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Pustam | पुस्तम | পুস্তম🇳🇵A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).Background: The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.In more detail: A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).<a href="https://mathstodon.xyz/tags/Pendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#Pendulum</a> <a href="https://mathstodon.xyz/tags/Cycloid" class="mention hashtag" rel="nofollow noopener" target="_blank">#Cycloid</a> <a href="https://mathstodon.xyz/tags/Period" class="mention hashtag" rel="nofollow noopener" target="_blank">#Period</a> <a href="https://mathstodon.xyz/tags/Frequency" class="mention hashtag" rel="nofollow noopener" target="_blank">#Frequency</a> <a href="https://mathstodon.xyz/tags/SHM" class="mention hashtag" rel="nofollow noopener" target="_blank">#SHM</a> <a href="https://mathstodon.xyz/tags/TimePeriod" class="mention hashtag" rel="nofollow noopener" target="_blank">#TimePeriod</a> <a href="https://mathstodon.xyz/tags/CycloidalPendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#CycloidalPendulum</a> <a href="https://mathstodon.xyz/tags/Lagrange" class="mention hashtag" rel="nofollow noopener" target="_blank">#Lagrange</a> <a href="https://mathstodon.xyz/tags/Cusp" class="mention hashtag" rel="nofollow noopener" target="_blank">#Cusp</a> <a href="https://mathstodon.xyz/tags/Energy" class="mention hashtag" rel="nofollow noopener" target="_blank">#Energy</a> <a href="https://mathstodon.xyz/tags/KineticEnergy" class="mention hashtag" rel="nofollow noopener" target="_blank">#KineticEnergy</a> <a href="https://mathstodon.xyz/tags/PotentialEnergy" class="mention hashtag" rel="nofollow noopener" target="_blank">#PotentialEnergy</a> <a href="https://mathstodon.xyz/tags/Lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Lagrangian</a> <a href="https://mathstodon.xyz/tags/Length" class="mention hashtag" rel="nofollow noopener" target="_blank">#Length</a> <a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#Math</a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mathstodon.xyz/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mathstodon.xyz/tags/Mechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mechanics</a> <a href="https://mathstodon.xyz/tags/ClassicalMechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#ClassicalMechanics</a> <a href="https://mathstodon.xyz/tags/Amplitude" class="mention hashtag" rel="nofollow noopener" target="_blank">#Amplitude</a> <a href="https://mathstodon.xyz/tags/CircularFrequency" class="mention hashtag" rel="nofollow noopener" target="_blank">#CircularFrequency</a> <a href="https://mathstodon.xyz/tags/Motion" class="mention hashtag" rel="nofollow noopener" target="_blank">#Motion</a> <a href="https://mathstodon.xyz/tags/Vibration" class="mention hashtag" rel="nofollow noopener" target="_blank">#Vibration</a> <a href="https://mathstodon.xyz/tags/HarmonicMotion" class="mention hashtag" rel="nofollow noopener" target="_blank">#HarmonicMotion</a> <a href="https://mathstodon.xyz/tags/Parameter" class="mention hashtag" rel="nofollow noopener" target="_blank">#Parameter</a> <a href="https://mathstodon.xyz/tags/ParemeterizedEquation" class="mention hashtag" rel="nofollow noopener" target="_blank">#ParemeterizedEquation</a> <a href="https://mathstodon.xyz/tags/GoverningEquations" class="mention hashtag" rel="nofollow noopener" target="_blank">#GoverningEquations</a> <a href="https://mathstodon.xyz/tags/Equation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Equation</a> <a href="https://mathstodon.xyz/tags/Equations" class="mention hashtag" rel="nofollow noopener" target="_blank">#Equations</a> <a href="https://mathstodon.xyz/tags/DifferentialEquations" class="mention hashtag" rel="nofollow noopener" target="_blank">#DifferentialEquations</a> <a href="https://mathstodon.xyz/tags/Calculus" class="mention hashtag" rel="nofollow noopener" target="_blank">#Calculus</a>

