Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum

Geometric phase can explain the rotation of a dynamical system independent of angular momentum. The canonical example of such is a cat (a non-rigid body with an inbuilt control system), falling from an inverted position, being able to re-orient itself with negligible total angular momentum so as to land on its feet. The system of three bodies moving under mutual gravitation is similarly non-rigid, capable of changing size and shape under the dynamics of that force. Using coordinates that reduce by translations and rotations and simultaneously regularise all binary collisions, which separate shape dynamics from rotational dynamics, we show how certain discrete symmetries (including both reversing and non-reversing symmetries of the equations of motion) can force the geometric phase of motion periodic to vanish. This result is illustrated with periodic orbits discovered in a numerical survey, many of which are heretofore unknown, and the findings of this survey are discussed in detail, including stability, geometric phase, and classification of orbits

Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum

Geometric phase can explain the rotation of a dynamical system independent of angular momentum. The canonical example of such is a cat (a non-rigid body with an inbuilt control system), falling from an inverted position, being able to re-orient itself with negligible total angular momentum so as to land on its feet. The system of three bodies moving under mutual gravitation is similarly non-rigid, capable of changing size and shape under the dynamics of that force. Using coordinates that reduce by translations and rotations and simultaneously regularise all binary collisions, which separate shape dynamics from rotational dynamics, we show how certain discrete symmetries (including both reversing and non-reversing symmetries of the equations of motion) can force the geometric phase of motion periodic to vanish. This result is illustrated with periodic orbits discovered in a numerical survey, many of which are heretofore unknown, and the findings of this survey are discussed in detail, including stability, geometric phase, and classification of orbits

Leveraging Place for Critical Sustainability Education:The Promise of Participatory Action Research

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The science of complex systems is needed to ameliorate the impacts of COVID-19 on mental health

To assist with proactive and effective responses to the global COVID-19 crisis, the scientific community has been rapidly deploying our most advanced analytic tools to model the dynamics of disease transmission based on existing (albeit imperfect) knowledge, data, and available human and material resources. The multifactorial, multilevel influences on transmission dynamics and the disease’s pervasive impact at the individual, community, and global levels have required the use of the analytic techniques of complex systems science, namely, systems modeling and simulation, to forecast the trajectory of the disease under different conditions, to quantify uncertainty, and to inform effective responses (1–3). These methods have been deployed by infectious disease epidemiologists for over a century (4), maturing into a robust interdisciplinary field intersecting mathematics, computational epidemiology, ecology, evolutionary biology, immunology, behavioral science, and public health (5). As a result, there have been numerous advances that have informed policies to control infectious diseases, facilitate epidemic and bioterrorism preparedness, and provide governments with critical tools for managing complexity and weighing alternative responses in the midst of the confusion of an evolving crisis (6–14). The field’s commitment to achieving rapid response capability in the face of changing conditions has led to advances in rapid assessment of the impact of the pandemic, and data assimilation methods that combine theory with empirical observations in a continuous knowledge feedback process facilitating continuous hypothesis development, testing, and refinement in the service of more effective decision making (15–19)

