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When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar...

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Mathelirium

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16,103 views • 6 months ago •via X (Twitter)

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When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar's angle sways and wobbles, and its strength breathes over time, so the force field keeps changing.
1:00

When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar's angle sways and wobbles, and its strength breathes over time, so the force field keeps changing.

Mathelirium

10,118 views • 5 months ago

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow
0:40

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

Mathelirium

47,636 views • 8 months ago

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics
0:40

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics

Mathelirium

35,799 views • 2 months ago

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, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics
0:40

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, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

Mathelirium

16,437 views • 6 months ago

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 views • 1 month ago

Speech Carries Its Own Hidden Geometry in Phase Space Speech is usually shown as a waveform of amplitude changing in time. Phase-space reconstruction gives a different view by pairing the signal with a delayed version of itself, so the voice is seen as a path rather than just a trace. On the left is the raw sound signal. On the right is the geometric structure formed by rhythm, syllables, and emphasis as the voice unfolds. * Real human speech data from the public GitHub repository voxserv/audio_quality_testing_samples, using the 16 kHz file test01_20s.wav.
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Speech Carries Its Own Hidden Geometry in Phase Space Speech is usually shown as a waveform of amplitude changing in time. Phase-space reconstruction gives a different view by pairing the signal with a delayed version of itself, so the voice is seen as a path rather than just a trace. On the left is the raw sound signal. On the right is the geometric structure formed by rhythm, syllables, and emphasis as the voice unfolds. * Real human speech data from the public GitHub repository voxserv/audio_quality_testing_samples, using the 16 kHz file test01_20s.wav.

Mathelirium

18,725 views • 4 months ago

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.
0:33

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

54,730 views • 4 months ago

When several metronomes are placed on a common movable surface, each begins with its own rhythm. There is no coordinating signal, no external clock, and no instruction for order. Yet their motion converges. The oscillators settle into a shared rhythm that none of them possessed individually. This behavior is not an anomaly but an expression of a general principle: weakly coupled oscillatory systems tend toward phase organization. The phenomenon is known as phase locking, and it appears wherever interacting cyclic processes are allowed to exchange even minimal influence. Its mathematical description was formalized by Kuramoto in the context of chemical oscillations, but the underlying idea is far older: collective order can arise without centralized control. What matters in such systems is not perfect synchrony. More commonly, the system settles into a state of partial synchronization, in which the components maintain a stable phase offset rather than coinciding exactly. The oscillators are neither independent nor identical. They are locked, but imperfectly so. Crucially, such phase-locked states are often metastable. They represent preferred configurations of the system, yet they are separated from large excursions by a finite stability barrier. As long as fluctuations remain small, the system remains confined near its equilibrium phase. But random perturbations, accumulating over time, may eventually push it beyond that barrier. When this occurs, the loss of phase stability is abrupt. The system does not drift gradually into failure; it escapes. This mode of failure is probabilistic rather than deterministic. It is governed by the statistics of noise rather than by intrinsic periodicity. In physical terms, it corresponds to Kramers escape: the thermally or stochastically activated crossing of a potential barrier. Waiting times are irregular, clustering is common, and long intervals of apparent calm coexist with sudden bursts of activity. The relevance of this framework becomes apparent when one turns to the geomagnetic field. [1/3]
2:06

