Constraints on the wing morphology of pterosaurs

Abstract

Animals that fly must be able to do so over a huge range of aerodynamic conditions, determined by weather, wind speed and the nature of their environment. No single parameter can be used to determine-let alone measure-optimum flight performance as it relates to wing shape. Reconstructing the wings of the extinct pterosaurs has therefore proved especially problematic: these Mesozoic flying reptiles had a soft-tissue membranous flight surface that is rarely preserved in the fossil record. Here, we review basic mechanical and aerodynamic constraints that influenced the wing shape of pterosaurs, and, building on this, present a series of theoretical modelling results. These results allow us to predict the most likely wing shapes that could have been employed by these ancient reptiles, and further show that a combination of anterior sweep and a reflexed proximal wing section provides an aerodynamically balanced and efficient theoretical pterosaur wing shape, with clear benefits for their flight stability.

Travelling waves and a fruitful ‘time’ reparametrization in relativistic electrodynamics

“…The above corollary is essentially the generalized Lawson-Woodward theorem of [28] 7 . This is partly more and partly less general than the so-called Lawson-Woodward (LW) or (General) Acceleration Theorem [29,33,31,30,32] (an outgrowth of the original Woodward-Lawson Theorem [41,42]). The LW theorem states that, in spite of the large energy variations during the interaction, the final energy gain of a charged particle interacting with an electromagnetic field in vacuum is zero if:…”

“…Condition 2 ensures that the motion is along a straight line (chosen as the z-axis) with constant velocity c, independently of E; the theorem was proven extending the claim from a monochromatic plane wave E to general E by linearity (the work done by the total electric force was the sum of the works done by its Fourier components), which was justified by condition (4). The claim can be justified also invoking quantum arguments (impossibility of absorption of a single real photon by 4-momentum conservation [32]), without need of assuming condition 2.…”

“…More precisely: we (re)derive in few lines the solutions [4,25,26] when the static electric and magnetic fields E s , B s are zero (section 3.1), or have only uniform longitudinal components (one or both: sections 3.2, 3.3, 3.4), or beside the latter have uniform transverse components fulfilling B ⊥ s = k ∧ E ⊥ s (section 3.5); here ⊥ denotes the component orthogonal to the direction k of propagation of the pulse. Section 3.1 includes a rigorous statement (Corollary 2) and proof of a generalized version [28] of the socalled Lawson-Woodward no-go theorem [29,30,31,32,33]; the latter states that the final energy variation of a charged particle induced by an EM pulse is zero under some rather general conditions (motion in vacuum, zero static fields, etc), in spite of the large energy variations during the interaction. To obtain large final energy variations one has thus to violate one of these general conditions.…”

On a ‘time’ reparametrization in relativistic electrodynamics with travelling waves

“…Both are very small if the pulse modulation is slow [extremely small if ∈ S(R) or ∈ C ∞ c (R)]. Recall the Lawson-Woodward Theorem [10][11][12][13] (an outgrowth of the original Woodward-Lawson Theorem [14,15]): in spite of large energy variations during the interaction, the final energy gain E f of a charged particle P interacting with an EM field is zero if: i) the interaction occurs in R 3 vacuum (no boundaries); ii) E s = B s = 0 and ⊥ is slowly modulated; iii) v z c along the whole acceleration path; iv) nonlinear (in ⊥ ) effects qβ∧B are negligible; v) the power radiated by P is negligible. Our Corollary, as Ref.…”

Travelling waves and light-front approach in relativistic electrodynamics

“…By (6), both are very small if the pulse modulation is slow [extremely small if ∈ S(R) or ∈ C ∞ c (R)]. This can be seen as a rigouros version of the Lawson-Woodward Theorem [11,12,13,14] (an outgrowth of the original Woodward-Lawson Theorem [15,16]): this theorem states that, in spite of large energy variations during the interaction, the final energy gain E f of a charged particle interacting with an EM field is zero if: i) the interaction occurs in R 3 vacuum (no boundaries); ii) E s = B s = 0 and ⊥ is slowly modulated; iii) v z c along the whole acceleration path; iv) nonlinear (in ⊥ ) effects qβ∧B are negligible; v) the power radiated by the particle is negligible. Our Corollary, as Ref.…”