103 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
Fractional Hamilton formalism within Caputo's derivative
In this paper we develop a fractional Hamiltonian formulation for dynamic
systems defined in terms of fractional Caputo derivatives. Expressions for
fractional canonical momenta and fractional canonical Hamiltonian are given,
and a set of fractional Hamiltonian equations are obtained. Using an example,
it is shown that the canonical fractional Hamiltonian and the fractional
Euler-Lagrange formulations lead to the same set of equations.Comment: 8 page
Fractional conservation laws in optimal control theory
Using the recent formulation of Noether's theorem for the problems of the
calculus of variations with fractional derivatives, the Lagrange multiplier
technique, and the fractional Euler-Lagrange equations, we prove a Noether-like
theorem to the more general context of the fractional optimal control. As a
corollary, it follows that in the fractional case the autonomous Hamiltonian
does not define anymore a conservation law. Instead, it is proved that the
fractional conservation law adds to the Hamiltonian a new term which depends on
the fractional-order of differentiation, the generalized momentum, and the
fractional derivative of the state variable.Comment: The original publication is available at http://www.springerlink.com
Nonlinear Dynamic
Fractional variational calculus of variable order
We study the fundamental problem of the calculus of variations with variable
order fractional operators. Fractional integrals are considered in the sense of
Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the
authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted
13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The
Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck),
Operator Theory: Advances and Applications, Birkh\"auser Verlag
(http://www.springer.com/series/4850
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
Fractional Euler-Lagrange differential equations via Caputo derivatives
We review some recent results of the fractional variational calculus.
Necessary optimality conditions of Euler-Lagrange type for functionals with a
Lagrangian containing left and right Caputo derivatives are given. Several
problems are considered: with fixed or free boundary conditions, and in
presence of integral constraints that also depend on Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form will
appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu
et al. (eds.), Springer New York, 2012, DOI:10.1007/978-1-4614-0457-6_9, in
pres
Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes
We study the fractional gravity for spacetimes with non-integer dimensions.
Our constructions are based on a geometric formalism with the fractional Caputo
derivative and integral calculus adapted to nonolonomic distributions. This
allows us to define a fractional spacetime geometry with fundamental
geometric/physical objects and a generalized tensor calculus all being similar
to respective integer dimension constructions. Such models of fractional
gravity mimic the Einstein gravity theory and various Lagrange-Finsler and
Hamilton-Cartan generalizations in nonholonomic variables. The approach
suggests a number of new implications for gravity and matter field theories
with singular, stochastic, kinetic, fractal, memory etc processes. We prove
that the fractional gravitational field equations can be integrated in very
general forms following the anholonomic deformation method for constructing
exact solutions. Finally, we study some examples of fractional black hole
solutions, fractional ellipsoid gravitational configurations and imbedding of
such objects in fractional solitonic backgrounds.Comment: latex2e, 11pt, 40 pages with table of conten
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
Variational Problems Involving a Caputo-Type Fractional Derivative
We study calculus of variations problems, where the Lagrange function depends on the
Caputo-Katugampola fractional derivative. This type of fractional operator is a generalization
of the Caputo and the Caputo–Hadamard fractional derivatives, with dependence on a real
parameter ρ. We present sufficient and necessary conditions of first and second order to
determine the extremizers of a functional. The cases of integral and holomonic constraints are also considered
- …