advanced_tools:legendre_transformation
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advanced_tools:legendre_transformation [2017/12/04 08:01] 127.0.0.1 external edit |
advanced_tools:legendre_transformation [2018/05/04 16:45] jakobadmin |
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| ====== Legendre Transformation ====== | ====== Legendre Transformation ====== | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Intuitive> |
| - | <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. | + | |
| - | + | ||
| - | 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> | + | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | <tabbox Layman?> | + | |
| <note tip> | <note tip> | ||
| Line 33: | Line 7: | ||
| </note> | </note> | ||
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| - | <tabbox Student> | + | <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 | * 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 | ||
| Line 43: | Line 17: | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
| Line 49: | Line 23: | ||
| </note> | </note> | ||
| | | ||
| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| + | <blockquote> | ||
| + | The Legendre transform shows up whenever we minimize or maximize something subject to constraints. That happens a lot. | ||
| - | --> Example1# | + | <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. | |
| - | <-- | + | |
| - | --> Example2:# | + | 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> | ||
| - | |||
| - | <-- | ||
| - | | ||
| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
advanced_tools/legendre_transformation.txt · Last modified: 2022/09/16 23:24 by 82.174.118.41
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