When several metronomes are placed on a common movable surface, each begins with its own rhythm. There is no coordinating signal, no external clock, and no instruction for order. Yet their motion converges. The oscillators settle into a shared rhythm that none of them possessed individually. This behavior is not an anomaly but an expression of a general principle: weakly coupled oscillatory systems tend toward phase organization. The phenomenon is known as phase locking, and it appears wherever interacting cyclic processes are allowed to exchange even minimal influence. Its mathematical description was formalized by Kuramoto in the context of chemical oscillations, but the underlying idea is far older: collective order can arise without centralized control. What matters in such systems is not perfect synchrony. More commonly, the system settles into a state of partial synchronization, in which the components maintain a stable phase offset rather than coinciding exactly. The oscillators are neither independent nor identical. They are locked, but imperfectly so. Crucially, such phase-locked states are often metastable. They represent preferred configurations of the system, yet they are separated from large excursions by a finite stability barrier. As long as fluctuations remain small, the system remains confined near its equilibrium phase. But random perturbations, accumulating over time, may eventually push it beyond that barrier. When this occurs, the loss of phase stability is abrupt. The system does not drift gradually into failure; it escapes. This mode of failure is probabilistic rather than deterministic. It is governed by the statistics of noise rather than by intrinsic periodicity. In physical terms, it corresponds to Kramers escape: the thermally or stochastically activated crossing of a potential barrier. Waiting times are irregular, clustering is common, and long intervals of apparent calm coexist with sudden bursts of activity. The relevance of this framework becomes apparent when one turns to the geomagnetic field. [1/3]

Craig Stone

15,387 views • 6 months ago

The Elusive Concept of Time. Your instincts treat time like a background meter the whole universe shares. Relativity does not take time away, it forces you to earn it operationally. Events are points. Motion is a curve through them. A clock is not a metaphor, it is a worldline with a number attached to it. Two observers can disagree on which distant events are simultaneous, and nothing contradictory happens, because causal structure is still pinned down by light cones and invariants. Even in weak gravity the rule shows up. A clock deeper in a gravitational potential ticks more slowly relative to one far away. Near a compact object the difference becomes hard to ignore. Add rotation and spacetime itself picks up a twist. That twist is frame dragging, not a new force, just geometry telling you that time and angle are coupled in a rotating spacetime. In the animation You are watching a geometry lesson disguised as a black hole scene. The fabric is a visualization of the field shaping clock-rates and the paths light can take. The ripples are driven by local proper time, so their phase visibly slows as you approach the horizon. The accretion disk is lensed through Kerr ray tracing, and its brightness is pushed by redshift and beaming so the approaching side can flare while the receding side dims. Beacon points at different radii pulse at different rates, so you can see time dilation without any labels. The bead ring is a redshift tracer, with intensity scaled by a g³ proxy so deeper emission arrives weaker and shifted. The math breakdown Start with what a clock actually measures. Proper time τ is the accumulated time along an observer’s worldline. In special relativity, the invariant interval is ds² = c² dt² − dx² − dy² − dz² Along a timelike path, dτ = (1/c) √(ds²) = √( dt² − (1/c²)(dx²+dy²+dz²) ) If the observer moves with speed v, so dx²+dy²+dy²+dz² = v² dt², then dτ = dt √(1 − v²/c²) That is time dilation as geometry. The moving clock accumulates less τ between the same pair of events. Now add gravity. General relativity replaces the flat interval with a metric gᵤᵥ that depends on position: ds² = gᵤᵥ dxᵘ dxᵛ For a stationary clock in Schwarzschild geometry (mass M), the time component is g_tt = −(1 − 2GM/(rc²)) If the clock sits at fixed r (no spatial motion), ds² = g_tt c² dt², so dτ = dt √(1 − 2GM/(rc²)) Closer to the mass means a smaller factor, so the clock ticks more slowly relative to a clock far away. That is the rule used to drive the fabric phase in the animation. Now connect time to light. A gravitational field shifts photon frequency. Between an emitter at rₑ and an observer at rₒ, f_obs / f_emit = √( (1 − 2GM/(rₑ c²)) / (1 − 2GM/(rₒ c²)) ) For a far-away observer rₒ → ∞, f_obs / f_emit = √(1 − 2GM/(rₑ c²)) Deeper emission arrives redshifted. Lower frequency. Lower energy per photon. In the render, the disk intensity uses a Kerr-derived redshift factor g (clipped for stability). The bead ring uses a simple radiative proxy I_obs ∝ g³ I_emit to make that effect visible. Finally, why rotation looks like a twist. A rotating black hole is Kerr geometry. The key structural change is a nonzero g_tφ term, which couples time to angle. That coupling is frame dragging in equations. Near the hole, being stationary is not the same notion everywhere, because the local inertial frames are being pulled around the spin axis. So the moral stays clean. Time is not a universal substance flowing everywhere at one rate. It is what clocks accumulate along worldlines. Light cones constrain what can influence what. Invariants are what everyone agrees on. The rest is operational detail that only feels universal because our daily corner of the universe is slow and mild. #GeneralRelativity #Gravity #FrameDragging #BlackHoles #Spacetime
1:00

