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A physical system, its phase-space state, and the energy function that governs the flow, all linked together. The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve... show more

Mathelirium

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Stellarix6 ay önce

This is a classic way to unify mechanics through phase space in Classical mechanics a pendulum’s physical swing, its phase-space trajectory (position–momentum evolution), and its energy function are different representations of the same underlying dynamical system governed by conservation laws and differential equations.👍🏻

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Yang Yang (John)6 ay önce

symplectic

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Julio Ruiz6 ay önce

Beautiful!

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Just Another Opinion6 ay önce

Woahh.. that is a excellent insight. The connection between energy landscape and phase space and physical system is neat. If i am understanding correctly the hamiltonian described the energy space? Does the energy surface have some correlation to the fields, like EM or Gravity

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Henning Sittler6 ay önce

👍

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Don Ariel (a proud dyslexic)6 ay önce

Hey- if any of your followers publish on arXiv- I need an endorser to become one. It would be huge! Endorsement Code: LT6YI3

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Lindsay Rex6 ay önce

Absolutely love this!

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Sergey Popach6 ay önce

it’s classical physics example

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Teo The Money East6 ay önce

@QuantumDom Another reason for Compute is the new currency

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Delcio Henrique da Silva Borges da Cunha6 ay önce

If entanglement takes 232 attoseconds, it's not "spooky" or instant. It's mediated. X Theory proposes the medium: the X field. L_X = X * R sets the propagation speed through spacetime structure. Entanglement = geometry, not magic. ER=EPR via X field. 10.5281/zenodo.19657030

Thechaingang profil fotoğrafı

Thechaingang6 ay önce

📐 Just submitted to viXra: "Non-Closure of Doubling Vesica Chains: A Constructive Proof of Irreducible Remainder in Self-Similar Geometric Expansions" On an isometric dot lattice with compass & straightedge, doubling V6 chains (R, 2R, 4R, …) always leave an exact irreducible remainder of 1·R. The generating radius itself cannot be consumed. viXra ref: 17963181 (processing now) Part of my Geosectometry framework. Feedback welcome! #Geosectometry #VesicaPiscis #Geometry #SelfSimilarity #ConstructiveMath

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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.

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

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

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

45,091 görüntüleme • 5 ay 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

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

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

Mathelirium

10,806 görüntüleme • 6 ay önce

Visualization of the harmonic oscillator. Showing how x(t) = A cos(ωt + φ) is the horizontal projection of uniform circular motion. The spring–mass system and the closed ellipse in (x, v) phase space represent the same evolving state, while kinetic and elastic potential energy continuously exchange as the total energy remains constant.

Visualization of the harmonic oscillator. Showing how x(t) = A cos(ωt + φ) is the horizontal projection of uniform circular motion. The spring–mass system and the closed ellipse in (x, v) phase space represent the same evolving state, while kinetic and elastic potential energy continuously exchange as the total energy remains constant.

Philosophy Of Physics

10,605 görüntüleme • 3 ay önce

Dr. Sue Morter: "The foundation of our reality is a collective consciousness that we have created together" “This energy rises up and cycles through a toric field flow, then it projects a movie out in front of us, and we walk into that movie every day of our lives.” All physical matter is compressed energy. Research traces that energy with no interruption all the way back to the unified field of consciousness. There is nothing that proves we are separate from source. With each layer of compression we move closer to physical form. Energy hits the earth, rises, cycles, and projects the reality we experience. Science shows the foundation of that reality is a collective consciousness we have created and agreed upon together. We treat it as solid and fixed, yet we are the consciousness of the entire system. We stand as the doorway between the infinite and the finite, the formless and the form. What we allow to pass through us without interruption determines the quality of the movie we walk into. When we conclude we are not enough, that life is unfair, or that nothing will work, that conclusion becomes the reality we inhabit. We have input. The movie is not fixed.

