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#classicalmechanics — Public Fediverse posts

Live and recent posts from across the Fediverse tagged #classicalmechanics, aggregated by home.social.

  1. Noether’s Theorem: Symmetry→Conservation Law.

    Symmetries of the action imply conserved quantities per Noether's theorem.

    A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
    Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

    Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

    #NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

  2. Noether’s Theorem: Symmetry→Conservation Law.

    Symmetries of the action imply conserved quantities per Noether's theorem.

    A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
    Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

    Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

    #NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

  3. ⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
    Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
    This explainer breaks it down really well. Worth a few minutes of your Monday.
    🔗 buff.ly/gEr6nQj

    #Physics #ClassicalMechanics #Science #Mechanics #STEM

  4. ⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
    Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
    This explainer breaks it down really well. Worth a few minutes of your Monday.
    🔗 buff.ly/gEr6nQj

    #Physics #ClassicalMechanics #Science #Mechanics #STEM

  5. Bumblebees and Ellipsoids (Poinsot’s Paper Spaceship #3)

    In the movie Ice Age, Manny the Mammoth says he is not fat - just poofy! It is just the fur! I feel the same must be true for bumblebees - they are fluffy, but not really fat! And yes, these were the things going through my mind, when I created this geometric drawing: A sphere is intersected by a "fat" ellipsoid that is neither an oblate UFO nor an elongated zeppelin. Poinsot's Paper Spaceship #3 - Bumblebee by elkement 2025. Geometric drawing of an ellipsoid intersecting a sphere […]

    elkement.art/2025/07/25/bumble

  6. Here we see three identical pendulums, oscillating independently. The red and purple ones are vibrating with small amplitudes and so their periods are nearly the same. But the blue one is undergoing what would be considered to be very large amplitude oscillations and has a significantly longer period. In fact, as the amplitude approaches π radians, the period increases without bound and approaches infinity.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  7. Here we see three identical pendulums, oscillating independently. The red and purple ones are vibrating with small amplitudes and so their periods are nearly the same. But the blue one is undergoing what would be considered to be very large amplitude oscillations and has a significantly longer period. In fact, as the amplitude approaches π radians, the period increases without bound and approaches infinity.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  8. The above is only strictly true for “small” amplitudes. If the #pendulum is subject to large amplitudes, it is no longer governed by the simple linear differential equations alluded to above. Given that the rod remains rigid and no energy is dissipated, the pendulum #equations may still be solved exactly, though they are now #nonlinear.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  9. The above is only strictly true for “small” amplitudes. If the #pendulum is subject to large amplitudes, it is no longer governed by the simple linear differential equations alluded to above. Given that the rod remains rigid and no energy is dissipated, the pendulum #equations may still be solved exactly, though they are now #nonlinear.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  10. In elementary #mechanics we are taught about the #SimplePendulum, which is modelled as a point #mass hanging via a rigid #rod or #string of a given length, under uniform #gravity. This simple model is also useful in introducing #OrdinaryDifferentialEquations and helps us to understand #SimpleHarmonicMotion.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  11. In elementary #mechanics we are taught about the #SimplePendulum, which is modelled as a point #mass hanging via a rigid #rod or #string of a given length, under uniform #gravity. This simple model is also useful in introducing #OrdinaryDifferentialEquations and helps us to understand #SimpleHarmonicMotion.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  12. In elementary #mechanics we are taught about the #SimplePendulum, which is modelled as a point #mass hanging via a rigid #rod or #string of a given length, under uniform #gravity. This simple model is also useful in introducing #OrdinaryDifferentialEquations and helps us to understand #SimpleHarmonicMotion.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  13. In elementary #mechanics we are taught about the #SimplePendulum, which is modelled as a point #mass hanging via a rigid #rod or #string of a given length, under uniform #gravity. This simple model is also useful in introducing #OrdinaryDifferentialEquations and helps us to understand #SimpleHarmonicMotion.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  14. In elementary #mechanics we are taught about the #SimplePendulum, which is modelled as a point #mass hanging via a rigid #rod or #string of a given length, under uniform #gravity. This simple model is also useful in introducing #OrdinaryDifferentialEquations and helps us to understand #SimpleHarmonicMotion.

    #MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation

  15. 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).

    #Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus

  16. 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).

    #Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus

  17. New #classicalmechanics video - finding normal modes of oscillation for the double pendulum. #python included of course

    youtu.be/fCUQRRScRHQ

  18. New #classicalmechanics video - finding normal modes of oscillation for the double pendulum. #python included of course

    youtu.be/fCUQRRScRHQ

  19. An Article in the Annual Review of Condensed Matter Physics on Turbulence by KR Sreenivasan and J Schumacher
    annualreviews.org/content/jour

    What is the turbulence problem, and when can we say it’s solved? 🌪️ This deep dive by Sreenivasan & Schumacher explores the math, physics, and engineering challenges of turbulence—from Navier-Stokes equations to intermittency and beyond. A must-read for anyone fascinated by chaos, complexity, and the unsolved mysteries of fluid dynamics! 🌀

    A summary of the talk presented by KR Sreenivasan in December 2023 at the International Center for Theoretical Sciences (ICTS-TIFR) in Bengaluru, as part of a program on field theory and turbulence.
    youtube.com/watch?v=fwVSBYh-KC

    "Field Theory and Turbulence" program link: icts.res.in/discussion-meeting

    #FluidDynamics #Physics #NavierStokes #UnsolvedMystery #Mechanics #Dynamics #FluidMechanics #Science #Chaos #TurbulentMotion #Randomness #Chaotic #Fluid #ClassicalMechanics
    #Turbulence

  20. New #physics #classicalMechanics video - finding the normal modes for coupled oscillators using the eigenvalue problem (and #python of course)

    youtu.be/f1v1Hm8N8mU

  21. New #physics #classicalMechanics video - finding the normal modes for coupled oscillators using the eigenvalue problem (and #python of course)

    youtu.be/f1v1Hm8N8mU

  22. This video by Up and Atom (also available on YouTube) is a great intro to John Norton’s 2008 paper ‘The dome: an unexpectedly simple failure of determinism’:

    nebula.tv/videos/upandatom-the

    The paper itself: doi.org/10.1086/594524

    #NewtonianPhysics #ClassicalMechanics #DomeParadox

  23. This video by Up and Atom (also available on YouTube) is a great intro to John Norton’s 2008 paper ‘The dome: an unexpectedly simple failure of determinism’:

    nebula.tv/videos/upandatom-the

    The paper itself: doi.org/10.1086/594524

    #NewtonianPhysics #ClassicalMechanics #DomeParadox