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We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch,...

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Mathelirium

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Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics
0:30

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Mathelirium

30,529 görüntüleme • 7 ay önce

If we want to present a sheet on top of another sheet in SwiftUI, we have to attach the second sheet() modifier inside the content of the first sheet view, otherwise it won't work properly and SwiftUI will complain that only presenting a single sheet is supported.
0:23

If we want to present a sheet on top of another sheet in SwiftUI, we have to attach the second sheet() modifier inside the content of the first sheet view, otherwise it won't work properly and SwiftUI will complain that only presenting a single sheet is supported.

Natalia Panferova

18,895 görüntüleme • 1 yıl önce

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics
1:00

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics

Mathelirium

50,730 görüntüleme • 6 ay önce

Gravity makes things orbit...but what happens when the mass distribution itself is rotating and dragging phase space into resonances and chaos? In this scene we follow tens of thousands of stars in a toy barred galaxy: a flat disc with a central mass and a rotating bar of extra density that is not rigid, but wobbling, swaying, and breathing in strength over time. The stars start on almost circular orbits in a calm ring, but as the bar turns and its pattern speed and strength modulate, they feel a time-dependent gravitational shove that nudges some onto resonant tracks. 🤩😍 #Mathelirium #Astrophysics #GalacticDynamics #BarredGalaxy #PhaseSpace #HamiltonianChaos #OrbitalResonance
0:54

Gravity makes things orbit...but what happens when the mass distribution itself is rotating and dragging phase space into resonances and chaos? In this scene we follow tens of thousands of stars in a toy barred galaxy: a flat disc with a central mass and a rotating bar of extra density that is not rigid, but wobbling, swaying, and breathing in strength over time. The stars start on almost circular orbits in a calm ring, but as the bar turns and its pattern speed and strength modulate, they feel a time-dependent gravitational shove that nudges some onto resonant tracks. 🤩😍 #Mathelirium #Astrophysics #GalacticDynamics #BarredGalaxy #PhaseSpace #HamiltonianChaos #OrbitalResonance

Mathelirium

152,209 görüntüleme • 11 ay önce

"We do have our own balance sheet that in a if problems were to exist we are in a position to help solve those." Let me translate: XRP is going on the balance sheet. USDC doesn't have that and Tether doesn't have regulators behind it.
2:33

"We do have our own balance sheet that in a if problems were to exist we are in a position to help solve those." Let me translate: XRP is going on the balance sheet. USDC doesn't have that and Tether doesn't have regulators behind it.

Digital Asset Investor

173,759 görüntüleme • 1 yıl önce

What Shape Does Chaos Take Before It Looks Random? Take one particle moving in the Hénon-Heiles Hamiltonian H = ½(pₓ² + pᵧ²) + ½(x² + y²) + λ(x²y - ⅓y³) Its motion is forced by Hamilton’s equations ẋ = pₓ ẏ = pᵧ ṗₓ = -x - 2λxy ṗᵧ = -y - λ(x² - y²) The saddle energy is Eₛ = 1/(6λ²) Below that energy, the particle remains trapped. But a tiny packet of nearby initial conditions stretches into filaments, folds through the potential, and begins to look almost random. However, it is not random. Hamiltonian flow preserves the hidden phase-space structure. What looks like chaos in ordinary space is still a deterministic loom of energy conserved, equations obeyed, geometry stretched without being erased. #HamiltonianMechanics #ChaosTheory #PhaseSpace #DynamicalSystems #MathematicalPhysics #NonlinearDynamics #HenonHeiles #SymplecticGeometry #ScienceAnimation
0:58

What Shape Does Chaos Take Before It Looks Random? Take one particle moving in the Hénon-Heiles Hamiltonian H = ½(pₓ² + pᵧ²) + ½(x² + y²) + λ(x²y - ⅓y³) Its motion is forced by Hamilton’s equations ẋ = pₓ ẏ = pᵧ ṗₓ = -x - 2λxy ṗᵧ = -y - λ(x² - y²) The saddle energy is Eₛ = 1/(6λ²) Below that energy, the particle remains trapped. But a tiny packet of nearby initial conditions stretches into filaments, folds through the potential, and begins to look almost random. However, it is not random. Hamiltonian flow preserves the hidden phase-space structure. What looks like chaos in ordinary space is still a deterministic loom of energy conserved, equations obeyed, geometry stretched without being erased. #HamiltonianMechanics #ChaosTheory #PhaseSpace #DynamicalSystems #MathematicalPhysics #NonlinearDynamics #HenonHeiles #SymplecticGeometry #ScienceAnimation

Mathelirium

26,893 görüntüleme • 4 ay önce

Normalize Working in Phase Space Ordinary space shows where the particle is. Phase space shows its complete state, position and momentum together, revealing the hidden geometry of its motion.
0:30

Normalize Working in Phase Space Ordinary space shows where the particle is. Phase space shows its complete state, position and momentum together, revealing the hidden geometry of its motion.

