![[NEW VIDEO!] The REAL Three Body Problem in Physics](https://img5.xaiju.com/storage/11/bl/tv/d38796-019e7bd1-2ad2-7c2e-a10a-c05be26dd285.jpg)
[NEW VIDEO!] The REAL Three Body Problem in Physics
Added 2024-07-15 11:37:15 +0000 UTCHello! I know I'm super late to the party with this video but it took me way longer than expected to understand! Thank you for your patience and I hope you enjoy the explanation :)
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Three body problems come in many guises. Poincare gave birth to modern Optimal Control Theory in which a controlling function effects a system simplified as two bodies. The controlling operator can be a matrix of functions that interacts with a pair of matrices representing a system. The linear operator mechanics nested within Dirac brackets has a higher Cantor aleph rating than the continuum. The manifolds within which the dynamics plays out can have a Riemannian curvature with saddle points and aspects of negative or positive curvature spread through out. A control function can change the pattern for geodesics, however it will have physical limits and everything obeys the conservation of energy and momentum. In going from the 2-body quantized angular momentum model of hydrogen given by Niels Bohr in 1911 to a 3-body model with one more electron or a compound nucleus, one has to grasp the multi-body difficulties that underlie fermions, those mass-endowed particles that refuse to occupy the same place at the same time, unless they have diametrically opposing spins. The atomic science may seem to be abstract and complicated, and yet, we deal with many aspects of 3-body problems in our daily frustrations in trying to find certain objects. An object (or creature) can hide by slipping behind another object. Herein we have the birth of the three dimensional world. Entangled fermions map out the geometry of a room and create the third dimension. When looking for a missing key, we have a 3-body problem in the form of the looker, the key, and everything else which is not of immediate interest. With a Dirac comb you search the landscape, sweep away distractions and hopefully pull off a sequence of steps to form an operator that acts on a state, and if you are lucky, gives it right back as the key in your hand. To wit, you solve an eigenvalue problem in a complicated three dimensional world with a history and a perpetual desire to try out every possible arrangement or mapping of a room. Feynman says sum over all of the possible sequences of steps in going from A to B as though they were quantum amplitudes. When a 3-body problem gets parsed into more components and greater detail, a Law of Large Numbers takes over and one sees the familiar classical world that minimizes the action S at a saddle point (in a space of functions). Variations of sequences (paths) away from the minimum paths tend to cancel each other out. A first order shift away from a saddle point introduces no more than a second order change in the action S for a path close to satisfying dS = 0. This tiny power-packed equation comes from Euler and Lagrange, after the time of Newton, and long before Poincare. The little d for an infinitesimal shift from one alternative to another usually gets written as the delta of Variational Calculus (a predecessor of Optimal Control Theory). The S stands for the action cost of a sequence of steps. Elsewhere in mathematical physics, S stands for entropy, which has its own tally depending on how many entangled bodies one happens to have. Each fermion brings its own space so when we start to accumulate them, we do not add them like dots into an already given space. Instead each particle adds to the number of cross-products of a full 3-dimensional space we have to manage. The dynamics lives in the cross-product of as many Hilbert spaces as we have particles. The number of particles and cross products, the number of dynamical dimensions can expand or collapse down to a point, a singularity. A tremendous amount of sophisticated mathematics (and computer programming) comes into play.
Scott Ready
2024-07-27 18:28:18 +0000 UTCLovely animations in this one, well done!
Martin S
2024-07-15 19:50:06 +0000 UTCThis is a really great video. What I like most about it is that you've used history, maths and phyics to explain the problem.
Armin Quast
2024-07-15 15:02:36 +0000 UTCHi Jade! Great to see a new video from you! Thanks for the very nice explanation of the three body problem and the origin of chaos 😃
Tim Ludwig
2024-07-15 12:13:31 +0000 UTC
