Orbits in a central force field: Bounded orbits

The nature of boundedness of orbits of a particle moving in a central force field is investigated. General conditions for circular orbits and their stability are discussed. In a bounded central field orbit, a particle moves clockwise or anticlockwise, depending on its angular momentum, and at the same time oscillates between a minimum and a maximum radial distance, defining an inner and an outer annulus. There are generic orbits suggested in popular texts displaying the general features of a central orbit. In this work it is demonstrated that some of these orbits, seemingly possible at the first glance, are not compatible with a central force field. For power law forces, the general nature of boundedness and geometric shape of orbits are investigated.Comment: 11 pages, 15 figures, submitted to Am. J. Phys. Nov 14 2003 (ms # 17211

Feature-rich bifurcations in a simple electronic circuit

A simple electronic circuit with a voltage controlled current source is investigated. The circuit exhibits rich dynamics upon varying the circuit elements such as L,C and R, and the control factor of the current source. Among several other interesting features, the circuit demonstrates two local bifurcations, namely, node to spiral and Hopf bifurcation, and a global homoclinic bifurcation. Phase-portraits corresponding to these bifurcations are presented and the implications of these bifurcations on system stability are discussed. In particular, the circuit parameters corresponding to the onset of Hopf bifurcation may be exploited to design an oscillator with stable frequency and amplitude. The circuit may be easily implemented with nonlinear resistive elements such as diodes or transistors in saturation and a gyrator block as the voltage controlled current source.Comment: 5 pages, 15 figure

Eigenvalues of the Anti-periodic Calogero - Sutherland Model

The U(1) Calogero Sutherland Model (CSM) with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially ordered state space is obtained in an upper triangular form. Consequently the diagonal elements become the energy eigenvalues.Comment: 1 figur

Virtual Displacement in Lagrangian Dynamics

The confusion and ambiguity encountered by students, in understanding virtual displacement and virtual work, is addressed in this article. A definition of virtual displacement is presented that allows one to express them explicitly for both time independent and time dependent constraints. It is observed that for time independent constraints the virtual displacements are the displacements allowed by the constraints. However this is not so for a general time dependent case. For simple physical systems, it is shown that, the work done on virtual displacements by the constraint forces is zero in both the situations. For allowed displacements however, this is not always true. It is also demonstrated that when constraint forces do zero work on virtual displacement, as defined here, we have a solvable mechanical problem. We identify this special class of constraints, physically realized and solvable, as the ideal constraints. The concept of virtual displacement and the principle of zero virtual work by constraint forces are central to both Lagrange's method of undetermined multipliers, and Lagrange's equations in generalized coordinates.Comment: 8 pages, 4 figure

Scaling of Rough Surfaces: Effects of Surface Diffusion on Growth and Roughness Exponents

Random deposition model with surface diffusion over several next nearest neighbours is studied. The results agree with the results obtained by Family for the case of nearest neighbour diffusion [F. Family, J. Phys. A 19(8), L441, 1986]. However for larger diffusion steps, the growth exponent and the roughness exponent show interesting dependence on diffusion length.Comment: 5 pages, 11 figures, 4 table

Calogero-Sutherland Model with Anti-periodic Boundary Conditions: Eigenvalues and Eigenstates

The U(1) Calogero Sutherland Model with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially ordered state space is obtained in an upper triangular form. Consequently the diagonal elements become the energy eigenvalues. The eigenstates are constructed using Young diagram and represented in terms of Jack symmetric polynomials. The eigenstates so obtained are orthonormalized.Comment: 9 pages, 4 figure

Damped bead on a rotating circular hoop - a bifurcation zoo

The evergreen problem of a bead on a rotating hoop shows a multitude of bifurcations when the bead moves with friction. This motion is studied for different values of the damping coefficient and rotational speeds of the hoop. Phase portraits and trajectories corresponding to all different modes of motion of the bead are presented. They illustrate the rich dynamics associated with this simple system. For some range of values of the damping coefficient and rotational speeds of the hoop, linear stability analysis of the equilibrium points is inadequate to classify their nature. A technique involving transformation of coordinates and order of magnitude arguments is presented to examine such cases. This may provide a general framework to investigate other complex systems.Comment: 20 pages, 17 figure

Quasi-solvability of Calogero-Sutherland model with Anti-periodic Boundary Condition

The U(1) Calogero-Sutherland Model with anti-periodic boundary condition is studied. This model is obtained by applying a vertical magnetic field perpendicular to the plane of one dimensional ring of particles. The trigonometric form of the Hamiltonian is recast by using a suitable similarity transformation. The transformed Hamiltonian is shown to be integrable by constructing a set of momentum operators which commutes with the Hamiltonian and amongst themselves. The function space of monomials of several variables remains invariant under the action of these operators. The above properties imply the quasi-solvability of the Hamiltonian under consideration.Comment: 2 figure

Surface properties and scaling behavior of a generalized ballistic deposition model in (1+1)-dimension

The surface exponents, the scaling behavior and the bulk porosity of a generalized ballistic deposition (GBD) model are studied. In nature, there exist particles with varying degrees of stickiness ranging from completely non-sticky to fully sticky. Such particles may adhere to any one of the successively encountered surfaces, depending on a sticking probability %should have the possibility of sticking to any of the %allowed points of contact on the surface with a sticking probability that is governed by the underlying stochastic mechanism. The microscopic configurations possible in this model are much larger than those allowed in existing models of ballistic deposition and competitive growth models that seek to mix ballistic and random deposition processes. In this article, we find the scaling exponents for surface width and porosity for the proposed GBD model. In terms of scaled width $\widetilde{W}$ and scaled time $\tilde{t}$, the numerical data collapse on to a single curve, demonstrating successful scaling with sticking probability p and system size L. Similar scaling behavior is also found for the porosity.Comment: 7 pages, 18 figures, To appear in Physical Review E, Accepted on 27 Jan 201

Gauge momentum operators for the Calogero-Sutherland model with anti-periodic boundary condition

The integrability of a classical Calogero systems with anti-periodic boundary condition is studied. This system is equivalent to the periodic model in the presence of a magnetic field. Gauge momentum operators for the anti-periodic Calogero system are constructed. These operators are hermitian and simultaneously diagonalizable with the Hamiltonian. A general scheme for constructing such momentum operators for trigonometric and hyperbolic Calogero-Sutherland model is proposed. The scheme is applicable for both periodic and anti-periodic boundary conditions. The existence of these momentum operators ensures the integrability of the system. The interaction parameter $\lambda$ is restricted to a certain subset of real numbers. This restriction is in fact essential for the construction of the hermitian gauge momentum operators.Comment: 2 figures, detailed calculation of commutation in general case added in the appendi