Chaos | Chapter 2 : Vector fields - The lego race

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Chaos | Chapter 2 : Vector fields - The lego race
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École Normale Supérieure de Lyon (ENS-Lyon)

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Point (geometry) Newton, Isaac Differential calculus State of matter INTEGRAL Latin square Forcing (mathematics) Multiplication sign Moment (mathematics) Mortality rate Variable (mathematics) Mereology 2 (number) Number Derivation (linguistics) Different (Kate Ryan album) Nichtlineares Gleichungssystem Differential equation Nichtlineares Gleichungssystem Game theory Position operator Physical system
4 (number)
Point (geometry) Statistical hypothesis testing Trajectory Group action Divisor Differential (mechanical device) State of matter Direction (geometry) Chaos (cosmogony) Goodness of fit Plane (geometry) Causality Different (Kate Ryan album) Velocity Mathematician Arrow of time Körper <Algebra> Determinant Position operator Initial value problem Condition number Physical system Predictability Mass flow rate Moment (mathematics) Curve Physicalism Maxima and minima Line (geometry) Sequence Measurement Voting Vector space Vector field Hydraulic motor Figurate number Bounded variation Spacetime
Mathematics Quantum state Model theory Chaos (cosmogony)
Mathematics Chaos (cosmogony)
it is October 26 1676 and Isaac Newton is writing a letter was great rivals like he wants to talk about his greatest discovery by same time wants to keep it a secret so instead he sends a rate of 6 8 2 C D 8 EU 13 to add 7 on 3 9 4 4 2 2 0 4 4 she will be X like it's not code the science historian to quit the Latin phrase there was hiding in Tikrit the that westerly through at this point that this would scrupulous that please in English given equation any number of variables find its derivatives and vise versa this is the differential and integral part crystal ball that is incredibly effective for predicting the future you calculate the Future Movement system we know its present state the forces that act on it is about solving differential equation and find let's look at this world made of like after all the mathematical world is a bit like a game where everything is simpler to real world that sometimes you child footprints indicate have followed by fiat they trajectories I suppose there legal athletes take steps at a steady pace but somehow steps there lot while some have stepped ashore the AppleTalk quickly 2 consecutive footprints are far apart for the 1 because slowly the steps across together Pharaoh joining 2 consecutive steps if the speed is the difference between the 2 positions to physically moments in the phrase differential calculus and journalists who were not . 5 seconds Bravo now movement really consists of discrete steps consider these moving race car they role they don't take steps but then we mean by speed 1 idea is to recall this song consists of 25 frames per 2nd so we can think of the motorbike 25 steps every sector
and we can talk about its just like
we're talking about the speed of political runner rimmed approximates 80
continuous motion as a
sequence of stepwise movements but what steps the becomes so small that they can't visible just like in the Kathleen speed of emotion is called calculating derivative it is here are some things in as at these arrows indicate speed and mathematicians column vectors as
amount magic opposite problem look at heroes on the floor of mathematicians call this a vector field a measure in a field of but instead stocks of wheat we have sectors the legal people's mission is to move with speed is indicated by the effective EU look under their feet and they see a factor which tells them how they they set off in that direction and that speed the a brief moment later they've arrived at a new point with a new direction and speed and off they go again as they do this over and over walking it's not difficult just put 1 foot from the others and do it again actually we should explain what we mean by a brief moments later dudes answer would be I know it's brief we only saw that a continuous movement is not the same thing succession of steps it but you get this test become smaller and smaller so rather than popular political person in Cardiff rise continuously you will follow what is called a trajectory but curve is tangent to the vector field everywhere if you have a vector field plane and 2 points better than initial positions of the 2 motors the fear of kosher elections summarizes the concept of charges he claims at the starting point determine future trajectories from each point there is a unique trajectory whose initial position is forgiven each point has destiny different for everyone the legal man faces is all he can to follow his trajectory noted to trajectories and never crossed for determining the trajectory from the knowledge of the velocity field it is the work of interval that's a cousin opposite direction from differential tactics they will hold E a little is a group of Lake people neatly ordered are little soldiers ready to start work on off they go we see that nice original ordered gets destroyed it is sometimes said that went determines the flow of feel as if these characters were floating on a river each following their course see the flow of humanity 0 7 billion labeled people moving all over the earth think of the flow of billions of billions of molecules in the Earth's atmosphere him and here's a simple example almost naive which will show us a weakness determined what watch the spectra fail the figures move forward and as you can see those on the left of the central line turned left and those on the right centralized turned the right In a way to terrorism is that everyone follows his destiny over which he has no control but on the other hand chilean starting very close to each other they have very different best it's a little thing can completely change the future all of you here In his little book about her emotion published in 1876 With his Maxwell stresses the sensitivity of physical phenomenon to initial conditions it's a good thing if life there is what U.S. so there is a Max but the same causes will always produce the same effects effects there is another Max of which must not confounded with the 1st which is that like causes produce like this is only true when small variations in initial circumstances produced only small variations in the final state of the system In agreed many physical phenomena this condition is satisfied that there are other cases in which a small initial variation of the great for instance the small difference in the speed of the car may cause an accident even while but he is dependence of the future on initial conditions is only 1 aspect of chaos but they're much more complex situation a mansion for example the vector field which no longer draw on the ground but is drawn in space is 1 for shown on a vertical plane moving back and forth now outwardly goes don't vote instead they fly in their spaceship every moment their speed is determined by the fact see what happens toward for the U.S. this is much more chaotic to we imagine a fortune teller who to divine the future position the spacious it is impossible on such a rollercoaster His predictions would be mere deception where will the ship in an hour known it is difficult to predict the future label astronaut just imagine predicting the future of a human being if


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