mathkaveli

In our previous posts, we’ve explored elliptical motion, energy, 3D orbits, rotating frames, and even complex numbers. Now we arrive at a thrilling conclusion: what happens when we disturb the system just a little? Today we explore the edge of predictability — welcome to the world of chaos. --- What Is Chaos in Physics? Chaos doesn’t mean randomness. In mathematics and physics, chaos means: The system is deterministic (it follows fixed rules) But it is **extremely sensitive** to initial conditions This is sometimes called the **butterfly effect**: a tiny change in one part of a system leads to large differences later. --- From Ellipses to Chaos Let’s start with elliptical motion again: r(t) = Now imagine that something disturbs the motion — a nearby object’s gravity, a small push, or even a tiny variation in mass. We’ll simulate this by introducing a **perturbation**: a small oscillation added to one of the axes. x(t) = a * cos(ωt) y(t) = b * sin(ωt) + ε * sin(μ...

So far in this series, we’ve described elliptical motion using position vectors, trigonometry, and rotation matrices.  But what if I told you there’s a more elegant way to represent all of it — using complex numbers?  Today, we’ll explore how complex numbers and Euler’s formula allow us to model elliptical motion, circular motion, and rotation with surprising simplicity. Along the way, we’ll translate our real-valued vector math into the complex plane — and simulate it all in Octave.  Complex Numbers as Vectors  A complex number is written as: z = x + i·y, where: x is the real part, y is the imaginary part, and i is the square root of -1.  You can think of this as a 2D vector: z = (x, y) Euler’s Formula: exp(iθ) = cos(θ) + i·sin(θ) This means we can represent circular motion using the exponential expression: z(t) = R · e^(i·ω·t) This traces a circle of radius R, rotating counterclockwise at angular speed ω. What About Ellipses?  To model an ellipse, we scal...

In our last post, we lifted ellipses into 3D space to explore planetary orbits. But here’s a twist — what if the ellipse itself spins while the particle moves along it? In other words, what happens if the entire orbital plane rotates over time? This is not just a fun thought experiment — it reflects real-world effects, like the precession of orbits, gyroscopes, and the Coriolis effect on Earth. Today, we’ll rotate our coordinate system and simulate the result in Octave.  Combining Two Motions Let’s start with a basic elliptical path: r(t) = < a*cos(ωt), b*sin(ωt) > Now imagine the entire coordinate system spins slowly — like placing the ellipse on a turntable. We apply a rotation angle φ(t) to the whole position vector. So the new position becomes: x_rot(t) = x(t)*cos(φ(t)) - y(t)*sin(φ(t)) y_rot(t) = x(t)*sin(φ(t)) + y(t)*cos(φ(t)) This is a classic 2D rotation matrix applied to a moving object. If φ(t) increases over time, we get compound motion: the object moves ...

We've seen how a particle can move in an elliptical path and how energy varies throughout the motion. But here's something exciting — planets also move in ellipses. This is where mathematics meets astronomy. In this post, we’ll explore Kepler’s Laws, simulate a simple planetary orbit in 3D using Octave, and discover how a flat ellipse turns into a space-filling orbit. --- Kepler’s First Law: The Ellipse of Orbits In the early 1600s, Johannes Kepler studied the planets and found something surprising: Kepler’s First Law: The orbit of a planet is an ellipse, with the sun at one focus. This means that planetary motion is not circular — but elliptical, just like the math we’ve been studying! --- Position in 3D: Tilting the Ellipse Let’s say the 2D elliptical orbit is: r(t) = < a*cos(θ), b*sin(θ) > To lift this into 3D, we rotate the ellipse out of the xy-plane. That’s done by applying a tilt angle (called inclination) and optional rotation around the z-axis. W...