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advanced_tools:legendre_transformation [2017/04/12 14:18]
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 ======
  
-===== 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
 +  * 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
 +
 + 
 +<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. ​
 +
 +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>​
  
-===== Beginner ===== 
  
-===== Examples =====+</​tabbox>​
  
-===== Advanced ===== 
  
-===== History ===== 
advanced_tools/legendre_transformation.1491999524.txt.gz · Last modified: 2017/12/04 08:01 (external edit)