Dr. Sue Morter: "The foundation of our reality is a collective consciousness that we have created together" “This energy rises up and cycles through a toric field flow, then it projects a movie out in front of us, and we walk into that movie every day of our lives.” All physical matter is compressed energy. Research traces that energy with no interruption all the way back to the unified field of consciousness. There is nothing that proves we are separate from source. With each layer of compression we move closer to physical form. Energy hits the earth, rises, cycles, and projects the reality we experience. Science shows the foundation of that reality is a collective consciousness we have created and agreed upon together. We treat it as solid and fixed, yet we are the consciousness of the entire system. We stand as the doorway between the infinite and the finite, the formless and the form. What we allow to pass through us without interruption determines the quality of the movie we walk into. When we conclude we are not enough, that life is unfair, or that nothing will work, that conclusion becomes the reality we inhabit. We have input. The movie is not fixed.

Interstellar

13,747 görüntüleme • 2 ay önce

According to Bentov, consciousness evolves to the 'absolute' which is the source of all consciousness. Matter, composed of quanta of energy, is the vibrating, changing component of pure consciousness. The absolute is fixed, manifest and invisible. Bentov proposed that the brain acts as a transducer, converting the energy of consciousness into the electrical and chemical signals that are used to communicate information within the body. He also believed that the brain has the ability to tune into different frequencies of consciousness, allowing individuals to experience different states of awareness. Bentov's theory is based on the idea that consciousness is a fundamental aspect of the universe and is not limited to the brain or the body. He argued that consciousness is non-local, meaning it is not bound by space and time and can exist independently of the physical body. Bentov's theory has been influential in the field of consciousness studies, but it remains a subject of debate and is not widely accepted by mainstream science.

According to Bentov, consciousness evolves to the 'absolute' which is the source of all consciousness. Matter, composed of quanta of energy, is the vibrating, changing component of pure consciousness. The absolute is fixed, manifest and invisible. Bentov proposed that the brain acts as a transducer, converting the energy of consciousness into the electrical and chemical signals that are used to communicate information within the body. He also believed that the brain has the ability to tune into different frequencies of consciousness, allowing individuals to experience different states of awareness. Bentov's theory is based on the idea that consciousness is a fundamental aspect of the universe and is not limited to the brain or the body. He argued that consciousness is non-local, meaning it is not bound by space and time and can exist independently of the physical body. Bentov's theory has been influential in the field of consciousness studies, but it remains a subject of debate and is not widely accepted by mainstream science.

Vicky Verma

390,608 görüntüleme • 2 yıl önce

#WATCH | Delhi: Adani Group Chairman Gautam Adani says, "What has become even more obvious from the recent Middle East conflict and targeted attacks on infrastructure is that energy security and digital security are no longer independent dimensions. They are now the twin foundations of national power. And it is increasingly evident that the country that controls its energy will drive its industrial future. The country that controls its compute will drive its intelligence future, and the country that controls both will shape the century ahead. This is the new geometry of power. If 1947 was about our political freedom, true independence in the 21st century will mean owning the energy that lights our homes and the intelligence that guides our minds. This is not a theory. The two largest economies in the world, the United States and China, have understood this deeply. Their systems may be different. Their politics may be different. Their institutions may be different. But their strategic objectives are the same. No nation can call itself truly self-reliant if it depends on others for the energy that powers its economy or the intelligence that shapes its destiny." (Video source: Adani Group)

#WATCH | Delhi: Adani Group Chairman Gautam Adani says, "What has become even more obvious from the recent Middle East conflict and targeted attacks on infrastructure is that energy security and digital security are no longer independent dimensions. They are now the twin foundations of national power. And it is increasingly evident that the country that controls its energy will drive its industrial future. The country that controls its compute will drive its intelligence future, and the country that controls both will shape the century ahead. This is the new geometry of power. If 1947 was about our political freedom, true independence in the 21st century will mean owning the energy that lights our homes and the intelligence that guides our minds. This is not a theory. The two largest economies in the world, the United States and China, have understood this deeply. Their systems may be different. Their politics may be different. Their institutions may be different. But their strategic objectives are the same. No nation can call itself truly self-reliant if it depends on others for the energy that powers its economy or the intelligence that shapes its destiny." (Video source: Adani Group)