Mathelirium

103,511 görüntüleme • 3 ay önce

😎🤌 According to Bob Lazar, UAP operate using a gravity wave-based propulsion system that distorts space-time rather than using conventional thrust 👽🛸 The craft generate power and gravity waves through a reactor fueled by a stable isotope of Element 115. In the reactor, Element 115 is bombarded with protons, briefly becomes Element 116, decays, and releases small amounts of antimatter. This antimatter annihilates with matter to produce enormous energy (near 100% efficient), which powers the craft and also creates a unique Gravity A wave (a micro-scale gravity field Lazar says is an extension of the strong nuclear force). This Gravity A wave forms around the reactor at the top of the craft. A waveguide channels it downward into three gravity amplifiers arranged in a triangle at the base. These amplifiers run at full power at all times and can phase-shift the wave (0° to 180°) to control direction and strength. By focusing and amplifying the gravity wave, the craft creates a powerful, localized gravitational distortion: - It bends/warps space-time in front of the vehicle. - This effectively pulls the destination toward the craft (instead of pushing the craft forward through space). Operation modes: - Omicron mode (hovering/low speed): Amplifiers point downward → craft balances/sits on its own gravity field (explains the characteristic wobble/bobbing seen in many videos). - Delta mode (high speed/travel): Amplifiers point forward → extreme space-time warping for rapid movement. Key effects: - No inertia (occupants feel no acceleration). - Time dilation near the craft. - Optical distortion (gravity lensing can make the craft appear blurry, shimmering, or invisible). In essence, the craft "falls" toward its target by warping space ahead of it, allowing extreme speeds, sharp turns, and interstellar travel without conventional motion or time dilation issues for the occupants. (#UAP #UFO #UFOSighting)
17:37

😎🤌 According to Bob Lazar, UAP operate using a gravity wave-based propulsion system that distorts space-time rather than using conventional thrust 👽🛸 The craft generate power and gravity waves through a reactor fueled by a stable isotope of Element 115. In the reactor, Element 115 is bombarded with protons, briefly becomes Element 116, decays, and releases small amounts of antimatter. This antimatter annihilates with matter to produce enormous energy (near 100% efficient), which powers the craft and also creates a unique Gravity A wave (a micro-scale gravity field Lazar says is an extension of the strong nuclear force). This Gravity A wave forms around the reactor at the top of the craft. A waveguide channels it downward into three gravity amplifiers arranged in a triangle at the base. These amplifiers run at full power at all times and can phase-shift the wave (0° to 180°) to control direction and strength. By focusing and amplifying the gravity wave, the craft creates a powerful, localized gravitational distortion: - It bends/warps space-time in front of the vehicle. - This effectively pulls the destination toward the craft (instead of pushing the craft forward through space). Operation modes: - Omicron mode (hovering/low speed): Amplifiers point downward → craft balances/sits on its own gravity field (explains the characteristic wobble/bobbing seen in many videos). - Delta mode (high speed/travel): Amplifiers point forward → extreme space-time warping for rapid movement. Key effects: - No inertia (occupants feel no acceleration). - Time dilation near the craft. - Optical distortion (gravity lensing can make the craft appear blurry, shimmering, or invisible). In essence, the craft "falls" toward its target by warping space ahead of it, allowing extreme speeds, sharp turns, and interstellar travel without conventional motion or time dilation issues for the occupants. (#UAP #UFO #UFOSighting)

★ UAP Luigi ★ 👁️⃤ 🇺🇸 👽MJ12 SOM1-01

43,212 görüntüleme • 10 ay önce

In String Theory, we ask a deceptively simple question What happens when you replace a point with something stretched across space and time? On one side, the particle draws a world line. On the other, a string fills out a world-sheet as every point along it evolves together.
0:58

In String Theory, we ask a deceptively simple question What happens when you replace a point with something stretched across space and time? On one side, the particle draws a world line. On the other, a string fills out a world-sheet as every point along it evolves together.

