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#lagrangian — Public Fediverse posts

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

  1. 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

  2. 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

  3. Tracking Ice Floes

    To understand why some sea ice melts and other sea ice survives, researchers tracked millions of floes over decades. This herculean undertaking combined satellite data, weather reports, and buoy data into a database covering nearly 20 years of data. With all of that information, the team could track the changes to specific pieces of ice rather than lumping data into overall averages.

    They found that an ice floe’s fate depended strongly on the route it took: ice that slipped from its starting region into warmer, more southern regions was likely to melt. They also saw region-specific effects, like that thick sea ice was more likely to melt in the East Siberian Sea’s summer, possibly due to warmer currents. The comprehensive, fine-grained analyses possible with this ice-tracking technique offer a chance to understand why some Arctic regions are more vulnerable to warming than others. (Image credit: D. Cantelli; research credit: P. Taylor et al.; via Eos)

    #climateChange #Eulerian #fluidDynamics #Lagrangian #melting #physics #planetaryScience #science #seaIce

  4. Tracking Ice Floes

    To understand why some sea ice melts and other sea ice survives, researchers tracked millions of floes over decades. This herculean undertaking combined satellite data, weather reports, and buoy data into a database covering nearly 20 years of data. With all of that information, the team could track the changes to specific pieces of ice rather than lumping data into overall averages.

    They found that an ice floe’s fate depended strongly on the route it took: ice that slipped from its starting region into warmer, more southern regions was likely to melt. They also saw region-specific effects, like that thick sea ice was more likely to melt in the East Siberian Sea’s summer, possibly due to warmer currents. The comprehensive, fine-grained analyses possible with this ice-tracking technique offer a chance to understand why some Arctic regions are more vulnerable to warming than others. (Image credit: D. Cantelli; research credit: P. Taylor et al.; via Eos)

    #climateChange #Eulerian #fluidDynamics #Lagrangian #melting #physics #planetaryScience #science #seaIce

  5. Tracking Ice Floes

    To understand why some sea ice melts and other sea ice survives, researchers tracked millions of floes over decades. This herculean undertaking combined satellite data, weather reports, and buoy data into a database covering nearly 20 years of data. With all of that information, the team could track the changes to specific pieces of ice rather than lumping data into overall averages.

    They found that an ice floe’s fate depended strongly on the route it took: ice that slipped from its starting region into warmer, more southern regions was likely to melt. They also saw region-specific effects, like that thick sea ice was more likely to melt in the East Siberian Sea’s summer, possibly due to warmer currents. The comprehensive, fine-grained analyses possible with this ice-tracking technique offer a chance to understand why some Arctic regions are more vulnerable to warming than others. (Image credit: D. Cantelli; research credit: P. Taylor et al.; via Eos)

    #climateChange #Eulerian #fluidDynamics #Lagrangian #melting #physics #planetaryScience #science #seaIce

  6. Tracking Ice Floes

    To understand why some sea ice melts and other sea ice survives, researchers tracked millions of floes over decades. This herculean undertaking combined satellite data, weather reports, and buoy data into a database covering nearly 20 years of data. With all of that information, the team could track the changes to specific pieces of ice rather than lumping data into overall averages.

    They found that an ice floe’s fate depended strongly on the route it took: ice that slipped from its starting region into warmer, more southern regions was likely to melt. They also saw region-specific effects, like that thick sea ice was more likely to melt in the East Siberian Sea’s summer, possibly due to warmer currents. The comprehensive, fine-grained analyses possible with this ice-tracking technique offer a chance to understand why some Arctic regions are more vulnerable to warming than others. (Image credit: D. Cantelli; research credit: P. Taylor et al.; via Eos)

    #climateChange #Eulerian #fluidDynamics #Lagrangian #melting #physics #planetaryScience #science #seaIce

  7. Tracking Ice Floes

    To understand why some sea ice melts and other sea ice survives, researchers tracked millions of floes over decades. This herculean undertaking combined satellite data, weather reports, and buoy data into a database covering nearly 20 years of data. With all of that information, the team could track the changes to specific pieces of ice rather than lumping data into overall averages.