ANI

161,632 görüntüleme • 5 ay önce

Quantum Probability Surface Goes Negative A Schrödinger-Cat state is built from two separated coherent wavepackets, |ψ⟩=N(|α⟩+|−α⟩), and its Wigner function reveals the part of Quantum Mechanics that ordinary probability cannot show. The two bright lobes behave like classical possibilities, while the interference between them carves negative valleys into phase space, where W(x,p)<0. In this animation, the surface height is the signed Wigner function. So, gold rises above zero, cyan sinks below it, and the blades between them expose the nonclassical structure of the wavefunction. #QuantumMechanics #WignerFunction #SchrodingerCat #Physics #QuantumArt #ScienceVisualization

Quantum Probability Surface Goes Negative A Schrödinger-Cat state is built from two separated coherent wavepackets, |ψ⟩=N(|α⟩+|−α⟩), and its Wigner function reveals the part of Quantum Mechanics that ordinary probability cannot show. The two bright lobes behave like classical possibilities, while the interference between them carves negative valleys into phase space, where W(x,p)<0. In this animation, the surface height is the signed Wigner function. So, gold rises above zero, cyan sinks below it, and the blades between them expose the nonclassical structure of the wavefunction. #QuantumMechanics #WignerFunction #SchrodingerCat #Physics #QuantumArt #ScienceVisualization

Mathelirium

12,603 görüntüleme • 4 ay önce

Plasma is the fourth state of matter and the most abundant in the universe. It is living, conscious energy that carries information and responds to thought and emotion. Your blood, your nervous system, and the space between stars are all plasma. You are a walking plasma being of light.

Plasma is the fourth state of matter and the most abundant in the universe. It is living, conscious energy that carries information and responds to thought and emotion. Your blood, your nervous system, and the space between stars are all plasma. You are a walking plasma being of light.

Ra 𐦝

32,247 görüntüleme • 3 ay önce

Our general understanding of the characteristics of the physical world are largely restricted by the limited range of our senses. The world appears to be comprised of tangible objects positioned within empty space, separate and distinct. Neither of these characterizations are true. Our unaided senses generate a false interpretation of the true state of reality. Space is note empty, it is substantive with a quantifiable and measurable energy density. Space, in terms of quantum vacuum fluctuations, can be considered as a veritable sea of oscillating energy, like a fluid, quantized at the Planck scale (a billion trillion trillion times smaller than a centimeter) as Planck spherical units, and these tiny oscillators make up the fluid medium of space and comprise the "material" stuff as well. Imagine being able to directly perceive this level of reality. Our photodetector proteins in our eyes are sensitive to electromagnetic radiation in the frequency range of 400 to 800 terahertz (trillions of oscillations per second), and we call this "visible light". If, however, we could see light at the Planck scale, were photons oscillate at the Planck frequency— a mass-energy value that makes the electromagnetic component at order of unity with spacetime curvature— then we would theoretically see directly the substantive fluid medium of space and our sight would relay a world that is integrally interconnected and all one substance; objects would not appear as separate and distinct or even fundamentally different than the substance comprising the bulk space. "Material" objects would appear just as patterned vortices of the fluid that is the very substance of space. So, tangible objects are made of the same substance as space, and only seem physical to our limited senses because of electromagnetic repulsive forces. The electromagnetic repulsive forces are generated by how these PSUs circulate and flow within the structured patterns of space that we call particles and atoms. In this way, the coherent phases, circulation, flow, and pressure forces of this Planck plasma fluid are the source of mass, force, and charge. We are now coming to an understanding of these dynamics at a fundamental level, exemplified in the publication The Origin of Mass and the Nature of Gravity 🔗