Pustam | पुस्तम | পুস্তম🇳🇵An aerospace engineering student from the Pennsylvania State University refines a 100-year-old math/aerodynamic (wind energy equation) problem, expanding wind energy possibilities.Article published in Wind Energy Science: <a href="https://wes.copernicus.org/articles/10/451/2025/" rel="nofollow noopener" translate="no" target="_blank">https://wes.copernicus.org/articles/10/451/2025/</a>Read more: [1] <a href="https://timesofindia.indiatimes.com/education/news/meet-divya-tyagi-the-penn-state-student-who-cracked-a-100-year-old-wind-energy-equation-boosting-turbine-efficiency/articleshow/119260883.cms" rel="nofollow noopener" translate="no" target="_blank">https://timesofindia.indiatimes.com/education/news/meet-divya-tyagi-the-penn-state-student-who-cracked-a-100-year-old-wind-energy-equation-boosting-turbine-efficiency/articleshow/119260883.cms</a> [2] <a href="https://www.psu.edu/news/engineering/story/student-refines-100-year-old-math-problem-expanding-wind-energy-possibilities" rel="nofollow noopener" translate="no" target="_blank">https://www.psu.edu/news/engineering/story/student-refines-100-year-old-math-problem-expanding-wind-energy-possibilities</a><a href="https://mathstodon.xyz/tags/Aerospace" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerospace</a> <a href="https://mathstodon.xyz/tags/Science" class="mention hashtag" rel="nofollow noopener" target="_blank">#Science</a> <a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#Math</a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mathstodon.xyz/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a> <a href="https://mathstodon.xyz/tags/Wind" class="mention hashtag" rel="nofollow noopener" target="_blank">#Wind</a> <a href="https://mathstodon.xyz/tags/WindEnergy" class="mention hashtag" rel="nofollow noopener" target="_blank">#WindEnergy</a> <a href="https://mathstodon.xyz/tags/Equation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Equation</a> <a href="https://mathstodon.xyz/tags/Lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Lagrangian</a> <a href="https://mathstodon.xyz/tags/Aerodynamic" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamic</a> <a href="https://mathstodon.xyz/tags/WES" class="mention hashtag" rel="nofollow noopener" target="_blank">#WES</a> <a href="https://mathstodon.xyz/tags/Turbine" class="mention hashtag" rel="nofollow noopener" target="_blank">#Turbine</a> <a href="https://mathstodon.xyz/tags/Turbines" class="mention hashtag" rel="nofollow noopener" target="_blank">#Turbines</a> <a href="https://mathstodon.xyz/tags/Efficiency" class="mention hashtag" rel="nofollow noopener" target="_blank">#Efficiency</a> <a href="https://mathstodon.xyz/tags/Energy" class="mention hashtag" rel="nofollow noopener" target="_blank">#Energy</a> <a href="https://mathstodon.xyz/tags/Divya" class="mention hashtag" rel="nofollow noopener" target="_blank">#Divya</a> <a href="https://mathstodon.xyz/tags/DivyaTyagi" class="mention hashtag" rel="nofollow noopener" target="_blank">#DivyaTyagi</a> <a href="https://mathstodon.xyz/tags/Tyagi" class="mention hashtag" rel="nofollow noopener" target="_blank">#Tyagi</a> <a href="https://mathstodon.xyz/tags/Penn" class="mention hashtag" rel="nofollow noopener" target="_blank">#Penn</a> <a href="https://mathstodon.xyz/tags/PSU" class="mention hashtag" rel="nofollow noopener" target="_blank">#PSU</a> <a href="https://mathstodon.xyz/tags/PennState" class="mention hashtag" rel="nofollow noopener" target="_blank">#PennState</a> <a href="https://mathstodon.xyz/tags/AerospaceEngineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#AerospaceEngineering</a> <a href="https://mathstodon.xyz/tags/Engineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#Engineering</a>

Rhett AllainNew <a href="https://mastodon.social/tags/python" class="mention hashtag" rel="nofollow noopener" target="_blank">#python</a> <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> video - building a 3D animation of a half-atwood with a spring using <a href="https://mastodon.social/tags/Lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Lagrangian</a> mechanics and <a href="https://mastodon.social/tags/sympy" class="mention hashtag" rel="nofollow noopener" target="_blank">#sympy</a> (of course <a href="https://mastodon.social/tags/vpython" class="mention hashtag" rel="nofollow noopener" target="_blank">#vpython</a> too)<a href="https://youtu.be/wB_hs-Dhs3c" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/wB_hs-Dhs3c</a>

Rhett AllainNew <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> <a href="https://mastodon.social/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a> - flinging a bead from a rotating stick. Yes, of course there is a <a href="https://mastodon.social/tags/python" class="mention hashtag" rel="nofollow noopener" target="_blank">#python</a> model. Do you not even know me?<a href="https://youtu.be/IqRaVK5lQ5g" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/IqRaVK5lQ5g</a>

Rhett AllainNew <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> <a href="https://mastodon.social/tags/classicalmechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#classicalmechanics</a> video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using <a href="https://mastodon.social/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a> and <a href="https://mastodon.social/tags/python" class="mention hashtag" rel="nofollow noopener" target="_blank">#python</a><a href="https://youtu.be/DF6ixJakMDU" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/DF6ixJakMDU</a>

