The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange’s characteristics method (a new approach) for solving nonlinear fractional partial differential equations. The key of this results is the fractional Taylor’s seriesf(x + h) = E _α(h^αD^α)f(x) whereE _α(·) is the Mittag-Leffler function.     

A Lagrangian stochastic model for nonpassive particle diffusion in turbulent flows

In this paper, we present a Lagrangian stochastic model for heavy particle dispersion in turbulence. The model includes the equation of motion for a heavy particle and a stochastic approach to predicting the velocity of fluid elements along the heavy particle trajectory. The trajectory crossing effect of heavy particles is described by using an Ito type stochastic differential equation combined with a fractional Langevin equation. The comparison of the predicted dispersion of four heavy particles with the observations shows that the model is potentially useful but requires further development.     

Variationally trivial Lagrangians and locally variational differential equations of arbitrary order

We continue our investigation of the Lagrangian formalism on jet bundle extensions using Fock space methods. We are able to provide the most general form of a variationally trivial Lagrangian of arbitrary order and we also give a generic expression for the most general locally variational differential equation. As anticipated in the literature, these expressions involve some special combinations of the highest order derivatives, called hyper-Jacobians.     

Variational derivation of two-component Camassa–Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa–Holm system (1). We show that the two-component Camassa–Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.     

Fractionalization of the complex-valued Brownian motion of order n using Riemann–Liouville derivative. Applications to mathematical finance and stochastic mechanics

The (complex-valued) Brownian motion of order n is defined as the limit of a random walk on the complex roots of the unity. Real-valued fractional noises are obtained as fractional derivatives of the Gaussian white noise (or order two). Here one combines these two approaches and one considers the new class of fractional noises obtained as fractional derivative of the complex-valued Brownian motion of order n. The key of the approach is the relation between differential and fractional differential provided by the fractional Taylor’s series of analytic function , where E is the Mittag–Leffler function on the one hand, and the generalized Maruyama’s notation, on the other hand. Some questions are revisited such as the definition of fractional Brownian motion as integral w.r.t. (dt), and the exponential growth equation driven by fractional Brownian motion, to which a new solution is proposed. As a first illustrative example of application, in mathematical finance, one proposes a new approach to the optimal management of a stochastic portfolio of fractional order via the Lagrange variational technique applied to the state moment dynamical equations. In the second example, one deals with non-random Lagrangian mechanics of fractional order. The last example proposes a new approach to fractional stochastic mechanics, and the solution so obtained gives rise to the question as to whether physical systems would not have their own internal random times.     

Fractional variational problems with the Riesz-Caputo derivative

In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem.     

KAM theory in configuration space

A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser. This research has been supported by the Nuffields Foundation under grant SCI/180/173/G and by the Stiftung Volkswagenwerk.     

Conservation laws of KdV equation with time dependent coefficients

We determine conservation laws of the generalized KdV equation of time dependent variable coefficients of the linear damping and dispersion. The underlying equation is not derivable from a variational principle and hence one cannot use Noether’s theorem here to construct conservation laws as there is no Lagrangian. However, we show that by utilizing the new conservation theorem and the partial Lagrangian approach one can construct a number of local and nonlocal conservation laws for the underlying equation.     

Fractional variational problems from extended exponentially fractional integral

In this letter, we study the fractional variational problems from extended exponentially fractional integral. Both the Lagrangian and Hamiltonian formulations are explored and discussed in some details. Some interesting consequences are revealed.     

Fractional Nottale’s Scale Relativity and emergence of complexified gravity

Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale’s Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.     

On different geometric formulations of Lagrangian formalism

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the -spectral sequence. On one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.     

The Lorentz-invariant deformation of the Whitham system for the nonlinear Klein-Gordon equation

We consider a deformation of the Whitham system for the nonlinear Klein-Gordon equation. This deformation has a Lorentz-invariant form. Using the Lagrangian formalism of the original system, we obtain the first nontrivial correction to the Whitham system in the Lorentz-invariant approach.     

Non-differentiable embedding of Lagrangian systems and partial differential equations

We develop the non-differentiable embedding theory of differential operators and Lagrangian systems using a new operator on non-differentiable functions. We then construct the corresponding calculus of variations and we derive the associated non-differentiable Euler-Lagrange equation, and apply this formalism to the study of PDEs. First, we extend the characteristics method to the non-differentiable case. We prove that non-differentiable characteristics for the Navier-Stokes equation correspond to extremals of an explicit non-differentiable Lagrangian system. Second, we prove that the solutions of the Schrödinger equation are non-differentiable extremals of the Newton?s Lagrangian.     

Variational formulations of differential equations and asymmetric fractional embedding

Variational formulations for classical dissipative equations, namely friction and diffusion equations, are given by means of fractional derivatives. In this way, the solutions of those equations are exactly the extremal of some fractional Lagrangian actions. The formalism used is a generalization of the fractional embedding developed by Cresson [Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007) 033504], where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives. Moreover, this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle.     

Existence of solutions for a fractional advection-dispersion equation with impulsive effects via variational approach

In this paper, based on the variational approach and iterative technique, the existence of nontrivial weak solutions is derived for a fractional advection-dispersion equation with impulsive effects, and the nonlinear term of fractional advection-dispersion equation contain the fractional order derivative. In addition, an example is presented as an application of the main result.     

Existence of solutions for fractional differential equation with p-Laplacian through variational method

In this paper, a class of fractional differential equation with p-Laplacian operator is studied based on the variational approach. Combining the mountain pass theorem with iterative technique, the existence of at least one nontrivial solution for our equation is obtained. Additionally, we demonstrate the application of our main result through an example.     

Variational formulation and efficient implementation for solving the tempered fractional problems

Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power‐law instead of pure power‐law for the CTRW model in characterizing the waiting time and jump length of the motion of particles. This paper focuses on providing the variational formulation and efficient implementation for solving the corresponding deterministic/macroscopic models, including the space tempered fractional equation and time tempered fractional equation. The convergence, numerical stability, and a series of variational equalities are theoretically proved. And the theoretical results are confirmed by numerical experiments.     

Lagrangians for self-adjoint and non-self-adjoint equations

It is relatively easy to obtain a variational formulation for a self-adjoint equation, while it is difficult or impossible to do this for a non-self-adjoint one. This work suggests an alternative approach to the derivation of the Lagrangian for both cases. A trial-Lagrangian is constructed, and the exact variational formulation can be obtained after the identification of the unknown function involved in the trial-Lagrangian.     

The DuBois–Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler–Lagrange Equation Involving Only Derivatives of Caputo

Derivatives and integrals of noninteger order were introduced more than three centuries ago but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler–Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois–Reymond lemma and showing how Euler–Lagrange equations involving only Caputo derivatives can be obtained.