@misc{weisstein_double_nodate,
	type = {Text},
	title = {Double {Pendulum} -- from {Eric} {Weisstein}'s {World} of {Physics}},
	copyright = {Copyright 1996-2007 Eric W. Weisstein},
	url = {https://scienceworld.wolfram.com/physics/DoublePendulum.html},
	abstract = {A double pendulum consists of one pendulum attached to another.  Double pendula are an example of a simple physical system which can exhibit chaotic behavior.  Consider a double bob pendulum with masses m\_1 and m\_2 attached by rigid massless wires of lengths l\_1 and l\_2. Further, let the angles the two wires make with the vertical be denoted {\textbackslash}theta\_1 and {\textbackslash}theta\_2, as illustrated above.  Finally, let gravity be given by g. Then the positions of the bobs are given by  x\_1 = l\_1{\textbackslash}sin{\textbackslash}theta\_1 y\_1 =...},
	language = {en},
	urldate = {2021-04-29},
	author = {Weisstein, Eric W.},
	note = {Publisher: Wolfram Research, Inc.},
	file = {Snapshot:C\:\\Users\\fer_l\\Zotero\\storage\\6DNI9RIM\\DoublePendulum.html:text/html},
}

@misc{noauthor_double_nodate,
	title = {Double {Pendulum}},
	url = {https://www.math24.net/double-pendulum},
	abstract = {A double pendulum is undoubtedly an actual miracle of nature. The jump in complexity, which is observed at the transition from a simple pendulum to a double pendulum is amazing. The oscillations of a simple pendulum are regular. For small deviations from equilibrium, these oscillations are harmonic and can be described by sine or cosine ... Read more},
	language = {en-US},
	urldate = {2021-04-29},
	journal = {Math24},
	file = {Snapshot:C\:\\Users\\fer_l\\Zotero\\storage\\FKZP5Y65\\double-pendulum.html:text/html},
}

@article{stroup_dynamics_nodate,
	title = {The {Dynamics} of {Pendula}: {An} {Introduction} to {Hamiltonian} {Systems} and {Chaos}},
	abstract = {The Laser Interferometer Gravitational Wave Observatory (LIGO) was built to detect and observe gravity waves, whose existence was predicted by Einstein’s theory of general relativity. A critical part of the interferometer are the mirrors which reflect the laser beams, and whose motion determines if a gravity wave is present. As of 2001, LIGO was slated to update their facility by housing each mirror on the lowest bob of a quadruple pendulum to reduce thermal noise.},
	language = {en},
	author = {Stroup, Adrianne},
	pages = {18},
	file = {Stroup - The Dynamics of Pendula An Introduction to Hamilt.pdf:C\:\\Users\\fer_l\\Zotero\\storage\\X7TUBGDA\\Stroup - The Dynamics of Pendula An Introduction to Hamilt.pdf:application/pdf},
}

@book{blanchard_differential_2012,
	title = {Differential {Equations}},
	isbn = {978-1-133-38808-1},
	abstract = {Incorporating an innovative modeling approach, this book for a one-semester differential equations course emphasizes conceptual understanding to help users relate information taught in the classroom to real-world experiences. Certain models reappear throughout the book as running themes to synthesize different concepts from multiple angles, and a dynamical systems focus emphasizes predicting the long-term behavior of these recurring models. Users will discover how to identify and harness the mathematics they will use in their careers, and apply it effectively outside the classroom.Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.},
	language = {en},
	publisher = {Cengage Learning},
	author = {Blanchard, Paul and Devaney, Robert L. and Hall, Glen R.},
	month = jul,
	year = {2012},
	note = {Google-Books-ID: PdkIAAAAQBAJ},
	keywords = {Mathematics / Differential Equations / General},
}

@book{boyce_elementary_2017,
	title = {Elementary {Differential} {Equations} and {Boundary} {Value} {Problems}},
	isbn = {978-1-119-44376-6},
	abstract = {Elementary Differential Equations and Boundary Value Problems 11e, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two or three semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.},
	language = {en},
	publisher = {John Wiley \& Sons},
	author = {Boyce, William E. and DiPrima, Richard C. and Meade, Douglas B.},
	month = aug,
	year = {2017},
	note = {Google-Books-ID: SyaVDwAAQBAJ},
	keywords = {Mathematics / Differential Equations / General, Mathematics / Mathematical Analysis},
}

@book{wiggins_introduction_2006,
	title = {Introduction to {Applied} {Nonlinear} {Dynamical} {Systems} and {Chaos}},
	isbn = {978-0-387-21749-9},
	abstract = {Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in - search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as nume- cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface to the Second Edition This edition contains a signi?cant amount of new material. The main r- son for this is that the subject of applied dynamical systems theory has seen explosive growth and expansion throughout the 1990s. Consequently, a student needs a much larger toolbox today in order to begin research on signi?cant problems.},
	language = {en},
	publisher = {Springer Science \& Business Media},
	author = {Wiggins, Stephen},
	month = apr,
	year = {2006},
	note = {Google-Books-ID: YhXnBwAAQBAJ},
	keywords = {Mathematics / Mathematical Analysis, Mathematics / Applied, Mathematics / Linear \& Nonlinear Programming, Science / Physics / General, Science / Physics / Mathematical \& Computational, Technology \& Engineering / Engineering (General)},
}