Our general understanding of the characteristics of the physical world are largely restricted by the limited range of our senses. The world appears to be comprised of tangible objects positioned within empty space, separate and distinct. Neither of these characterizations are true. Our unaided senses generate a false interpretation of the true state of reality. Space is note empty, it is substantive with a quantifiable and measurable energy density. Space, in terms of quantum vacuum fluctuations, can be considered as a veritable sea of oscillating energy, like a fluid, quantized at the Planck scale (a billion trillion trillion times smaller than a centimeter) as Planck spherical units, and these tiny oscillators make up the fluid medium of space and comprise the "material" stuff as well. Imagine being able to directly perceive this level of reality. Our photodetector proteins in our eyes are sensitive to electromagnetic radiation in the frequency range of 400 to 800 terahertz (trillions of oscillations per second), and we call this "visible light". If, however, we could see light at the Planck scale, were photons oscillate at the Planck frequency— a mass-energy value that makes the electromagnetic component at order of unity with spacetime curvature— then we would theoretically see directly the substantive fluid medium of space and our sight would relay a world that is integrally interconnected and all one substance; objects would not appear as separate and distinct or even fundamentally different than the substance comprising the bulk space. "Material" objects would appear just as patterned vortices of the fluid that is the very substance of space. So, tangible objects are made of the same substance as space, and only seem physical to our limited senses because of electromagnetic repulsive forces. The electromagnetic repulsive forces are generated by how these PSUs circulate and flow within the structured patterns of space that we call particles and atoms. In this way, the coherent phases, circulation, flow, and pressure forces of this Planck plasma fluid are the source of mass, force, and charge. We are now coming to an understanding of these dynamics at a fundamental level, exemplified in the publication The Origin of Mass and the Nature of Gravity 🔗

Nassim Haramein

15,986 görüntüleme • 2 yıl önce

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Mathelirium

22,170 görüntüleme • 4 ay önce

A Triple Spring-Mass Pendulum Just for The Fun of Physics We model the mass positions as r₁ = (x₁, y₁), r₂ = (x₂, y₂), r₃ = (x₃, y₃) with y positive downward in the simulation. The three spring segments are r₁ = r₁ − (0,0) r₁₂ = r₂ − r₁ r₂₃ = r₃ − r₂ Each spring applies a Hooke force along its own direction, F(r; k, L) = −k (|r| − L) (r / |r|) and each mass also feels gravity, Fgᵢ = (0, mᵢ g) So the equations of motion are m₁ r̈₁ = Fg₁ + F(r₁; k₁, L₁) − F(r₁₂; k₂, L₂) m₂ r̈₂ = Fg₂ + F(r₁₂; k₂, L₂) − F(r₂₃; k₃, L₃) m₃ r̈₃ = Fg₃ + F(r₂₃; k₃, L₃) We integrate these ODEs forward with RK4 using a small dt. Different scenes come from changing the mass ratios and from giving the third mass a small initial kick. The yellow trail is the path of the third mass. The inset phase portraits show the same motion in state space: (x₃, vₓ₃) and (−y₃, vᵧ₃) When those curves stay tight and loop cleanly, the motion is more regular. When they spread, fold, and smear, energy is being redistributed across the whole chain between gravity, spring stretch, and kinetic motion. #NonlinearDynamics #CoupledOscillators #SpringPendulum #ChaosInMotion #ComputationalPhysics #MathematicalVisualization

A Triple Spring-Mass Pendulum Just for The Fun of Physics We model the mass positions as r₁ = (x₁, y₁), r₂ = (x₂, y₂), r₃ = (x₃, y₃) with y positive downward in the simulation. The three spring segments are r₁ = r₁ − (0,0) r₁₂ = r₂ − r₁ r₂₃ = r₃ − r₂ Each spring applies a Hooke force along its own direction, F(r; k, L) = −k (|r| − L) (r / |r|) and each mass also feels gravity, Fgᵢ = (0, mᵢ g) So the equations of motion are m₁ r̈₁ = Fg₁ + F(r₁; k₁, L₁) − F(r₁₂; k₂, L₂) m₂ r̈₂ = Fg₂ + F(r₁₂; k₂, L₂) − F(r₂₃; k₃, L₃) m₃ r̈₃ = Fg₃ + F(r₂₃; k₃, L₃) We integrate these ODEs forward with RK4 using a small dt. Different scenes come from changing the mass ratios and from giving the third mass a small initial kick. The yellow trail is the path of the third mass. The inset phase portraits show the same motion in state space: (x₃, vₓ₃) and (−y₃, vᵧ₃) When those curves stay tight and loop cleanly, the motion is more regular. When they spread, fold, and smear, energy is being redistributed across the whole chain between gravity, spring stretch, and kinetic motion. #NonlinearDynamics #CoupledOscillators #SpringPendulum #ChaosInMotion #ComputationalPhysics #MathematicalVisualization

Mathelirium

36,226 görüntüleme • 7 ay önce

.James Talarico: The current political landscape is too small for Texas. The politics of this state is narrow-minded and short-sighted. The uniquely Texan ability to welcome new friends and new ideas has made us one of the most exciting and innovative states in the country. We're the state that put a man on the moon. We're the state that pioneered ranching and energy and computers. We're the state that gave this country Barbara Jordan, Ann Richards, LBJ and the Great Society. We're the state that put breakfast in a taco.