The Elusive Concept of Time. Your instincts treat time like a background meter the whole universe shares. Relativity does not take time away, it forces you to earn it operationally. Events are points. Motion is a curve through them. A clock is not a metaphor, it is a worldline with a number attached to it. Two observers can disagree on which distant events are simultaneous, and nothing contradictory happens, because causal structure is still pinned down by light cones and invariants. Even in weak gravity the rule shows up. A clock deeper in a gravitational potential ticks more slowly relative to one far away. Near a compact object the difference becomes hard to ignore. Add rotation and spacetime itself picks up a twist. That twist is frame dragging, not a new force, just geometry telling you that time and angle are coupled in a rotating spacetime. In the animation You are watching a geometry lesson disguised as a black hole scene. The fabric is a visualization of the field shaping clock-rates and the paths light can take. The ripples are driven by local proper time, so their phase visibly slows as you approach the horizon. The accretion disk is lensed through Kerr ray tracing, and its brightness is pushed by redshift and beaming so the approaching side can flare while the receding side dims. Beacon points at different radii pulse at different rates, so you can see time dilation without any labels. The bead ring is a redshift tracer, with intensity scaled by a g³ proxy so deeper emission arrives weaker and shifted. The math breakdown Start with what a clock actually measures. Proper time τ is the accumulated time along an observer’s worldline. In special relativity, the invariant interval is ds² = c² dt² − dx² − dy² − dz² Along a timelike path, dτ = (1/c) √(ds²) = √( dt² − (1/c²)(dx²+dy²+dz²) ) If the observer moves with speed v, so dx²+dy²+dy²+dz² = v² dt², then dτ = dt √(1 − v²/c²) That is time dilation as geometry. The moving clock accumulates less τ between the same pair of events. Now add gravity. General relativity replaces the flat interval with a metric gᵤᵥ that depends on position: ds² = gᵤᵥ dxᵘ dxᵛ For a stationary clock in Schwarzschild geometry (mass M), the time component is g_tt = −(1 − 2GM/(rc²)) If the clock sits at fixed r (no spatial motion), ds² = g_tt c² dt², so dτ = dt √(1 − 2GM/(rc²)) Closer to the mass means a smaller factor, so the clock ticks more slowly relative to a clock far away. That is the rule used to drive the fabric phase in the animation. Now connect time to light. A gravitational field shifts photon frequency. Between an emitter at rₑ and an observer at rₒ, f_obs / f_emit = √( (1 − 2GM/(rₑ c²)) / (1 − 2GM/(rₒ c²)) ) For a far-away observer rₒ → ∞, f_obs / f_emit = √(1 − 2GM/(rₑ c²)) Deeper emission arrives redshifted. Lower frequency. Lower energy per photon. In the render, the disk intensity uses a Kerr-derived redshift factor g (clipped for stability). The bead ring uses a simple radiative proxy I_obs ∝ g³ I_emit to make that effect visible. Finally, why rotation looks like a twist. A rotating black hole is Kerr geometry. The key structural change is a nonzero g_tφ term, which couples time to angle. That coupling is frame dragging in equations. Near the hole, being stationary is not the same notion everywhere, because the local inertial frames are being pulled around the spin axis. So the moral stays clean. Time is not a universal substance flowing everywhere at one rate. It is what clocks accumulate along worldlines. Light cones constrain what can influence what. Invariants are what everyone agrees on. The rest is operational detail that only feels universal because our daily corner of the universe is slow and mild. #GeneralRelativity #Gravity #FrameDragging #BlackHoles #Spacetime