@book{hirsch_differential_2004,
	title = {Differential {Equations}, {Dynamical} {Systems}, and an {Introduction} to {Chaos}},
	isbn = {978-0-12-349703-1},
	abstract = {Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second Edition, provides a rigorous yet accessible introduction to differential equations and dynamical systems. The original text by three of the world's leading mathematicians has become the standard textbook for graduate courses in this area. Thirty years in the making, this Second Edition brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra. The book explores the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. It presents the simplification of many theorem hypotheses and includes bifurcation theory throughout. It contains many new figures and illustrations; a simplified treatment of linear algebra; detailed discussions of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractor; and increased coverage of discrete dynamical systems. This book will be particularly useful to advanced students and practitioners in higher mathematics.  Developed by award-winning researchers and authorsProvides a rigorous yet accessible introduction to differential equations and dynamical systemsIncludes bifurcation theory throughoutContains numerous explorations for students to embark upon NEW IN THIS EDITION New contemporary material and updated applicationsRevisions throughout the text, including simplification of many theorem hypothesesMany new figures and illustrationsSimplified treatment of linear algebraDetailed discussion of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractorIncreased coverage of discrete dynamical systems},
	language = {en},
	publisher = {Academic Press},
	author = {Hirsch, Morris W. and Smale, Stephen and Devaney, Robert L.},
	year = {2004},
	note = {Google-Books-ID: INYJuKGmgd0C},
	keywords = {Mathematics / Differential Equations / General},
}

@book{strogatz_nonlinear_2018,
	title = {Nonlinear {Dynamics} and {Chaos}: {With} {Applications} to {Physics}, {Biology}, {Chemistry}, and {Engineering}},
	isbn = {978-0-429-97219-5},
	shorttitle = {Nonlinear {Dynamics} and {Chaos}},
	abstract = {This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.},
	language = {en},
	publisher = {CRC Press},
	author = {Strogatz, Steven H.},
	month = may,
	year = {2018},
	note = {Google-Books-ID: 1kpnDwAAQBAJ},
	keywords = {Mathematics / General},
}

@misc{nolte_ups_2020,
	title = {The {Ups} and {Downs} of the {Compound} {Double} {Pendulum}},
	url = {https://galileo-unbound.blog/2020/10/18/the-ups-and-downs-of-the-compound-double-pendulum/},
	abstract = {A chief principle of chaos theory states that even simple systems can display complex dynamics.  All that is needed for chaos, roughly, is for a system to have at least three dynamical variabl…},
	language = {en},
	urldate = {2021-04-30},
	journal = {Galileo Unbound},
	author = {Nolte, David D.},
	month = oct,
	year = {2020},
}

@misc{noauthor_double_nodate-1,
	title = {Double {Pendulum} {Java} {Application}},
	url = {http://www.physics.smu.edu/fattarus/double_pendulum.html},
	urldate = {2021-04-30},
	file = {Double Pendulum Java Application:C\:\\Users\\fer_l\\Zotero\\storage\\L9R3ALCY\\double_pendulum.html:text/html},
}

@article{myers_low-cost_2020,
	title = {Low-cost double pendulum for high-quality data collection with open-source video tracking and analysis},
	volume = {8},
	issn = {2468-0672},
	url = {https://www.sciencedirect.com/science/article/pii/S246806722030047X},
	doi = {10.1016/j.ohx.2020.e00138},
	abstract = {The double pendulum is a system that manifests fascinating non-linear behavior. This made it a popular tool in academic settings for illustrating the intricate response of a seemingly simple physical apparatus, or to validate tools for studying nonlinear phenomena. In addition, the double pendulum is also widely used in several modeling applications including robotics and human locomotion analysis. However, surprisingly, there is a lack of a thoroughly documented hardware that enables designing, building, and reliably tracking and collecting data from a double pendulum. This paper provides comprehensive documentation of a research quality bench top double pendulum. The contributions of our work include (1) providing detailed CAD drawings, part lists, and assembly instructions for building a low friction double pendulum. (2) A new tracking algorithm written in Python for tracking the position of both links of the double pendulum. This algorithm measures the angles of the links by examining each frame, and computes uncertainties in the measured angles by following several trackers on each link. Additionally, our tracking algorithm bypasses the data transmission difficulties caused by instrumenting the bottom link with physical sensors. (3) A derivation of the equations of motion of the actual physical system. (4) A description of the process (with provided Python code) for extracting the model parameters—e.g., damping—with error bounds from physical measurements.},
	language = {en},
	urldate = {2021-04-30},
	journal = {HardwareX},
	author = {Myers, Audun D. and Tempelman, Joshua R. and Petrushenko, David and Khasawneh, Firas A.},
	month = oct,
	year = {2020},
	keywords = {Double pendulum, Occlusions, Pendulum, Tracking},
	pages = {e00138},
	file = {ScienceDirect Full Text PDF:C\:\\Users\\fer_l\\Zotero\\storage\\433HCLTE\\Myers et al. - 2020 - Low-cost double pendulum for high-quality data col.pdf:application/pdf;ScienceDirect Snapshot:C\:\\Users\\fer_l\\Zotero\\storage\\S9K36XR4\\S246806722030047X.html:text/html},
}

@misc{noauthor_quantum_2021,
	title = {Quantum mechanics},
	copyright = {Creative Commons Attribution-ShareAlike License},
	url = {https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&oldid=1020414201},
	abstract = {Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.
Classical physics, the description of physics that existed before the theory of relativity and quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, while quantum mechanics explains the aspects of nature at small (atomic and subatomic) scales, for which classical mechanics is insufficient. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).
Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle's energy, momentum, and other physical properties may yield.},
	language = {en},
	urldate = {2021-04-30},
	journal = {Wikipedia},
	month = apr,
	year = {2021},
	note = {Page Version ID: 1020414201},
}