advanced_tools:legendre_transformation
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advanced_tools:legendre_transformation [2017/05/06 10:27] jakobadmin created |
advanced_tools:legendre_transformation [2022/09/16 23:24] (current) 82.174.118.41 [Concrete] |
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| - | ====== legendre transformation ====== | + | ====== Legendre Transformation ====== |
| + | <tabbox Intuitive> | ||
| + | |||
| + | <note tip> | ||
| + | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| + | </note> | ||
| + | | ||
| + | <tabbox Concrete> | ||
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| + | * See: [[http://aapt.scitation.org/doi/pdf/10.1119/1.3119512|Making sense of the Legendre transform]] by R. K. P. ZiaEdward, F. RedishSusan, R. McKay | ||
| + | * and Ryder - Quantum Field Theory page 260 | ||
| + | * see also https://blog.jessriedel.com/2017/06/28/legendre-transform/ | ||
| + | * http://blog.sigfpe.com/2005/10/quantum-mechanics-and-fourier-legendre.html | ||
| + | * https://www.andrew.cmu.edu/course/33-765/pdf/Legendre.pdf | ||
| + | * http://mathforum.org/kb/message.jspa?messageID=3868685 (Legendre transformation is "zero temperature limit" of the [[advanced_tools:laplace_transformation]]) | ||
| + | * http://blog.sigfpe.com/2012/01/some-parallels-between-classical-and.html | ||
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| + | <tabbox Abstract> | ||
| + | |||
| + | <blockquote>The Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings.<cite>http://blog.sigfpe.com/2005/10/quantum-mechanics-and-fourier-legendre.html</cite></blockquote> | ||
| + | | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| + | <blockquote> | ||
| + | The Legendre transform shows up whenever we minimize or maximize something subject to constraints. That happens a lot. | ||
| + | |||
| + | <cite>https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/</cite> | ||
| + | </blockquote> | ||
| The Legendre transformation is a useful mathematical tool that is used in thermodynamics, classical mechanics and quantum field theory. | The Legendre transformation is a useful mathematical tool that is used in thermodynamics, classical mechanics and quantum field theory. | ||
| - | Maybe the most famous application is that the Hamiltonian and the Lagrangian are connected by a Legendre transformation. | + | Maybe the most famous application is that in classical mechanics, quantum mechanics and quantum field theory the Hamiltonian and the Lagrangian are connected by a Legendre transformation. |
| Moreover, the Legendre transformation is used in thermodynamics to motivate the connection between the internal energy, enthalpy, and Gibbs and Helmholtz free energies. | Moreover, the Legendre transformation is used in thermodynamics to motivate the connection between the internal energy, enthalpy, and Gibbs and Helmholtz free energies. | ||
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| </blockquote> | </blockquote> | ||
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| - | <tabbox Layman?> | ||
| - | |||
| - | <note tip> | ||
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| - | </note> | ||
| - | | ||
| - | <tabbox Student> | ||
| - | |||
| - | See Ryder - Quantum Field Theory page 260 | ||
| - | |||
| - | <tabbox Researcher> | ||
| - | |||
| - | <note tip> | ||
| - | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| - | </note> | ||
| - | | ||
| - | <tabbox Examples> | ||
| - | |||
| - | --> Example1# | ||
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| - | |||
| - | <-- | ||
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| - | --> Example2:# | ||
| - | |||
| - | |||
| - | <-- | ||
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| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
advanced_tools/legendre_transformation.1494059277.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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