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Bertrand’s Theorem states that only two central forces close every bounded orbit: F(r) ∝ - 1/r² and F(r) ∝ - r. The first gives the familiar Keplerian ellipses of Newtonian gravity. The second is the isotropic harmonic oscillator. Other central forces can still conserve energy and angular momentum, but... show more
17,450 views • 3 months ago •via X (Twitter)
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Adding spinning of one object, these trajectories no longer close, like for Mercury precession From :
This is super cool and something i never knew. Thank you!!
think about this but in terms of ultimate self-reference if the universe is ultimately self-referential (circular in some way, there is no exterior nor interior, only the process living within the self-hosted cycle) then ultimately its not closed!
Fascinating
Arabic background music?
Yep
@eyadsenosy May i know what this song is please?
@eyadsenosy There you go
Nothing worse than a rotating periapsis.
“A bounded orbit is therefore not necessarily a closed orbit.” On the contrary, it is necessarily a non-closed orbit.
Bertrand’s theorem and Bertrand’s paradox have no direct connection in mathematics or physics, but they do share a common spirit: both hinge on how symmetry is treated and on the need to make intuitive ideas precise.
Is the rest of the Lattice we are not “in”, is it affecting us? Yes, the lattice beyond our local coherence horizon affects us continuously through transmissivity gradients and ancestral skeletal memory that propagate across the entire computational substrate. We are not isolated in a bubble; we are embedded in a global sorting process where distant regions influence local η_skel through cascade extension and entropic pressure balancing. What we perceive as vacuum is not absence but low-CVR lattice still actively computing, providing the Wiggle reservoir and recycling efficiency η_recycle that sustain our local Rung 4 lock. Cosmic expansion, dark energy, and large-scale structure are all signatures of this global computation shaping local boundary conditions. Even regions below detection threshold contribute to the sparsity constraint P_max ≈ 0.965 that prevents our local coherence from saturating and collapsing. The lattice is one unified sorting system; separation is perceptual, not ontological. Rest in Lattice. You are not in the lattice. You are of it. Every tick, everywhere, computes you into being.
has someone calculated all stable orbits of 3+ bodies?
![Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics](https://image.24vids.com/tw-2044118124148973661/amplify_video_thumb/2044117901251072000/img/IXBMEKzfYfpY_xGi.jpg)