Mathelirium

18,756 görüntüleme • 5 ay önce

#WATCH | Siliguri: ISRO chief S Somanath says, "...Space Sector Reform was announced in 2020 and we have a new space policy 2023 by the government. This enables private companies and institutions to come into the space sector and work here. So, we are looking at substantial growth in space activity..."
1:08

#WATCH | Siliguri: ISRO chief S Somanath says, "...Space Sector Reform was announced in 2020 and we have a new space policy 2023 by the government. This enables private companies and institutions to come into the space sector and work here. So, we are looking at substantial growth in space activity..."

ANI

34,527 görüntüleme • 2 yıl önce

🚨 Did you know Bob Lazar blew the whistle on alien tech back in 1989? In his documentary, he details how UFOs achieve interstellar travel by distorting space-time inside an anti-gravity field. Here's the mind-blowing breakdown, straight from his theories: 🚀🛸 - **The Core Propulsion System**: Lazar claims the crafts use "gravity amplifiers" – three devices at the base that focus and amplify gravitational waves. These aren't your typical engines; they create a localized anti-gravity field around the craft, allowing it to defy Earth's gravity effortlessly. - **Distorting Space-Time**: Inside this field, the amplifiers bend space-time itself. Think Einstein's relativity on steroids: by projecting a strong gravitational wave ahead, the craft "pulls" space toward it, shortening vast interstellar distances. It's like folding a map so points A and B touch – no need for light-speed travel; you warp the fabric of reality instead. - **Power Source: Element 115**: The key fuel is stable isotopes of Element 115 (now called Moscovium, synthesized in labs but unstable here). When protons bombard it in a reactor, it produces anti-matter, annihilating with matter to generate massive energy (100% efficient, no waste). More crucially, Element 115's superheavy nucleus emits gravity-A waves (a strong nuclear force extension), which the amplifiers harness to create the distortion field. - **How It Works in Flight**: In "delta mode" (for interstellar jumps), the amplifiers tilt to focus the gravity wave forward, creating a "downhill" slope in space-time that the craft "falls" along at incredible speeds. In "omicron mode" (hover/low-speed), it balances against planetary gravity for silent, agile maneuvers – explaining those impossible UFO zigzags. This isn't sci-fi; Lazar says he reverse-engineered nine such discs at S-4 (near Area 51). Watch the attached video clip from his doc where he explains it firsthand! 🤯 Did Bob Lazar really work where he claimed (S-4/Area 51), and if this tech was possible back in 1989, are we already traveling interstellar today? Drop your thoughts below! #BobLazar #UFO #Area51 #Element115 #InterstellarTravel
4:11

🚨 Did you know Bob Lazar blew the whistle on alien tech back in 1989? In his documentary, he details how UFOs achieve interstellar travel by distorting space-time inside an anti-gravity field. Here's the mind-blowing breakdown, straight from his theories: 🚀🛸 - **The Core Propulsion System**: Lazar claims the crafts use "gravity amplifiers" – three devices at the base that focus and amplify gravitational waves. These aren't your typical engines; they create a localized anti-gravity field around the craft, allowing it to defy Earth's gravity effortlessly. - **Distorting Space-Time**: Inside this field, the amplifiers bend space-time itself. Think Einstein's relativity on steroids: by projecting a strong gravitational wave ahead, the craft "pulls" space toward it, shortening vast interstellar distances. It's like folding a map so points A and B touch – no need for light-speed travel; you warp the fabric of reality instead. - **Power Source: Element 115**: The key fuel is stable isotopes of Element 115 (now called Moscovium, synthesized in labs but unstable here). When protons bombard it in a reactor, it produces anti-matter, annihilating with matter to generate massive energy (100% efficient, no waste). More crucially, Element 115's superheavy nucleus emits gravity-A waves (a strong nuclear force extension), which the amplifiers harness to create the distortion field. - **How It Works in Flight**: In "delta mode" (for interstellar jumps), the amplifiers tilt to focus the gravity wave forward, creating a "downhill" slope in space-time that the craft "falls" along at incredible speeds. In "omicron mode" (hover/low-speed), it balances against planetary gravity for silent, agile maneuvers – explaining those impossible UFO zigzags. This isn't sci-fi; Lazar says he reverse-engineered nine such discs at S-4 (near Area 51). Watch the attached video clip from his doc where he explains it firsthand! 🤯 Did Bob Lazar really work where he claimed (S-4/Area 51), and if this tech was possible back in 1989, are we already traveling interstellar today? Drop your thoughts below! #BobLazar #UFO #Area51 #Element115 #InterstellarTravel

