1,721,042 research outputs found
Computing with Hamiltonian operators
No full textHamiltonian operators for partial differential equations are ubiquitous in mathematical models of theoretical and applied physics. In this paper the new Reduce package cde for computations hl{with} Hamiltonian operators is presented. cde can verify the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators, and hl{it can compute} the Lie derivative of a Hamiltonian operator with respect to a vector field. hl{More generally, it can compute with} (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of mathematical or theoretical physics
Effect of tire slip losses on the energy demand and fuel consumption of a light-duty vehicle
No full textThe contribution of the tire-road slip of traction wheels to the total resistance opposing the motion of a light-duty commercial vehicle has been investigated through the simulation of several homologation and custom driving cycles. The calculation of the contribution of the tire slip losses was based on the estimation of the longitudinal tire slip, by means of Pacejka’s MF5.2 tire model. In this work, the computational steps required to evaluate this contribution were implemented in a previously developed fuel consumption simulation tool. Simulations were performed under several vehicle loading conditions and tire inflation pressures on traction and non-traction wheels, and considering different tire-road adherence conditions, in order to obtain a characterization of the tire slip losses over a wide range of working conditions. An analysis of the results shows that, although the contribution of tire slip losses to the total vehicle energy demand and fuel consumption may be relevant – especially under low-load, low adherence conditions – the sensitivity of the average on-cycle vehicle energy/fuel consumption to changes in the tire inflation pressure is only affected slightly by tire slip losses. Therefore, tire slip losses can be neglected in practice, when the aim of a simulation is to optimize the tire pressure to achieve average vehicle working conditions over a driving cycle
Bi-Hamiltonian structures of WDVV-type
Full text linkWe study a class of nonlinear partial differential equations (PDEs) that admit the same bi-Hamiltonian structure as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle-Pot & euml;min form, which are compatible. Using various equivalence groups, we classify such equations in two-component and three-component cases. In a four-component case we add further evidence to the conjecture that there exists only one integrable system of the above type. Finally, we give an example of the six-component system with required bi-Hamiltonian structure. To streamline the symbolic computation, we develop an algorithm to find the aforementioned Hamiltonian operators, which includes putting forward a conjecture on the structure of the metric parameterizing the first-order Hamiltonian operator
Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds
No full textPoisson brackets between conserved quantities are a fundamental tool in the theory of integrable systems. The subclass of weakly nonlocal Poisson brackets occurs in many significant integrable systems. Proving that a weakly nonlocal differential operator defines a Poisson bracket can be challenging. We propose a computational approach to this problem through the identification of such operators with superfunctions on supermanifolds
Homogeneous Hamiltonian operators and the theory of coverings
No full textA new method (by Kersten, Krasil'shchik and Verbovetsky), based on the theory of differential coverings, allows to relate a system of PDEs with a differential operator in such a way that the operator maps conserved quantities into symmetries of the system of PDEs. When applied to a quasilinear first-order system of PDEs and a Dubrovin–Novikov homogeneous Hamiltonian operator the method yields conditions on the operator and the system that have interesting differential and projective geometric interpretations
WDVV equations and invariant bi-Hamiltonian formalism
Full text linkAbstract The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository
Classification of bi-Hamiltonian pairs extended by isometries
No full textThe aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin-Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten-Dijkgraaf-Verlinde-Verlinde equations
Projective-geometric aspects of bi-Hamiltonian structures of KdV type
No full textWe introduce the problem of classification of bi-Hamiltonian structures of KdV type under projective reciprocal transformations. This problem
leads naturally to studying the compatibility of a first order localizable homogeneous Hamiltonian operator with a higher order homogeneous Hamiltonian operator. We study the simplest third-order case where the orbit contains a constant operator. Computations with weakly non local Hamiltonian operators have been made by techniques developed in a previous paper
A translation of the T. Levi-Civita paper “Interpretazione Gruppale degli Integrali di un Sistema Canonico” (Rend. Acc. Lincei, 1899, s. 3a, vol. VII, pp. 235–238)
No full textIn this paper we provide a translation of a paper by T. Levi-Civita,
published in 1899, about the correspondence between symmetries and
conservation laws for Hamilton’s equations. We discuss the results of
this paper and their relationship with the more general classical re-
sults by E. Noether
- …
