formalisms:lagrangian_formalism
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formalisms:lagrangian_formalism [2018/09/04 07:15] 92.225.203.36 [Concrete] |
formalisms:lagrangian_formalism [2018/12/30 00:35] thomas_abshier [Intuitive] |
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| - | To understand this let's consider a rescue swimmer who sees someone drowning in the water. Which path should he take to get to the swimmer as fast as possible? He is slow in the water and fast if he runs on the beach. This gives him two extreme options: | + | To understand this let's consider a rescue swimmer who sees someone drowning in the water. Which path should he take to get to the swimmer as fast as possible? He is slow in the water and fast running on the beach. This gives him two extreme options: |
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| OR is there some better option? | OR is there some better option? | ||
| - | There //is//. The optimal path is a trade off of the two choices above. While it is true that the rescuer is much faster on the sand, the path in Option 2, above, is much longer than if he runs and swims diagonal. | + | There //is//. The optimal path is a trade-off of the two choices above. While it is true that the rescuer is much faster on the sand, the path in Option 2, above, is much longer than if he runs and swims diagonally. |
| - | This is exactly the same behaviour light has. From Fermat's principle, we can now see why light gets broken across two media: Light is slower in of the media, and thus has to choose a trade off between a minimal total path length and a minimal length in slower medium. | + | This is exactly the same behaviour light has. From Fermat's principle, we can now see why light gets broken across two media: Light is slower in the media and thus has to choose a trade-off between a minimal total path length and a minimal length in the slower medium. |
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| To exploit the idea that nature is lazy, we need to define some quantity that measures "laziness". We name this quantity the "action" of a system. A system with bigger "action" is less lazy than a system with small "action". The "action" is the sum of the difference between the kinetic and potential energy for all "timesteps" in a given interval. The difference between the kinetic and the potential energy is called the Lagrangian. For every point in time $t$, the Lagrangian has some value $L(t)$. Let's say we want to analyze a system for 5 seconds. The action of the system is the sum of all values of the Lagrangian during these five seconds. In mathematical terms, the action is therefore the integral of the Lagrangian function $L(t)$, starting at some $t_0$ and ending at some $t_1$: | To exploit the idea that nature is lazy, we need to define some quantity that measures "laziness". We name this quantity the "action" of a system. A system with bigger "action" is less lazy than a system with small "action". The "action" is the sum of the difference between the kinetic and potential energy for all "timesteps" in a given interval. The difference between the kinetic and the potential energy is called the Lagrangian. For every point in time $t$, the Lagrangian has some value $L(t)$. Let's say we want to analyze a system for 5 seconds. The action of the system is the sum of all values of the Lagrangian during these five seconds. In mathematical terms, the action is therefore the integral of the Lagrangian function $L(t)$, starting at some $t_0$ and ending at some $t_1$: | ||
| - | $$ \text{Action} = \int_{t_0}^{t_1} L(t) dt .$$ | + | $$ \text{Action} = \int_{t_0}^{t_1} L(t) dt $$ |
| The statement that nature is lazy means now in mathematical terms that during some given time span, here from $t_0$ to $t_1$, the system behaves in such a way that the action is as small as possible. This is known as the principle of minimal action. | The statement that nature is lazy means now in mathematical terms that during some given time span, here from $t_0$ to $t_1$, the system behaves in such a way that the action is as small as possible. This is known as the principle of minimal action. | ||
formalisms/lagrangian_formalism.txt · Last modified: 2023/03/08 17:35 by 62.4.55.178
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