9 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
Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense
A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients
We present a direct approach to the construction of Lagrangians for a large
class of one-dimensional dynamical systems with a simple dependence (monomial
or polynomial) on the velocity. We rederive and generalize some recent results
and find Lagrangian formulations which seem to be new. Some of the considered
systems (e.g., motions with the friction proportional to the velocity and to
the square of the velocity) admit infinite families of different Lagrangian
formulations.Comment: 17 page
Generalized Hamilton's Principle with Fractional Derivatives
We generalize Hamilton's principle with fractional derivatives in Lagrangian
L(t,y(t),{}_0D_t^\al y(t),\alpha) so that the function and the order of
fractional derivative are varied in the minimization procedure. We
derive stationarity conditions and discuss them through several examples
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