Structural modelling of deformable screens for large door openings

The mathematical modelling of deformable, permeable screen doors was explored to assess their behaviour under wind loading. A load-response model was proposed whereby the wind load was modelled using a simplified approach that allowed it to be approximated as a uniformly distributed pressure load with empirical modification factors applied to relate it to the real case of a door on a building. Several approaches were adopted to model the mechanical behaviour of the door system in response to load, including discrete models based on mass-spring systems, continuum models based on the membrane equations (including tension modulation in some cases), and computational models using finite element packages. The primary aim of the work was to determine the distribution of wind load to the door's supporting `tabs' and estimate a failure wind speed. The mass-spring model and the membrane models without tension modulation both generated unrealistic deflection magnitudes in response to wind load, but could be calibrated in future work, and then used to obtain an estimate of the total force on the tabs. A tension-modulated version of the membrane model performed better with regards to deflected shape and magnitude, but time constraints meant that the load on the tabs was not calculated. Preliminary validation experiments were undertaken and deflected shape and magnitude were successfully measured in response to given wind speeds. References Standards Australia. AS/NZS1170.2–-Structural design actions part 2: Wind actions, 2011. F. Avanzini and R. Maronga. A modular physically based approach to the sound synthesis of membrane percussion instruments. IEEE Transactions on Audio, Speech, and Language Processing, 18(4):891–902, 2010. doi:10.1109/TASL.2009.2036903 B. Bank. Energy-based synthesis of tension modulation in strings. In Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09), 2009. http://dafx09.como.polimi.it/proceedings/papers/paper_76.pdf Bert Blocken. 50 years of computational wind engineering: Past, present and future. Journal of Wind Engineering and Industrial Aerodynamics, 129(0):69–102, 2014. doi:10.1016/j.jweia.2014.03.008 Demetres Briassoulis, Antonis Mistriotis, and Anastasios Giannoulis. Wind forces on porous elevated panels. Journal of Wind Engineering and Industrial Aerodynamics, 98(12):919–928, 2010. doi:10.1016/j.jweia.2010.09.006 John Cheung and William Melbourne. Wind loading on porous cladding. In 9th Australasian fluid mechanics conference, 1986. http://people.eng.unimelb.edu.au/imarusic/proceedings/9/CheungMelbourne.pdf John Holmes. Wind Loading of Structures. Spon Press, 2001. P. D. Howell, G. Kozyreff, and J. R. Ockendon. Applied Solid Mechanics. Cambridge University Press, 2009. C.W Letchford. Wind loads on rectangular signboards and hoardings. Journal of Wind Engineering and Industrial Aerodynamics, 89(2):135–151, 2001. doi:10.1016/S0167-6105(00)00068-4 Wojciech Okrasinski and \T1\L ukasz Plociniczak. A nonlinear mathematical model of the corneal shape. Nonlinear Analysis: Real World Applications, 13(3):1498–1505, 2012. doi:10.1016/j.nonrwa.2011.11.014 D. W. Oplinger. Frequency response of a nonlienar stretched string. Journal of the Acoustical Society of America, 32(12):1529–1538, 1960. doi:10.1121/1.1907948 Xavier Provot. Deformation constraints in a mass-spring model to describe rigid cloth behaviour. In Graphics interface, pages 147–147. Canadian Information Processing Society, 1995. http://kucg.korea.ac.kr/education/2005/CSCE352/paper/provot95.pdf P.J Richards and M Robinson. Wind loads on porous structures. Journal of Wind Engineering and Industrial Aerodynamics, 83(13):455–465, 1999. doi:10.1016/S0167-6105(99)00093-8 A.P. Robertson, Ph. Roux, J. Gratraud, G. Scarascia, S. Castellano, M. Dufresne de Virel, and P. Palier. Wind pressures on permeably and impermeably-clad structures. Journal of Wind Engineering and Industrial Aerodynamics, 90(45):461–474, 2002. Bluff Body Aerodynamics and Applications. doi:10.1016/S0167-6105(01)00210-0 T. Tolonen, V. Valimaki, and M. Karjalainen. Modeling of tension modulation nonlinearity in plucked strings. IEEE Transactions on Speech and Audio Processing, 8(3):300–310, 2000. doi:10.1109/89.84121

Sound Decision Making in Uncertain Times: Can Systems Modelling Be Useful for Informing Policy and Planning for Suicide Prevention?

The COVID-19 pandemic demonstrated the significant value of systems modelling in supporting proactive and effective public health decision making despite the complexities and uncertainties that characterise an evolving crisis. The same approach is possible in the field of mental health. However, a commonly levelled (but misguided) criticism prevents systems modelling from being more routinely adopted, namely, that the presence of uncertainty around key model input parameters renders a model useless. This study explored whether radically different simulated trajectories of suicide would result in different advice to decision makers regarding the optimal strategy to mitigate the impacts of the pandemic on mental health. Using an existing system dynamics model developed in August 2020 for a regional catchment of Western Australia, four scenarios were simulated to model the possible effect of the COVID-19 pandemic on levels of psychological distress. The scenarios produced a range of projected impacts on suicide deaths, ranging from a relatively small to a dramatic increase. Discordance in the sets of best-performing intervention scenarios across the divergent COVID-mental health trajectories was assessed by comparing differences in projected numbers of suicides between the baseline scenario and each of 286 possible intervention scenarios calculated for two time horizons; 2026 and 2041. The best performing intervention combinations over the period 2021-2041 (i.e., post-suicide attempt assertive aftercare, community support programs to increase community connectedness, and technology enabled care coordination) were highly consistent across all four COVID-19 mental health trajectories, reducing suicide deaths by between 23.9-24.6% against the baseline. However, the ranking of best performing intervention combinations does alter depending on the time horizon under consideration due to non-linear intervention impacts. These findings suggest that systems models can retain value in informing robust decision making despite uncertainty in the trajectories of population mental health outcomes. It is recommended that the time horizon under consideration be sufficiently long to capture the full effects of interventions, and efforts should be made to achieve more timely tracking and access to key population mental health indicators to inform model refinements over time and reduce uncertainty in mental health policy and planning decisions