    They found that an ice floe’s fate depended strongly on the route it took: ice that slipped from its starting region into warmer, more southern regions was likely to melt. They also saw region-specific effects, like that thick sea ice was more likely to melt in the East Siberian Sea’s summer, possibly due to warmer currents. The comprehensive, fine-grained analyses possible with this ice-tracking technique offer a chance to understand why some Arctic regions are more vulnerable to warming than others. (Image credit: D. Cantelli; research credit: P. Taylor et al.; via Eos)

    #climateChange #Eulerian #fluidDynamics #Lagrangian #melting #physics #planetaryScience #science #seaIce

  8. New #python #physics video - building a 3D animation of a half-atwood with a spring using #Lagrangian mechanics and #sympy (of course #vpython too)

    youtu.be/wB_hs-Dhs3c

  9. New #python #physics video - building a 3D animation of a half-atwood with a spring using #Lagrangian mechanics and #sympy (of course #vpython too)

    youtu.be/wB_hs-Dhs3c

  10. New #python #physics video - building a 3D animation of a half-atwood with a spring using #Lagrangian mechanics and #sympy (of course #vpython too)

    youtu.be/wB_hs-Dhs3c

  11. New #python #physics video - building a 3D animation of a half-atwood with a spring using #Lagrangian mechanics and #sympy (of course #vpython too)

    youtu.be/wB_hs-Dhs3c

  12. New #python #physics video - building a 3D animation of a half-atwood with a spring using #Lagrangian mechanics and #sympy (of course #vpython too)

    youtu.be/wB_hs-Dhs3c

  13. New #physics #lagrangian - flinging a bead from a rotating stick. Yes, of course there is a #python model. Do you not even know me?

    youtu.be/IqRaVK5lQ5g

  14. New #physics #lagrangian - flinging a bead from a rotating stick. Yes, of course there is a #python model. Do you not even know me?

    youtu.be/IqRaVK5lQ5g

  15. New #physics #lagrangian - flinging a bead from a rotating stick. Yes, of course there is a #python model. Do you not even know me?

    youtu.be/IqRaVK5lQ5g

  16. New #physics #lagrangian - flinging a bead from a rotating stick. Yes, of course there is a #python model. Do you not even know me?

    youtu.be/IqRaVK5lQ5g

  17. New #physics #lagrangian - flinging a bead from a rotating stick. Yes, of course there is a #python model. Do you not even know me?

    youtu.be/IqRaVK5lQ5g

  18. New #physics #classicalmechanics video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using #lagrangian and #python

    youtu.be/DF6ixJakMDU

  19. New #physics #classicalmechanics video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using #lagrangian and #python

    youtu.be/DF6ixJakMDU

  20. New #physics #classicalmechanics video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using #lagrangian and #python

    youtu.be/DF6ixJakMDU

  21. New #physics #classicalmechanics video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using #lagrangian and #python

    youtu.be/DF6ixJakMDU

  22. New #physics #classicalmechanics video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using #lagrangian and #python

    youtu.be/DF6ixJakMDU

  23. There are just too many #lagrangian #physics problems - but I had to do a bead on a rotating hoop. Of course #python is included.

    youtu.be/gQFTs1oc78o

  24. There are just too many #lagrangian #physics problems - but I had to do a bead on a rotating hoop. Of course #python is included.

    youtu.be/gQFTs1oc78o

  25. There are just too many #lagrangian #physics problems - but I had to do a bead on a rotating hoop. Of course #python is included.

    youtu.be/gQFTs1oc78o

  26. There are just too many #lagrangian #physics problems - but I had to do a bead on a rotating hoop. Of course #python is included.

    youtu.be/gQFTs1oc78o

  27. There are just too many #lagrangian #physics problems - but I had to do a bead on a rotating hoop. Of course #python is included.

    youtu.be/gQFTs1oc78o

  28. Yes. I have ANOTHER #Lagrangian mechanics problem for you - a pendulum on an oscillating pivot point. #physics. #python
    youtu.be/ITlSFagJPAo

  29. Yes. I have ANOTHER #Lagrangian mechanics problem for you - a pendulum on an oscillating pivot point. #physics. #python
    youtu.be/ITlSFagJPAo

  30. Yes. I have ANOTHER #Lagrangian mechanics problem for you - a pendulum on an oscillating pivot point. #physics. #python
    youtu.be/ITlSFagJPAo

  31. Yes. I have ANOTHER #Lagrangian mechanics problem for you - a pendulum on an oscillating pivot point. #physics. #python
    youtu.be/ITlSFagJPAo

  32. Yes. I have ANOTHER #Lagrangian mechanics problem for you - a pendulum on an oscillating pivot point. #physics. #python
    youtu.be/ITlSFagJPAo