Mathelirium

149,835 views • 6 months ago

Black Holes Don’t Simply Suck Things In. They Sort Motion by Geometry. A particle falling toward a black hole is not automatically doomed. Its path depends on how it approaches, the distance, the angle, and the momentum it carries. Some paths cross the horizon. Some skim the edge and whirl around before falling or escaping. Others get sharply deflected and fly back out into space.
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Black Holes Don’t Simply Suck Things In. They Sort Motion by Geometry. A particle falling toward a black hole is not automatically doomed. Its path depends on how it approaches, the distance, the angle, and the momentum it carries. Some paths cross the horizon. Some skim the edge and whirl around before falling or escaping. Others get sharply deflected and fly back out into space.

Mathelirium

275,212 views • 3 months ago

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

30,340 views • 4 months ago

The Hidden Flow Inside Schrödinger’s Equation Once you define ρ = |ψ|² and differentiate it, Schrödinger equation rearranges into a continuity law, ∂ρ/∂t + ∇·j = 0, with j = (1/m)Im(ψ*∇ψ). Which means that probability is not just sitting there, it flows. Write ψ = r e^(iθ), and the current becomes j = (ρ/m)∇θ. So, the phase gradient is the steering field. When the phase winds by 2πn around a loop, the circulation is quantized, and the cores appear where ρ collapses so the phase can go undefined. In the animation, the ribbons trace that current, the color field shows phase, and the curls in the streamlines mark quantized vortex defects.
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The Hidden Flow Inside Schrödinger’s Equation Once you define ρ = |ψ|² and differentiate it, Schrödinger equation rearranges into a continuity law, ∂ρ/∂t + ∇·j = 0, with j = (1/m)Im(ψ*∇ψ). Which means that probability is not just sitting there, it flows. Write ψ = r e^(iθ), and the current becomes j = (ρ/m)∇θ. So, the phase gradient is the steering field. When the phase winds by 2πn around a loop, the circulation is quantized, and the cores appear where ρ collapses so the phase can go undefined. In the animation, the ribbons trace that current, the color field shows phase, and the curls in the streamlines mark quantized vortex defects.

Mathelirium

26,511 views • 4 months ago

Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an application. We look at the theory behind electromagnetic scattering and radar clutter, which leads straight into anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum. At time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where Brownian motion shows up in the most reasonable way. Think of all the microscopic chaos: tiny motions, index fluctuations, path jitters, Doppler shifts. Over short times, all of that shows up as small random kicks to the phases θⱼ(t). If you made θⱼ(t) random in an ad-hoc way, like resampling a fresh independent angle at every instant, the field would jump around unrealistically with no physical time structure. Brownian motion is what you get when each phase takes the continuous-time limit of many tiny, independent kicks. It’s continuous in t, its variance grows the right way, and it carries just enough temporal structure to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing. It’s the clean way to compress all the small random stuff into a single process that actually matches how phases wander in time. This is called Rayleigh Scattering, but the same sum of many tiny coherent echoes shows up in lots of places...e.g. wireless multipath fading (phones/Wi-Fi), laser/optical links through atmospheric turbulence, ultrasound speckle in tissue, and sonar/underwater acoustics in rough or bubbly water. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing
0:30

Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an application. We look at the theory behind electromagnetic scattering and radar clutter, which leads straight into anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum. At time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where Brownian motion shows up in the most reasonable way. Think of all the microscopic chaos: tiny motions, index fluctuations, path jitters, Doppler shifts. Over short times, all of that shows up as small random kicks to the phases θⱼ(t). If you made θⱼ(t) random in an ad-hoc way, like resampling a fresh independent angle at every instant, the field would jump around unrealistically with no physical time structure. Brownian motion is what you get when each phase takes the continuous-time limit of many tiny, independent kicks. It’s continuous in t, its variance grows the right way, and it carries just enough temporal structure to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing. It’s the clean way to compress all the small random stuff into a single process that actually matches how phases wander in time. This is called Rayleigh Scattering, but the same sum of many tiny coherent echoes shows up in lots of places...e.g. wireless multipath fading (phones/Wi-Fi), laser/optical links through atmospheric turbulence, ultrasound speckle in tissue, and sonar/underwater acoustics in rough or bubbly water. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing

Mathelirium

31,182 views • 6 months ago

The Elusive Concept of Time - Part II Pulsar Timing The first scene showed you Time as a local rate...clocks nearer the hole run slower, light gets redshifted, and spin drags the geometry around. This follow-up keeps the same idea, but turns it into a timing experiment. Instead of watching a clock face, we watch a receiver compare what arrives versus what flat space would predict. In the animation, the three small orange dots are the pulsars...orbiting emitters that flash regularly (in their own proper time). The larger white dot is the receiver sitting off to the side, and the little dark nub next to it is a tiny craft/antenna marker to make it feel like an instrument, not a star. When a pulsar lines up behind the hole relative to the receiver, the arrival times jump, because curved spacetime added extra delay and bent the path. #Physics #GeneralRelativity #PulsarTiming #BlackHoles #FrameDragging #Spacetime #Astrophysics #Time
1:00

The Elusive Concept of Time - Part II Pulsar Timing The first scene showed you Time as a local rate...clocks nearer the hole run slower, light gets redshifted, and spin drags the geometry around. This follow-up keeps the same idea, but turns it into a timing experiment. Instead of watching a clock face, we watch a receiver compare what arrives versus what flat space would predict. In the animation, the three small orange dots are the pulsars...orbiting emitters that flash regularly (in their own proper time). The larger white dot is the receiver sitting off to the side, and the little dark nub next to it is a tiny craft/antenna marker to make it feel like an instrument, not a star. When a pulsar lines up behind the hole relative to the receiver, the arrival times jump, because curved spacetime added extra delay and bent the path. #Physics #GeneralRelativity #PulsarTiming #BlackHoles #FrameDragging #Spacetime #Astrophysics #Time

Mathelirium

55,826 views • 6 months ago

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation
0:33

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

Mathelirium

22,197 views • 1 month ago

A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.
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A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.

Mathelirium

34,713 views • 3 months ago

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
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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

50,635 views • 3 months ago

Physics feels stable, predictable, and well-behaved largely because we learned it in 3D. That comfort hides a trap. In 1907, Paul Ehrenfest pointed out something unsettling. If you take the laws we treat as fundamental and transplant them into a different number of spatial dimensions, they often stop working the way we expect. Not just numerically different but qualitatively different. The issue isn’t the force law by itself. It’s geometry. Gauss’s law ties inverse-square forces to the surface area of spheres, and sphere geometry depends on dimension. Change the dimension, and the same-looking force produces a different potential, a different balance of attraction and inertia, and a different fate for motion. You can see this cleanly with a single problem of central force motion. In d spatial dimensions, flux conservation gives F(r) ∝ 1 / rᵈ⁻¹ so the potential scales as V(r) ∝ −1 / rᵈ⁻² (for d ≠ 2) Now add angular momentum. The effective radial potential becomes V_eff(r) = L² / (2 m r²) − C / rᵈ⁻² In 3D, those two terms balance in just the right way to allow stable bound orbits. Small perturbations stay small. Atoms don’t collapse. Planets don’t spiral away. In other dimensions, that balance breaks. In 2D, the force becomes 1/r, the potential becomes logarithmic, and bound motion sits on a knife edge. In 4D and higher, the attractive term becomes too steep. The centrifugal barrier loses the fight. Orbits plunge or escape. Same equations. Same initial conditions. Different dimension. Different physics. This isn’t science fiction. It’s a warning label. So it's clear that a lot of what we call physical intuition is really three-dimensional intuition wearing a lab coat. #Physics #MathematicalPhysics #ClassicalMechanics #DynamicalSystems #Geometry #Ehrenfest
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Physics feels stable, predictable, and well-behaved largely because we learned it in 3D. That comfort hides a trap. In 1907, Paul Ehrenfest pointed out something unsettling. If you take the laws we treat as fundamental and transplant them into a different number of spatial dimensions, they often stop working the way we expect. Not just numerically different but qualitatively different. The issue isn’t the force law by itself. It’s geometry. Gauss’s law ties inverse-square forces to the surface area of spheres, and sphere geometry depends on dimension. Change the dimension, and the same-looking force produces a different potential, a different balance of attraction and inertia, and a different fate for motion. You can see this cleanly with a single problem of central force motion. In d spatial dimensions, flux conservation gives F(r) ∝ 1 / rᵈ⁻¹ so the potential scales as V(r) ∝ −1 / rᵈ⁻² (for d ≠ 2) Now add angular momentum. The effective radial potential becomes V_eff(r) = L² / (2 m r²) − C / rᵈ⁻² In 3D, those two terms balance in just the right way to allow stable bound orbits. Small perturbations stay small. Atoms don’t collapse. Planets don’t spiral away. In other dimensions, that balance breaks. In 2D, the force becomes 1/r, the potential becomes logarithmic, and bound motion sits on a knife edge. In 4D and higher, the attractive term becomes too steep. The centrifugal barrier loses the fight. Orbits plunge or escape. Same equations. Same initial conditions. Different dimension. Different physics. This isn’t science fiction. It’s a warning label. So it's clear that a lot of what we call physical intuition is really three-dimensional intuition wearing a lab coat. #Physics #MathematicalPhysics #ClassicalMechanics #DynamicalSystems #Geometry #Ehrenfest