.James Talarico: The current political landscape is too small for Texas. The politics of this state is narrow-minded and short-sighted. The uniquely Texan ability to welcome new friends and new ideas has made us one of the most exciting and innovative states in the country. We're the state that put a man on the moon. We're the state that pioneered ranching and energy and computers. We're the state that gave this country Barbara Jordan, Ann Richards, LBJ and the Great Society. We're the state that put breakfast in a taco.

Team Talarico

16,866 görüntüleme • 4 ay önce

The Probability Density of a Brownian Particle Solves the. Heat Equation. Einstein’s 1905 paper on Brownian motion doesn’t get the same spotlight as General Relativity, but it’s still an astonishing piece of work. It takes a messy, noisy phenomenon and extracts three hard facts from it: First, the randomness of the particles is physical. The motion is molecular impacts made visible. Second, one path looks like chaos, but the cloud has a rigid scaling law...mean-square displacement grows like t, so the width grows like √t. Third, the cloud evolves deterministically. Its density solves the heat equation PDE! So Brownian probability spreads the same way heat spreads!

The Probability Density of a Brownian Particle Solves the. Heat Equation. Einstein’s 1905 paper on Brownian motion doesn’t get the same spotlight as General Relativity, but it’s still an astonishing piece of work. It takes a messy, noisy phenomenon and extracts three hard facts from it: First, the randomness of the particles is physical. The motion is molecular impacts made visible. Second, one path looks like chaos, but the cloud has a rigid scaling law...mean-square displacement grows like t, so the width grows like √t. Third, the cloud evolves deterministically. Its density solves the heat equation PDE! So Brownian probability spreads the same way heat spreads!

Mathelirium

21,543 görüntüleme • 8 ay önce

Is this the Syria that they claim has become safe, and where weapons are confined to the hands of the state??? These are the tribes (supporters of the government) and its backers. They are firing randomly and attacking the residents of Damascus, without any restraint or deterrent… #NewSyria

Is this the Syria that they claim has become safe, and where weapons are confined to the hands of the state??? These are the tribes (supporters of the government) and its backers. They are firing randomly and attacking the residents of Damascus, without any restraint or deterrent… #NewSyria

Judy Berxwedan

12,742 görüntüleme • 2 ay önce

All that is left of 2nd Street in Pithole City, a true ghost town in Pennsylvania where the entire grid of the city, some of the foundations and a well are all that remains of one of the fastest growing cities in America that vanished within three years of its inception.

All that is left of 2nd Street in Pithole City, a true ghost town in Pennsylvania where the entire grid of the city, some of the foundations and a well are all that remains of one of the fastest growing cities in America that vanished within three years of its inception.

ZitoSalena

24,069 görüntüleme • 2 ay önce

Sir John Curtice, "The public have not reacted against the expansion of the state in the way both the Conservatives and Labour imagine they are doing" "All the evidence shows that the public are much more concerned about the state of the NHS, than they are about lowering taxation" "And frankly the Conservatives would be better spending money on increasing funding for the NHS that they spent on tax cuts, ending the junior doctor strike and getting waiting times down" "The public's priority is with a better NHS, not with the level of taxation"

Sir John Curtice, "The public have not reacted against the expansion of the state in the way both the Conservatives and Labour imagine they are doing" "All the evidence shows that the public are much more concerned about the state of the NHS, than they are about lowering taxation" "And frankly the Conservatives would be better spending money on increasing funding for the NHS that they spent on tax cuts, ending the junior doctor strike and getting waiting times down" "The public's priority is with a better NHS, not with the level of taxation"

Farrukh

675,835 görüntüleme • 2 yıl önce