formalisms:lagrangian_formalism
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formalisms:lagrangian_formalism [2018/09/04 07:15] 92.225.203.36 [Concrete] |
formalisms:lagrangian_formalism [2018/11/08 23:13] 49.2.181.170 [Concrete] |
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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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