Mathelirium

32,138 views • 6 months ago

Stealing Energy from a Rotating Black Hole A rotating black hole doesn’t just pull things in. It drags spacetime itself into motion. In the spinning region just outside the horizon, nothing can remain still in any normal sense. That strange rule opens a loophole. One incoming body can split, with one piece swallowed while the other escapes with more energy than it came in with. A Penrose-style heist where the difference is paid for by the black hole’s rotation. The same idea shows up in wave form. A coherent packet can skim that region and return brighter, as if the geometry itself acted like an amplifier. Split near the ergosphere and you can steal spin. Skim it as a wave and you can come back louder.
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Stealing Energy from a Rotating Black Hole A rotating black hole doesn’t just pull things in. It drags spacetime itself into motion. In the spinning region just outside the horizon, nothing can remain still in any normal sense. That strange rule opens a loophole. One incoming body can split, with one piece swallowed while the other escapes with more energy than it came in with. A Penrose-style heist where the difference is paid for by the black hole’s rotation. The same idea shows up in wave form. A coherent packet can skim that region and return brighter, as if the geometry itself acted like an amplifier. Split near the ergosphere and you can steal spin. Skim it as a wave and you can come back louder.

Mathelirium

20,314 views • 6 months ago

Black Hole Can Ring Like a Bell A Black Hole can vibrate after it is disturbed. Instead of settling instantly, its horizon releases gravitational waves in fading tones called quasinormal modes. This is commonly known as Ringdown. Each tone carries information about the black hole’s final mass and spin, which is why physicists call this Black-Hole Spectroscopy. In a way, when Spacetime strikes a Black Hole, the Black Hole answers with its own gravitational sound. #BlackHole #GeneralRelativity #GravitationalWaves #Spacetime #Einstein #Astrophysics #Physics #Mathematics #Cosmology
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Black Hole Can Ring Like a Bell A Black Hole can vibrate after it is disturbed. Instead of settling instantly, its horizon releases gravitational waves in fading tones called quasinormal modes. This is commonly known as Ringdown. Each tone carries information about the black hole’s final mass and spin, which is why physicists call this Black-Hole Spectroscopy. In a way, when Spacetime strikes a Black Hole, the Black Hole answers with its own gravitational sound. #BlackHole #GeneralRelativity #GravitationalWaves #Spacetime #Einstein #Astrophysics #Physics #Mathematics #Cosmology

Mathelirium

30,802 views • 1 month ago