73 research outputs found
Extending Bauer's corollary to fractional derivatives
We comment on the method of Dreisigmeyer and Young [D. W. Dreisigmeyer and P.
M. Young, J. Phys. A \textbf{36}, 8297, (2003)] to model nonconservative
systems with fractional derivatives. It was previously hoped that using
fractional derivatives in an action would allow us to derive a single retarded
equation of motion using a variational principle. It is proven that, under
certain reasonable assumptions, the method of Dreisigmeyer and Young fails.Comment: Accepted Journal of Physics A at www.iop.org/EJ/journal/JPhys
Nonconservative Lagrangian mechanics II: purely causal equations of motion
This work builds on the Volterra series formalism presented in [D. W.
Dreisigmeyer and P. M. Young, J. Phys. A \textbf{36}, 8297, (2003)] to model
nonconservative systems. Here we treat Lagrangians and actions as `time
dependent' Volterra series. We present a new family of kernels to be used in
these Volterra series that allow us to derive a single retarded equation of
motion using a variational principle
Time-Fractional KdV Equation: Formulation and Solution using Variational Methods
In this work, the semi-inverse method has been used to derive the Lagrangian
of the Korteweg-de Vries (KdV) equation. Then, the time operator of the
Lagrangian of the KdV equation has been transformed into fractional domain in
terms of the left-Riemann-Liouville fractional differential operator. The
variational of the functional of this Lagrangian leads neatly to Euler-Lagrange
equation. Via Agrawal's method, one can easily derive the time-fractional KdV
equation from this Euler-Lagrange equation. Remarkably, the time-fractional
term in the resulting KdV equation is obtained in Riesz fractional derivative
in a direct manner. As a second step, the derived time-fractional KdV equation
is solved using He's variational-iteration method. The calculations are carried
out using initial condition depends on the nonlinear and dispersion
coefficients of the KdV equation. We remark that more pronounced effects and
deeper insight into the formation and properties of the resulting solitary wave
by additionally considering the fractional order derivative beside the
nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure
Kinematics and hydrodynamics of spinning particles
In the first part (Sections 1 and 2) of this paper --starting from the Pauli
current, in the ordinary tensorial language-- we obtain the decomposition of
the non-relativistic field velocity into two orthogonal parts: (i) the
"classical part, that is, the 3-velocity w = p/m OF the center-of-mass (CM),
and (ii) the so-called "quantum" part, that is, the 3-velocity V of the motion
IN the CM frame (namely, the internal "spin motion" or zitterbewegung). By
inserting such a complete, composite expression of the velocity into the
kinetic energy term of the non-relativistic classical (i.e., newtonian)
lagrangian, we straightforwardly get the appearance of the so-called "quantum
potential" associated, as it is known, with the Madelung fluid. This result
carries further evidence that the quantum behaviour of micro-systems can be
adirect consequence of the fundamental existence of spin. In the second part
(Sections 3 and 4), we fix our attention on the total 3-velocity v = w + V, it
being now necessary to pass to relativistic (classical) physics; and we show
that the proper time entering the definition of the four-velocity v^mu for
spinning particles has to be the proper time tau of the CM frame. Inserting the
correct Lorentz factor into the definition of v^mu leads to completely new
kinematical properties for v_mu v^mu. The important constraint p_mu v^mu = m,
identically true for scalar particles, but just assumed a priori in all
previous spinning particle theories, is herein derived in a self-consistent
way.Comment: LaTeX file; needs kapproc.st
Fractional Dynamics of Relativistic Particle
Fractional dynamics of relativistic particle is discussed. Derivatives of
fractional orders with respect to proper time describe long-term memory effects
that correspond to intrinsic dissipative processes. Relativistic particle
subjected to a non-potential four-force is considered as a nonholonomic system.
The nonholonomic constraint in four-dimensional space-time represents the
relativistic invariance by the equation for four-velocity u_{\mu}
u^{\mu}+c^2=0, where c is a speed of light in vacuum. In the general case, the
fractional dynamics of relativistic particle is described as non-Hamiltonian
and dissipative. Conditions for fractional relativistic particle to be a
Hamiltonian system are considered
Geometry and field theory in multi-fractional spacetime
We construct a theory of fields living on continuous geometries with
fractional Hausdorff and spectral dimensions, focussing on a flat background
analogous to Minkowski spacetime. After reviewing the properties of fractional
spaces with fixed dimension, presented in a companion paper, we generalize to a
multi-fractional scenario inspired by multi-fractal geometry, where the
dimension changes with the scale. This is related to the renormalization group
properties of fractional field theories, illustrated by the example of a scalar
field. Depending on the symmetries of the Lagrangian, one can define two
models. In one of them, the effective dimension flows from 2 in the ultraviolet
(UV) and geometry constrains the infrared limit to be four-dimensional. At the
UV critical value, the model is rendered power-counting renormalizable.
However, this is not the most fundamental regime. Compelling arguments of
fractal geometry require an extension of the fractional action measure to
complex order. In doing so, we obtain a hierarchy of scales characterizing
different geometric regimes. At very small scales, discrete symmetries emerge
and the notion of a continuous spacetime begins to blur, until one reaches a
fundamental scale and an ultra-microscopic fractal structure. This fine
hierarchy of geometries has implications for non-commutative theories and
discrete quantum gravity. In the latter case, the present model can be viewed
as a top-down realization of a quantum-discrete to classical-continuum
transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and
improved (especially section 4.5), typos corrected, references added; v4:
further typos correcte
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