scholarly journals Preserving first integrals and volume forms of additively split systems

2007 ◽  
Vol 27 (2) ◽  
pp. 381-405 ◽  
Author(s):  
Philippe Chartier ◽  
Ander Murua
2010 ◽  
Vol 32 (2) ◽  
pp. 107-120
Author(s):  
Pham Chi Vinh ◽  
Trinh Thi Thanh Hue ◽  
Dinh Van Quang ◽  
Nguyen Thi Khanh Linh ◽  
Nguyen Thi Nam

The method of first integrals (MFI) based on the equation of motion for the displacement vector, or  based on the one for the traction vector was introduced  recently in order to find explicit secular equations of Rayleigh waves whose characteristic equations (i.e the equations determining the attenuation factor) are fully quartic or are of higher order (then the classical approach is not applicable). In this paper it is shown that, not only to Rayleigh waves,  the MFI can be applicable also to other waves by running it on the equations for mixed vectors. In particular: (i) By applying the MFI  to the equations for the displacement-traction vector we get the explicit dispersion equations of Stoneley waves in twinned crystals (ii)  Running the MFI on the equations for the traction-electric induction vector and the traction-electrical potential vector provides the explicit dispersion equations of SH-waves in piezoelastic materials. The obtained dispersion equations are identical with the ones previously derived using the method of polarization vector, but the procedure of driving them is more simple.


Author(s):  
Peter Mann

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.


Probus ◽  
2021 ◽  
Vol 33 (1) ◽  
pp. 57-93
Author(s):  
Ana Lívia Agostinho ◽  
Larry M. Hyman

Abstract Creole languages have generally not figured prominently in cross-linguistic studies of word-prosodic typology. In this paper, we present a phonological analysis of the prosodic system of Lung’Ie or Principense (ISO 639-3 code: pre), a Portuguese-lexifier creole language spoken in São Tomé and Príncipe. Lung’Ie has produced a unique result of the contact between the two different prosodic systems common in creolization: a stress-accent lexifier and tone language substrates. The language has a restrictive privative H/Ø tone system, in which the /H/ is culminative, but non-obligatory. Since rising and falling tones are contrastive on long vowels, the tone must be marked underlyingly. While it is clear that tonal indications are needed, Lung’Ie reveals two properties more expected of an accentual system: (i) there can only be one heavy syllable per word; (ii) this syllable must bear a H tone. This raises the question of whether syllables with a culminative H also have metrical prominence, i.e. stress. However, the problem with equating stress with H tone is that Lung’Ie has two kinds of nouns: those with a culminative H and those which are toneless. The nouns with culminative H are 87% of Portuguese origin, incorporated through stress-to-tone alignment, while the toneless ones are 92% of African origin. Although other creole languages have been reported with split systems of “accented” vs. fully specified tonal lexemes, and others with mixed systems of tone and stress, Lung’Ie differs from these cases in treating African origin words as toneless, a quite surprising result. We consider different analyses and conclude that Lung’Ie has a privative /H/ tone system with the single unusual stress-like property of weight-to-tone.


2018 ◽  
Vol 24 (2) ◽  
pp. 175-183
Author(s):  
Jean-Claude Ndogmo

Abstract Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.


2019 ◽  
Vol 23 (3) ◽  
pp. 1281-1304 ◽  
Author(s):  
Ben R. Hodges

Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.


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