What can the answer be?

This final article in the series first addresses the problem of calculating reciprocal vectors inn-dimensions. The deeper definition of vectors (and other quantities such as tensors) in terms of the transformation properties under rotations is brought out. Some examples which generalise this notion to other kinds of transformations are given.     

Symmetries and conservation laws in classical and quantum mechanics 1. Classical mechanics

We describe the connection between continuous symmetries and conservation laws in classical mechanics. This is done at successively more sophisticated levels, bringing out important features at each level: the Newtonian, the Euler-Lagrange, and the Hamiltonian phase-space forms of mechanics. The role of the Action Principle is emphasised, and many examples are given.     

Einstein’s miraculous year

With each passing year, the young Albert Einstein’s achievements in physics in the year 1905 seem to be ever more miraculous. We describe why the centenary of this remarkable year is worthy of celebration.     

Whatcan the answer be?

In Part 2 of this series, we saw how dual vectors arose very naturally even in elementary vector analysis. At the end of that article, I stated that dual vectors and the reciprocal basis were very far-reaching concepts. They appear in many different contexts, some of which will be described in the sequel. In this part, we begin with a situation that might appear to be simpler than the case worked out in part 2-but we are in for a surprise!     

Using working agreements in participatory action research: working through moral problems with Malaysian students

In this paper we propose that taking time before the work begins to develop agreements about how to work together in participatory action research enables researchers to directly address several ethical issues that can be problematic in this type of research: gaining fully informed consent; in-group confidentiality; cultural misconceptions; and protecting participants from risky self-disclosures. We begin by discussing some of these ethical challenges that can arise in participatory work, especially with young people. We argue that these issues are magnified and politicised in certain politico-cultural settings, and therefore are all the more important when working cross-culturally. Drawing on the findings of the lead writer’s doctoral thesis, which sought to find a more relevant way of teaching moral education in a Malaysian setting by including the voices of young people, we show how participants responded to attempts to address these issues by creating a safe space in which to discuss sensitive topics through the use of a working agreement. Responses indicate that when such an agreement was in place, the young people in this Malaysian study experienced a greater sense of safety, greater encouragement to participate, and were more confident in their ability to solve ethical problems than in situations where there was no working agreement. Furthermore, the agreement enabled the researcher to be more aware of, and responsive to, the cultural context of the participants.     

The transonic dip in aeroelastic divergence speed—An explicit formula

Using continuum models we develop a closed form analytical formula for the divergence speed as a function of the Mach number and in particular its dependence on wing camber. There is a transonic dip even when the angle of attack is zero and it depends on the wing thickness ratio. For nonzero camber the angle of attack in fact plays a lesser role. The main analytical tool is the Possio integral equation which is shown to have an explicit solution. Neither CFD nor FEM is employed.     

Integrated guidance and autopilot design for a chasing UAV via high-order sliding modes

Integrated guidance and control (IGC) approaches exploit the synergy between guidance and control designs. This study focuses on the integrated guidance and control (autopilot) design for a chasing Uninhabited Aerial Vehicle (UAV) against a target aircraft. A second-order sliding structure with a second-order sliding mode (SOSM) including a high-order sliding mode (HOSM) observer for the estimation of the uncertain sliding surfaces is selected to develop an integrated guidance and autopilot scheme. In order to make the design synthesis easier, intermediate control variables for partial derivatives of a sliding surface are carefully selected. The resulting sliding surface structure is simple and sufficient to relate the actuator input to the sliding surface. The potential of the proposed method is demonstrated through an aircraft application by comparing its simulation performance, number of tuning parameters used, and information needed for its implementation with an approach where the guidance law and the controller are designed separately.     

Flutter analysis of an articulated high aspect ratio wing in subsonic airflow

We present a methodology for calculating flutter speeds of a high aspect ratio flying wing articulated with point masses in inviscid air flow. This highly flexible wing configuration typically models a HALE (High Altitude Long Endurance) UAV (Unmanned Aerial Vehicle) type aircraft. To demonstrate the procedure, we perform flutter analysis on an actual articulated wing model and we investigate the dependence of the flutter speed on the number of loads mounted onto the structure and the number of panels comprising the flying wing for both varying and constant span. The results show that the flutter speed decreases as more panels and point masses are incorporated into the flying wing. On the other hand, the number of point masses mounted onto the structure has a small effect on the flutter speed if the wing span is kept constant.