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Principle Of Minimal Action And Kinetic Energy ... Physics (Episode 6): Principle of Minimal Action and Kinetic Energy. --- Please SUBSCRIBE to Science & Reason: • • • --- It all started with light smart light. Then it quickly spread to rocks and other projectiles. Now it is seemingly everywhere. But before I tell you more about that, consider how smart people react to certain situations. If I have a trial where you have to start at point A, run and touch any point on a wall and then run to point B. How do you decide what is the quickest path to take how do you decide where to touch the wall? You could run to here or here or here. But after moment you pick this point because you think it is the shortest path and results in the quickest journey. In a second trial, you are a lifeguard and you spot a swimmer in trouble out in the water. How do you decide on the quickest path to the swimmer? You run faster than you can swim, so do you run to this point and swim from there or maybe this point? If you choose correctly, your path will be the path of minimum time. And you will have joined ranks with Fermat, Maupertuis, Euler, and Lagrange in furthering one of the most universal principles of nature. Fermat: Living in the first half of the 17th century, Pierre Fermat developed the concept that light travels at different speeds in different media, and maintained that the path that light takes is always the one that takes the least time. His detractors at the time gleefully pointed out the obvious objection how does the light know which path will be the quickest? Smart light? Maupertuis and Euler: Pierre-Louis de Maupertuis, the son of a wealthy pirate, expanded Fermats least-time idea. He wrote that in ALL events in nature, there is a certain quantity called action which is always a minimum. His good friend and mathematician Leonhard Euler added the idea of conservation of energy, and made least action an exact dynamical theorem with this mathematical form. Euler also developed an ingenious, graphical method to find the path of minimum action for a system. Lagrange: Another young scientist of the time, Joseph-Louis Lagrange, eagerly applied these ideas to the tautochrone problem. This is the problem of finding a curve shape on which a weight started at any point on the curve will slide to the bottom in the same amount of time regardless of its starting point on the curve. He expanded the idea of Action to be a function of the kinetic energy minus the potential energy. This has proven to be a much more universal formulation. Now that we know all this, lets throw a smart baseball straight up with enough force that it returns to our outstretched hand three seconds later, and lets see if we can discover its motion using the principle of least action. When we first throw it, it has a lot of kinetic energy and its speed is at a maximum. But gravity slows it down as it rises. At its peak, its speed is zero and all of its energy has been transformed to potential energy. On the downward trip, all the potential energy gets reconverted into kinetic energy. Its kinetic energy is related to its speed and its potential energy is related to its height. Now on this graph, we have the baseballs height on the y-axis and the time on the horizontal axis. To begin we have divided the three seconds of flight time into a few time intervals. As we adjust the height of each point, the computer will calculate the total action for us. Remember we are looking for the minimum action possible while keeping the ball in the air for three seconds. The height of the dot is directly related to the potential energy and the slope of the line gives the speed which is related to the kinetic energy. The steeper the slope, the faster the baseball is moving. As you can see it is an iterative process, it will take several cycles before we can no longer make the total action any lower that is ... less negative. Deductions: Here is the same problem with more sections in the graph. To do this perfectly the number of sections must go toward infinity and the width of each section must go toward zero. But this approximation with forty sections is good enough to show all the features. CONCLUSION: While everyone has heard of Isaac Newton and Force = mass times acceleration, F=ma it is this other, equivalent formulation of kinematics that is more versatile. It has direct extensions into relativity and quantum mechanics. So how does the light know which path to take to satisfy the principle of least action? Richard Feynman gave us the answer, and we will explore that answer in a video on QED. --- The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind. • .
Length: 08:36


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