Rhett AllainNew <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> video - using Lagrange Multipliers to find forces of constraint. <a href="https://mastodon.social/tags/classicalmechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#classicalmechanics</a> <a href="https://mastodon.social/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a> <a href="https://youtu.be/CLckOYdTZyo" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/CLckOYdTZyo</a>

Rhett AllainNew <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> <a href="https://mastodon.social/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a> video - a chain sliding off a table (yes, animated in <a href="https://mastodon.social/tags/webvpython" class="mention hashtag" rel="nofollow noopener" target="_blank">#webvpython</a> of course)<a href="https://youtu.be/WkpUp4bGwIY" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/WkpUp4bGwIY</a>

Rhett AllainThere are just too many <a href="https://mastodon.social/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a> <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> problems - but I had to do a bead on a rotating hoop. Of course <a href="https://mastodon.social/tags/python" class="mention hashtag" rel="nofollow noopener" target="_blank">#python</a> is included.<a href="https://youtu.be/gQFTs1oc78o" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/gQFTs1oc78o</a>

Rhett AllainYes. I have ANOTHER <a href="https://mastodon.social/tags/Lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Lagrangian</a> mechanics problem for you - a pendulum on an oscillating pivot point. <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a>. <a href="https://mastodon.social/tags/python" class="mention hashtag" rel="nofollow noopener" target="_blank">#python</a> <a href="https://youtu.be/ITlSFagJPAo" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/ITlSFagJPAo</a>

Rhett AllainNew <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> video - my introduction to <a href="https://mastodon.social/tags/Lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Lagrangian</a> mechanics<a href="https://youtu.be/Hq-1TDKWGgw" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/Hq-1TDKWGgw</a>

Jitse NiesenCongratulations to dr (*) Jacob Richardson, who successfully (*) defended his PhD thesis "On Discrete Lagrangian Multiform Structures and their Reductions" in the face of persistent questioning from Andy Hone and myself.Well done to his supervisors Frank Nijhoff and Vincent Caudrelier. Some results are described in the preprint of Jacob and <a href="https://mathstodon.xyz/@mathmats" class="u-url mention" rel="nofollow noopener" target="_blank">@mathmats</a>, "Discrete Lagrangian multiforms for quad equations, tetrahedron equations, and octahedron equations", <a href="https://arxiv.org/abs/2403.16845" rel="nofollow noopener" translate="no" target="_blank">https://arxiv.org/abs/2403.16845</a> .(*) subject to editorial and presentational corrections<a href="https://mathstodon.xyz/tags/IntegrableSystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#IntegrableSystems</a> <a href="https://mathstodon.xyz/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a>

Rhett AllainI saw this interesting <a href="https://mastodon.social/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> question with 4 masses and springs on @aapt communities - and I just HAD to solve it THREE different ways (<a href="https://mastodon.social/tags/lagrangian" class="mention hashtag" rel="nofollow noopener" target="_blank">#lagrangian</a> and <a href="https://mastodon.social/tags/python" class="mention hashtag" rel="nofollow noopener" target="_blank">#python</a> included)<a href="https://youtu.be/1RzM8SkNL3Q" rel="nofollow noopener" translate="no" target="_blank">https://youtu.be/1RzM8SkNL3Q</a>

Published papers at TMLRProvably Convergent Policy Optimization via Metric-aware Trust Region MethodsJun Song, Niao He, Lijun Ding, Chaoyue ZhaoAction editor: Amir-massoud Farahmand.<a href="https://openreview.net/forum?id=jkTqJJOGMS" target="_blank" rel="nofollow noopener" translate="no">https://openreview.net/forum?id=jkTqJJOGMS</a><a href="https://sigmoid.social/tags/optimality" class="mention hashtag" rel="tag">#optimality</a> <a href="https://sigmoid.social/tags/reinforcement" class="mention hashtag" rel="tag">#reinforcement</a> <a href="https://sigmoid.social/tags/lagrangian" class="mention hashtag" rel="tag">#lagrangian</a>

New Submissions to TMLRProvably Convergent Policy Optimization via Metric-aware Trust Region Methods<a href="https://openreview.net/forum?id=jkTqJJOGMS" target="_blank" rel="nofollow noopener" translate="no">https://openreview.net/forum?id=jkTqJJOGMS</a><a href="https://sigmoid.social/tags/optimality" class="mention hashtag" rel="tag">#optimality</a> <a href="https://sigmoid.social/tags/reinforcement" class="mention hashtag" rel="tag">#reinforcement</a> <a href="https://sigmoid.social/tags/lagrangian" class="mention hashtag" rel="tag">#lagrangian</a>

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