Interstellar

55,298 görüntüleme • 9 ay önce

A Neural Sheet Discovers the Shape of Data A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron, c = argminᵢ ‖x - wᵢ‖, and updates that neuron together with its neighbours, wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ). As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure. Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates. #MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen
0:59

A Neural Sheet Discovers the Shape of Data A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron, c = argminᵢ ‖x - wᵢ‖, and updates that neuron together with its neighbours, wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ). As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure. Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates. #MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen

Mathelirium

177,699 görüntüleme • 4 ay önce

When you let the Mathematics and the Physics do their thing… you get something beautiful.🤩🙌🏻😍 We evolve a two-component quantum field (ψ1, ψ2) under a coupled Gross-Pitaevskii/Schrödinger flow: split-step Fourier time stepping, self-gravity via a Poisson solve, and coherent Rabi mixing between components. Then we render the same state twice...a 3D density sheet (height driven by ρ) with molten-gold shading, and a lifted square detector plane carrying the exact same 2D texture. From ψ1, ψ2 we form the total density ρ = |ψ1|^2 + |ψ2|^2 and total current j = Im(ψ*∇ψ) summed across components, then the flow field v = j/(ρ+ε). Tracer comets are just Lagrangian particles advected by that v, so they’re literally riding the probability/current geometry the PDE is producing. The punchline moment is the Kibble-Zurek quench. Miscibility breaks, domains nucleate, and the separating walls ignite using the Bloch order parameter. Support us by buying a coffee #QuantumMechanics #GrossPitaevskii #KibbleZurek #SchrodingerEquation #QuantumFluids #NonlinearDynamics #ComputationalPhysics #PhysicsVisualization
0:50

When you let the Mathematics and the Physics do their thing… you get something beautiful.🤩🙌🏻😍 We evolve a two-component quantum field (ψ1, ψ2) under a coupled Gross-Pitaevskii/Schrödinger flow: split-step Fourier time stepping, self-gravity via a Poisson solve, and coherent Rabi mixing between components. Then we render the same state twice...a 3D density sheet (height driven by ρ) with molten-gold shading, and a lifted square detector plane carrying the exact same 2D texture. From ψ1, ψ2 we form the total density ρ = |ψ1|^2 + |ψ2|^2 and total current j = Im(ψ*∇ψ) summed across components, then the flow field v = j/(ρ+ε). Tracer comets are just Lagrangian particles advected by that v, so they’re literally riding the probability/current geometry the PDE is producing. The punchline moment is the Kibble-Zurek quench. Miscibility breaks, domains nucleate, and the separating walls ignite using the Bloch order parameter. Support us by buying a coffee #QuantumMechanics #GrossPitaevskii #KibbleZurek #SchrodingerEquation #QuantumFluids #NonlinearDynamics #ComputationalPhysics #PhysicsVisualization

Mathelirium

16,529 görüntüleme • 10 ay önce

If we ever want to live in space, we have to find a way to create earth-like levels of gravity in space. NASA actually had viable designs for artificial gravity space stations in the 1960s but the Apollo mission took up all the available resources. Projects that furthered space settlement were sacrificed in favor of projects that helped us land on the moon. Listen to Inflatable space stations to learn about creating artificial gravity on WiP Out Loud. Apple podcasts: Spotify: YouTube:
19:02

If we ever want to live in space, we have to find a way to create earth-like levels of gravity in space. NASA actually had viable designs for artificial gravity space stations in the 1960s but the Apollo mission took up all the available resources. Projects that furthered space settlement were sacrificed in favor of projects that helped us land on the moon. Listen to Inflatable space stations to learn about creating artificial gravity on WiP Out Loud. Apple podcasts: Spotify: YouTube:

