595 research outputs found
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
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
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
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
Generalized ballistic deposition in 2 dimensions : scaling of surface width, porosity and conductivity
A deposition process with particles having realistic intermediate stickiness
is studied in 2+1 dimensions. At each stage of the deposition process, for any
given configuration, a newly depositing particle gives rise to allowed set of
configurations that are vastly larger than those for deposition of a mixture of
purely non-sticky (random like) and purely sticky (ballistic like) particles.
We obtain scaling behavior and demonstrate collapse of scaled data for surface
width and porosity. Scaling of conductivity, when a porous structure thus
formed, is saturated with conductive fluid, e.g. brine, is studied. The results
obtained are in good agreement with Archie's law for porous sedimentary rocks.Comment: 9 pages, 20 figure
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
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
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
and scaled time , 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
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
Continuous Time Random Walk with time-dependent jump probability : A Direct Probabilistic Approach
We investigate the dynamics of a particle executing a general Continuous Time
Random Walk (CTRW) in three dimensions under the influence of arbitrary
time-varying external fields. Contrary to the general approach in recent works,
our method invokes neither the Fractional Fokker-Planck equation (FFPE) nor the
Stochastic Langevin Equation (SLE). Rather, we use rigorous probability
arguments to derive the general expression for moments of all orders of the
position probability density of the random walker for arbitrary waiting time
density and jump probability density. Closed form expression for the position
probability density is derived for the memoryless condition. For the special
case of CTRW on a one-dimensional lattice with nearest neighbour jumps, our
equations confirm the phenomena of "death of linear response" and
"field-induced dispersion" for sub-diffusion pointed out in [I. M. Sokolov and
J. Klafter, Phys. Rev. Lett. {\bf 97}, 140602 (2006)]. However, our analysis
produces additional terms in the expressions for higher moments, which have
non-trivial consequences. We show that the disappearance of these terms result
from the approximation involved in taking the continuum limit to derive the
generalized Fokker-Planck equation. This establishes the incompleteness of the
FFPE formulation, especially in predicting the higher moments. We also discuss
how different predictions of the model alter if we allow jumps beyond nearest
neighbours and possible circumstances where this becomes relevant.Comment: 26 pages, 2 figure
Surface morphology of a modified ballistic deposition model
The surface and bulk properties of a modified ballistic deposition model are
investigated. The deposition rule interpolates between nearest and next-nearest
neighbor ballistic deposition and the random deposition models. The stickiness
of the depositing particle is controlled by a parameter and the type of
inter-particle force. Two such forces are considered - Coulomb and van der
Waals type. The interface width shows three distinct growth regions before
eventual saturation. The rate of growth depends more strongly on the stickiness
parameter than on the type of inter-particle force. However, the porosity of
the deposits is strongly influenced by the inter-particle force.Comment: 6 pages, 17 figure
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