formalisms:lagrangian_formalism
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formalisms:lagrangian_formalism [2018/06/20 20:47] 128.187.112.30 Typo "law physical laws" changed to "physical laws" |
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. | ||
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| The kinetic energy is a measure for how much is happening, i.e. how much activity is going on in the system. The potential energy, as the name indicates, is a measure for how much activity could potentially happen, but does not. A good example is a ball at the top of a cliff. At this point its potential energy is maximal, but could be converted to kinetic energy at any moment if it slides down the cliff. | The kinetic energy is a measure for how much is happening, i.e. how much activity is going on in the system. The potential energy, as the name indicates, is a measure for how much activity could potentially happen, but does not. A good example is a ball at the top of a cliff. At this point its potential energy is maximal, but could be converted to kinetic energy at any moment if it slides down the cliff. | ||
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| + | In other words, the Lagrangian measures how much is happening, minus how much could be happening but isn't. | ||
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| + | ---- | ||
| Let's consider an explicit example: We throw a ball and want to know what path it will follow between two given points $A$ and $B$ on the ground, where it starts at a fixed time $t_A$ and ends up on the ground at fixed time $t_B$. | Let's consider an explicit example: We throw a ball and want to know what path it will follow between two given points $A$ and $B$ on the ground, where it starts at a fixed time $t_A$ and ends up on the ground at fixed time $t_B$. | ||
formalisms/lagrangian_formalism.txt · Last modified: 2023/03/08 17:35 by 62.4.55.178
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