formalisms:lagrangian_formalism
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formalisms:lagrangian_formalism [2020/04/03 16:10] 95.168.180.148 Minor fixes and link added |
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| The basic idea of the Lagrangian formalism can be summarized by the statement: | The basic idea of the Lagrangian formalism can be summarized by the statement: | ||
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| //Nature is lazy.// | //Nature is lazy.// | ||
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| The laziness of nature is demonstrated nicely by how light behaves. | The laziness of nature is demonstrated nicely by how light behaves. | ||
| - | **The Principle of Least Time | + | **The Principle of Least Time** |
| - | ** | + | |
| Long before Joseph Lagrange invented the formalism now named after him, it was well known that light always takes the path between two points that requires the least travel time. This is known as **Fermat's Principle**. | Long before Joseph Lagrange invented the formalism now named after him, it was well known that light always takes the path between two points that requires the least travel time. This is known as **Fermat's Principle**. | ||
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| **Reading Recommendations** | **Reading Recommendations** | ||
| - | * [[http://nautil.us/blog/to-save-drowning-people-ask-yourself-what-would-light-do|To Save Drowning People, Ask Yourself “What Would Light Do?”]] by Aatish Bhatia | + | * [[https://nautil.us/to-save-drowning-people-ask-yourself-what-would-light-do-234852/|To Save Drowning People, Ask Yourself “What Would Light Do?”]] by Aatish Bhatia |
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| --> Why is the correct path given by the path with minimal action?# | --> Why is the correct path given by the path with minimal action?# | ||
| - | The answer can be found through the [[theories:quantum_mechanics:path_integral|path integral formulation]] of [[theories:quantum_mechanics:canonical|quantum mechanics]]. The thing is that a particle really has some probability to go all possible ways. However, the classical path is the most probable path, because paths close this this path infer constructively and hence yield a big probability. In contrast, for other paths far aways from the classical path the interference is destructive and hence the probability is tiny. The path with minimal action gives the biggest contribution to the path integral in the classical limit $\hbar \to 0$. | + | The answer can be found through the [[theories:quantum_mechanics:path_integral|path integral formulation]] of [[theories:quantum_mechanics:canonical|quantum mechanics]]. The thing is that a particle really has some probability to go all possible ways. However, the classical path is the most probable path, because paths close this this path interfere constructively and hence yield a big probability. In contrast, for other paths far aways from the classical path the interference is destructive and hence the probability is tiny. The path with minimal action gives the biggest contribution to the path integral in the classical limit $\hbar \to 0$. |
| This is explained nicely in Section 3 [[https://arxiv.org/pdf/quant-ph/0004090.pdf|here]]. | This is explained nicely in Section 3 [[https://arxiv.org/pdf/quant-ph/0004090.pdf|here]]. | ||
formalisms/lagrangian_formalism.1585923054.txt.gz · Last modified: 2020/04/03 14:10 (external edit)
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