Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity

Abstract

Nonstandard Lagrangians are generating functions of different equations of motion. They have gained increasing importance in many different fields. In fact, nonstandard Lagrangians date back to 1978, when Arnold entitled them "nonnatural" in his classic book, Mathematical Methods of Classical Mechanics (Springer, New York. 1978). In applied mathematics, most dynamical equations can be obtained by using generating Lagrangian functions (e.g., power-law and exponential Lagrangians), which has been shown by mathematicians, who have also demonstrated that there is an infinite number of such functions. Besides this interesting field, the topic of fractional calculus of variations has gained growing importance because of its wide application in different fields of science. In this paper, we generalize the fractional actionlike variational approach for the case of a nonstandard exponential Lagrangian. To appreciate this new approach, we explore some of its main consequences in Einstein's general relativity. Some results are revealed and discussed accordingly mainly the transition from general relativity to complex relativity and emergence of a discrete gravitational coupling constant. PACS Nos.: 02.30.Xx, 04.20.-q, 04.20.Fy. Résumé : Les Lagrangiens non standard sont des fonctions génératrices d'équations différentes du mouvement. Ils sont de plus en plus importants dans différents domaines scientifiques. En fait, les Lagrangiens non standard datent de 1978, lorsque Arnold les a baptisés de non naturels dans son livre Méthodes Mathématiques de la Mécanique Classique (Springer, New York. 1978). En mathématiques appliquées, la plupart des équations dynamiques peuvent être obtenues de fonctions génératrices de Lagrange (par exemple, des Lagrangiens en loi de puissance ou exponentiels), ce qui a été illustré par des mathématiciens qui ont aussi démontré qu'il y a un nombre infini de telles fonctions. À côté de ce champ, le calcul fractionnaire des variations croît en importance, à cause de son large domaine d'applications en sciences. Ici, nous généralisons l'approche variationnelle de type action fractionnelle pour un Lagrangien exponentiel non standard. Pour mieux apprécier cette nouvelle approche, nous explorons certaines de ses conséquences majeures en relativité générale (RG) d'Einstein. Quelques résultats sont présentés et nous analysons surtout le passage de la RG à la relativité complexe et une constante de couplage gravitationnel discrète. [Traduit par la Rédaction]

Barotropic FRW cosmologies with Chiellini damping

“…Recently, the Riccati framework has been developed for scalar field FRW cosmologies by Harko et al [12], while El-Nabulsi obtained a Riccati equation for the 'generalized time-dependent Hubble parameter' H g (t) = Γ(α)t 1−α H(t), where Γ(α) is the Euler Gamma function and α is an arbitrary real number. Notice that α = 1 corresponds to the standard FRW cosmology [13,14]. In fact, by introducing the modified Hubble parameter in conformal time H u (η) = u ′ /γu [6][7][8][9][10][11], the Riccati equation (3) can be reduced to the linear second order equation…”

Non-Standard Lagrangians with Higher-Order Derivatives and the Hamiltonian Formalism

“…For the case of a time-independent 3rd-order derivative NSL of the form Lðq ... ; € q; _ q; qÞ ¼ q ð3Þ , Eq. (19) gives q ð6Þ þ 3q ð5Þ q ð4Þ þ q ð4Þ…”

Non-standard Lagrangians in rotational dynamics and the modified Navier–Stokes equation

“…They were introduced by Arnold since 1978 [1], R. A. El-Nabulsi (B) College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641112, Sichuan, China e-mail:nabulsiahmadrami@yahoo.fr but they were ignored for a good period of time due to the lack of their Hamiltonian formalisms. Their roles in the theory of differential equations [2][3][4], dissipative systems [5][6][7][8][9][10][11][12] and theoretical physics [13][14][15][16][17] are well appreciated. Despite the fact that a solid and comprehensive Hamiltonian formulation is missed, NSL are considered to be good candidates to explain dissipative dynamical systems.…”