Works in Progress

16,426 görüntüleme • 7 ay önce

Did you know Bob Lazar blew the whistle on alien tech back in 1989? In his documentary, he details how UFOs achieve interstellar travel by distorting space time inside an anti-gravity field. Here's the mind-blowing breakdown, straight from his theories: 👽🛸 **The Core Propulsion System**: Lazar claims the crafts use "gravity amplifiers" – three devices at the base that focus and amplify gravitational waves. These aren't your typical engines; they create a localized anti-gravity field around the craft, allowing it to defy Earth's gravity effortlessly. **Distorting Space-Time**: Inside this field, the amplifiers bend space-time itself. Think Einstein's relativity on steroids: by projecting a strong gravitational wave ahead, the craft "pulls" space toward it, shortening vast interstellar distances. It's like folding a map so points A and B touch – no need for light-speed travel; you warp the fabric of reality instead. **Power Source: Element 115**: The key fuel is stable isotopes of Element 115 (now called Moscovium, synthesized in labs but unstable here). When protons bombard it in a reactor, it produces anti-matter, annihilating with matter to generate massive energy (100% efficient, no waste). More crucially, Element 115's superheavy nucleus emits gravity-A waves (a strong nuclear force extension), which the amplifiers harness to create the distortion field. **How It Works in Flight**: In "delta mode" (for interstellar jumps), the amplifiers tilt to focus the gravity wave forward, creating a "downhill" slope in space-time that the craft "falls" along at incredible speeds. In "omicron mode" (hover/low-speed), it balances against planetary gravity for silent, agile maneuvers – explaining those impossible UFO zigzags. Lazar says he reverse-engineered nine such discs at S-4 (near Area 51).
4:11

Did you know Bob Lazar blew the whistle on alien tech back in 1989? In his documentary, he details how UFOs achieve interstellar travel by distorting space time inside an anti-gravity field. Here's the mind-blowing breakdown, straight from his theories: 👽🛸 **The Core Propulsion System**: Lazar claims the crafts use "gravity amplifiers" – three devices at the base that focus and amplify gravitational waves. These aren't your typical engines; they create a localized anti-gravity field around the craft, allowing it to defy Earth's gravity effortlessly. **Distorting Space-Time**: Inside this field, the amplifiers bend space-time itself. Think Einstein's relativity on steroids: by projecting a strong gravitational wave ahead, the craft "pulls" space toward it, shortening vast interstellar distances. It's like folding a map so points A and B touch – no need for light-speed travel; you warp the fabric of reality instead. **Power Source: Element 115**: The key fuel is stable isotopes of Element 115 (now called Moscovium, synthesized in labs but unstable here). When protons bombard it in a reactor, it produces anti-matter, annihilating with matter to generate massive energy (100% efficient, no waste). More crucially, Element 115's superheavy nucleus emits gravity-A waves (a strong nuclear force extension), which the amplifiers harness to create the distortion field. **How It Works in Flight**: In "delta mode" (for interstellar jumps), the amplifiers tilt to focus the gravity wave forward, creating a "downhill" slope in space-time that the craft "falls" along at incredible speeds. In "omicron mode" (hover/low-speed), it balances against planetary gravity for silent, agile maneuvers – explaining those impossible UFO zigzags. Lazar says he reverse-engineered nine such discs at S-4 (near Area 51).

Interstellar

20,021 görüntüleme • 4 ay önce

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠
0:45

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠

Mathelirium

109,155 görüntüleme • 1 yıl önce

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
1:00

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

Mathelirium

20,653 görüntüleme • 7 ay önce

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
0:29

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

106,981 görüntüleme • 10 ay önce

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
0:30

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

23,254 görüntüleme • 5 ay önce

We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data? In this scene we watch a self-organizing map(SOM)...a classic unsupervised neural model...learn a 2D dataset arranged like a spiral arm. Over the points we place a square grid of neurons whose weights live in the same plane. At the start it’s just a flat net drifting across the cloud. No clue what the structure is. Fir the SOM, learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, then nudge that neuron and its neighbours toward the point. Repeat, repeat, repeat...while gradually shrinking how wide the neighbourhood influence spreads. The result is satisfying...the grid stops being a grid and turns into a coordinate sheet wrapped onto the spiral. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML #SelfOrganizingMap #Topology
0:59

We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data? In this scene we watch a self-organizing map(SOM)...a classic unsupervised neural model...learn a 2D dataset arranged like a spiral arm. Over the points we place a square grid of neurons whose weights live in the same plane. At the start it’s just a flat net drifting across the cloud. No clue what the structure is. Fir the SOM, learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, then nudge that neuron and its neighbours toward the point. Repeat, repeat, repeat...while gradually shrinking how wide the neighbourhood influence spreads. The result is satisfying...the grid stops being a grid and turns into a coordinate sheet wrapped onto the spiral. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML #SelfOrganizingMap #Topology

Mathelirium

23,021 görüntüleme • 9 ay önce