advanced_tools:legendre_transformation
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advanced_tools:legendre_transformation [2017/04/12 14:25] jakobadmin |
advanced_tools:legendre_transformation [2019/03/16 09:29] 77.181.188.79 [Abstract] |
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| ====== Legendre Transformation ====== | ====== Legendre Transformation ====== | ||
| - | ===== PopSci/Cocktail Party Explanation ===== | + | <tabbox Intuitive> |
| - | ===== Why is it interesting? ===== | + | <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> | ||
| + | * 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 | ||
| + | * 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 | ||
| + | |||
| + | |||
| + | <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?> | ||
| <blockquote> | <blockquote> | ||
| - | After introducing Lagrangian and Hamiltonian mechanics, Section 5 demonstrates how the two formulations are equivalent under the Legendre transform | + | The Legendre transform shows up whenever we minimize or maximize something subject to constraints. That happens a lot. |
| - | <cite>https://www.math.uwaterloo.ca/~karigian/training/herman-final-MMath-project.pdf</cite> | + | <cite>https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/</cite> |
| </blockquote> | </blockquote> | ||
| + | The Legendre transformation is a useful mathematical tool that is used in thermodynamics, classical mechanics and quantum field theory. | ||
| - | ===== Beginner ===== | + | 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. | ||
| + | |||
| + | <blockquote> | ||
| + | Both uses can be compactly motivated if the Legendre | ||
| + | transform is properly understood. Unfortunately, that transform is often relegated to a footnote in | ||
| + | a textbook, or worse is presented as a complicated mathematical procedure. [...] In a nutshell, a Legendre transform simply changes the | ||
| + | independent variables in a function of two variables by application of the product rule. | ||
| + | |||
| + | <cite>https://www.aapt.org/docdirectory/meetingpresentations/SM14/Mungan-Poster.pdf</cite> | ||
| + | </blockquote> | ||
| - | See Ryder - Quantum Field Theory page 260 | ||
| - | ===== Examples ===== | + | </tabbox> |
| - | ===== Advanced ===== | ||
| - | ===== History ===== | ||
advanced_tools/legendre_transformation.txt · Last modified: 2022/09/16 23:24 by 82.174.118.41
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