We show that only for a finite set of integral values for the five parameters quasi-doubly periodic eigenfunctions expressible in terms of generalized Jacobi functions exist. For this purpose we also establish a relation to the generalized Ince equation.Comment: 15 pages,1 table, accepted for publication in Journal of Physics

Thermodynamic instability and first-order phase transition in an ideal Bose gas

We conduct a rigorous investigation into the thermodynamic instability of ideal Bose gas confined in a cubic box, without assuming thermodynamic limit nor continuous approximation. Based on the exact expression of canonical partition function, we perform numerical computations up to the number of particles one million. We report that if the number of particles is equal to or greater than a certain critical value, which turns out to be 7616, the ideal Bose gas subject to Dirichlet boundary condition reveals a thermodynamic instability. Accordingly we demonstrate - for the first time - that, a system consisting of finite number of particles can exhibit a discontinuous phase transition featuring a genuine mathematical singularity, provided we keep not volume but pressure constant. The specific number, 7616 can be regarded as a characteristic number of 'cube' that is the geometric shape of the box.Comment: 1+21 pages; 3 figures (2 color and 1 B/W); Final version to appear in Physical Review A. Title changed from the previous one, "7616: Critical number of ideal Bose gas confined in a cubic box

Peer mentorship and positive effects on student mentor and mentee retention and academic success

This study examined how the introduction of peer mentorship in an undergraduate health and social welfare programme at a large northern university affected student learning. Using an ethnographic case study approach, the study draws upon data collected from a small group of mentors and their mentees over a period of one academic year using interviews, reflective journals, assessment and course evaluation data. Analysis of the data collected identified a number of key findings: peer mentorship improves assessment performance for both mentee and mentor; reduces stress and anxiety, enhances participation and engagement in the academic community, and adds value to student outcomes

Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.Comment: 11 page

Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems

An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We show here that this criterion can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved. Since our criterion for instability is local in coordinate space, these results provide a new and minimal method for achieving control of a chaotic system