Gauge Variant Symmetries for the Schr\"odinger Equation

The last multiplier of Jacobi provides a route for the determination of families of Lagrangians for a given system. We show that the members of a family are equivalent in that they differ by a total time derivative. We derive the Schr\"odinger equation for a one-degree-of-freedom system with a constant multiplier. In the sequel we consider the particular example of the simple harmonic oscillator. In the case of the general equation for the simple harmonic oscillator which contains an arbitrary function we show that all Schr\"odinger equations possess the same number of Lie point symmetries with the same algebra. From the symmetries we construct the solutions of the Schr\"odinger equation and find that they differ only by a phase determined by the gauge.Comment: 12 page

Linking academic emotions and student engagement: mature-aged distance students’ transition to university

This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of Further and Higher Education on 2013, available online: https://www.tandfonline.com/doi/full/10.1080/0309877X.2014.895305Research into both student engagement and student emotions is increasing, with widespread agreement that both are critical determinants of student success in higher education. Less researched are the complex, reciprocal relationships between these important influences. Two theoretical frameworks inform this paper: Pekrun’s taxonomy of academic emotions and Kahu’s conceptual framework of student engagement. The prospective qualitative design aims to allow a rich understanding of the fluctuating and diverse emotions that students experience during the transition to university and to explore the relationships between academic emotions and student engagement. The study follows 19 mature-aged (aged 24 and over) distance students throughout their first semester at university, using video diaries to collect data on their emotional experiences and their engagement with their study. Pre and post-semester interviews were also conducted. Findings highlight that different emotions have different links to engagement: as important elements in emotional engagement, as inhibitors of engagement and as outcomes that reciprocally influence engagement. There are two key conclusions. First, student emotions are the point of intersection between the university factors such as course design and student variables such as motivation and background. Second, the flow of influence between emotions, engagement, and learning is reciprocal and complex and can spiral upwards towards ideal engagement or downwards towards disengagement and withdrawal.Publishe

Space and time to engage: Mature-aged distance students learn to fit study into their lives

This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of Lifelong Education on 2014, available online: https://www.tandfonline.com/doi/full/10.1080/02601370.2014.884177Student engagement, a student’s emotional, behavioural and cognitive connection to their study, is widely recognized as important for student achievement. Influenced by a wide range of personal, structural and sociocultural factors, engagement is both unique and subjective. One important structural factor shown in past research to be a barrier for distance students is access to quality space and time. This qualitative study followed 19 mature-aged distance students and their families, exploring how they learned to manage their space and time throughout their first semester at university. Institutions often claim that distance study and the increased use of technology overcomes barriers of space and time; however, the findings from this study suggest it merely changes the nature of those barriers. The ideal space and time for these students was individual and lay at the intersection of three, sometimes competing, demands: study, self and family. A critical influence on success is family support, as is access to financial resources. Learning what constitutes ideal space and time for engagement is an important part of the transition to university. The institution has a vital role to play in aiding this process by ensuring flexibility of course design is maintained, providing more flexible advice and targeting support at this important issue.falsePublishe

The engagement of mature distance students

This is an Accepted Manuscript of an article published by Taylor & Francis in Higher Education Research and Development in 2013, available online: http://www.tandfonline.com/10.1080/07294360.2013.777036.Publishe

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic

Ermakov's Superintegrable Toy and Nonlocal Symmetries

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2,R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA