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Topic: Legendre transformation


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  Legendre transformation - Wikipedia, the free encyclopedia
A Legendre transformation results in a new function, in which one or more independent variables is replaced by the derivative of an original function with respect to this variable.
The strategy behind the use of Legendre transforms is to shift the dependence of a function from one independent variable to another (the derivative of the original function with regard to this independent variable) by taking the difference between the original function and their product.
A Legendre transform is used in classical mechanics to derive the Hamiltonian formulation from the Lagrangian one, and conversely.
en.wikipedia.org /wiki/Legendre_transformation   (1175 words)

  
 Springer Online Reference Works   (Site not responding. Last check: 2007-11-01)
A transformation in mathematical analysis that establishes a duality between objects in dual spaces (in parallel with projective duality in analytic geometry and polar duality in convex geometry, cf.
The Legendre transformation plays an important role in analysis, particularly in convex analysis (see [1], [2], [4]), in the theory of differential equations, in variational calculus (see [6]), and in classical mechanics, thermodynamics, the theory of elasticity and other branches of mathematical physics.
The application of the Legendre transformation to the Lagrangian of a problem in classical variational calculus reduces it to the Hamilton function.
eom.springer.de /L/l058080.htm   (471 words)

  
 Legendre Transformation
The Legendre Transformation allows to describe a function using a different set of variable.
Legendre Transformation are applied in many different areas of physics.
The formal description of the Legendre transformation is given below.
mit.fnal.gov /~paus/8.21-IAP2001/notes/notes/node6.html   (147 words)

  
 Lagrangian mechanics - Wikipedia, the free encyclopedia
However, this is a misnomer: the action only needs to be stationary, and the correct trajectory could be produced by a maximum, saddle point, or minimum in the action.
The Hamiltonian, denoted by H, is obtained by performing a Legendre transformation on the Lagrangian.
The Hamiltonian is the basis for an alternative formulation of classical mechanics known as Hamiltonian mechanics.
en.wikipedia.org /wiki/Lagrangian_mechanics   (956 words)

  
 [No title]
The new function g(p) is called the Legendre transform of the function f(x).
The Legendre transform of f(x) = m x + b is thus the single point (m,b) or { b, if p = m g(p) = { undefined, otherwise Let's try a slightly harder example: two half-lines meeting at a point.
Thus the Legendre transform of a differentiable function looks like g(p) = x p - f(x) where we've inserted the overall minus sign, so g(p) is _minus_ the intercept of the line with slope p which touches the graph of f(x).
www.math.niu.edu /~rusin/known-math/00_incoming/legendre   (1421 words)

  
 Hamiltonian
Use the Legendre transformation to express the action integral in terms of the Hamiltonian.
We start in the Hamiltonian picture and define finite "canonical transformations" as new positions (Q) and their canonical momenta (P) that are each functions of both old positions (q) and their canonical momenta (p) in such a way that the Hamiltonian equations retain the same form under the transformation.
The finite time (t) canonical transformation is determined from the condition that the new Hamiltonian (K) be exactly zero and its corresponding positions (Q) and momenta (P) are constant in time along the trajectory with vanishing time derivatives.
www.qedcorp.com /pcr/pcr/hamilton.html   (1214 words)

  
 Matematický ústav - Abstract   (Site not responding. Last check: 2007-11-01)
A reformulation, generalization and extension of basic concepts such as Hamiltonian system, Hamilton equations, regularity, and Legendre transformation, is presented.
A revision of the concepts of {\it regularity} and {\it Legendre transformation} is proposed, reflecting geometric properties of the related exterior differential system.
The new look is shown to lead to new regularity conditions and Legendre transformation formulas, and provides a procedure of {\it regularization} of variational problems.
www.math.slu.cz /Abstrakty/GA1100.php   (238 words)

  
 canonical transformations - Advanced Physics Forums   (Site not responding. Last check: 2007-11-01)
The number of ignorable variables is not only related to the the mechanical system but also relavent to the choices of cononical variables.
To construct a canonical transformation, we need the help of \"Generators\" which are functions of qj,pj, Qj, Pj and t.
With the help of Pfaffian equation and Legendre transformation, one can find out different canonical transformations given by the corresponding generators.
www.advancedphysics.org /forum/showthread.php?t=153   (441 words)

  
 [No title]   (Site not responding. Last check: 2007-11-01)
It says that for every continuous symmetry a system has, there exists a conserved charge Q. Specifically, for a symmetry transformation q'_j = q_j + \epsilon (\delta q_j), which changes the Lagrangian by L' = L + \epsilon dF/dt, the charge Q = (\sum_j p_j (\delta q_j)) - F is conserved.
We noted why the Lagrangian is unchanged by this transformation, to first order in \epsilon.
(This transformation from L(q_i, \dot{q}_i) to H(p_i, q_i) is an example of a Legendre transformation, which you will frequently run into in thermodynamics).
www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/4.txt   (479 words)

  
 J W Gibbs, Gibbs Models: computer-visualized thermodynamic surfaces
I believe that the attention paid in that book to the Legendre transform and to the crucial role played by stability theory in any thorough understanding of classical thermodynamics makes it an essential resource for the serious student.
He developed the generating equations (for multicomponent systems, using extensive variables), wrote the convergence schemes to solve those equations, and created a transformation matrix that enabled him to plot any combination of thermodynamic variables attainable through the Legendre transformation.
The trick then is to somehow picture the manner in which those surfaces are transformed in shape by an additional Legendre transformation.
www.public.iastate.edu /~jolls/dedication.html   (957 words)

  
 CONTROL and DYNAMICAL SYSTEMS - California Institute of Technology
Then, we introduce a generalized Legendre transformation to define a Hamiltonian $H_{P}$ on a constraint momentum space $P=\mathbb{F}L(\Delta_{Q})$ and also define a generalized Hamiltonian $H$ on the Pontryagin bundle $TQ \oplus T^{\ast}Q$ by incorporating primary constraints in the sense of Dirac.
Further, we illustrate the equivalence between implicit Lagrangian and Hamiltonian systems in the context of the generalized Legendre transformation, where we also clarify the duality relation between the Legendre map and the primary constraints.
We shall illustrate an example of L-C circuits, which is a typical physical system of a degenerate Lagrangian with constraints in both contexts of implicit Lagrangian and Hamiltonian systems.
www.cds.caltech.edu /research/seminars/seminar.php?id='431'&date=2006-05-01   (363 words)

  
 PT0035   (Site not responding. Last check: 2007-11-01)
The transformed problem is a fixed nonlinear boundary value problem.
Nevertheless, the transformed problem still suffers from a singularity in the origin.
Only that, due to the general properties of contact transformations, potential and gravity change their roles, i.e.
www.olympus.net /biz/IAPSO/abstracts05/posters/PT0035.html   (275 words)

  
 Conan Leung Abstract
Mirror symmetry is a duality transformation between the complex geometry and the symplectic geometry of two Calabi-Yau manifolds.
In this talk, we explain this conjectural duality in terms of the Fouriertransformation and the Legendre transformation.
We will discuss such duality transformations on G2 manifolds and Hyperkahler manifolds.In the latter case, we obtain a general Plucker formula for dual varieties.
www.mth.msu.edu /~mccarthy/colloq.01-2/leung-abstract.html   (75 words)

  
 Degenerate Lagrangians and Legendre transformation   (Site not responding. Last check: 2007-11-01)
Transition from Lagrangian to Hamiltonian mechanics is performed by Legendre transformation.
Therefore, Legendre transformation is not invertible, we are not able to express the velocity via momentum:
Note that these transformations (conserving point in the phase space) correspond to reparametrization of the trajectory:
sim.ol.ru /~nikitin/course/node2.html   (612 words)

  
 Methods of Mathematical Physics, Volume 1,:0471504475:R. Courant; D. Hilbert:eCampus.com
Transformation to principal axes of quadratic and Hermitian forms
Transformation to principal axes on the basis of a maximum principle
Transformation of variational problems to canonical and involutory form
www.ecampus.com /bk_detail.asp?isbn=0471504475   (623 words)

  
 Legendre Transformations (II, An example)
Figure 1: E(S) is parabolic in S, as is A(T) in T, via the Legendre Transformations
We wish to perform a Legendre Transformation on E(S,V) to change the S variable to something else (it's going to be the temperature).
That is, we have started with a function of S and V and ended up with a function of T and V. Just to prove that we know what we're doing, we will do it again, this time transforming A(T,V) into E(S,V).
web.uconn.edu /~cdavid/latex2html/thermo2/thermo2.html   (197 words)

  
 Gibbs and Helmholtz Free Energies, Equilibrium, Maxwell's Relations
Equations 6a and 6b give us another thermodynamic definition of T and a thermodynamic definition of V (which is curious since we have always regarded V as a purely mechanical variable).
We could make the Legendre transformation from U to H by adding the pV term to U only because V is related to p and U through Equation 3b.
Using this as a guide it would seem reasonable to use Equation 3a to change the variable S to T in the function U, and use Equation 6a to change the variable S to T in the function H.
www.chem.arizona.edu /~salzmanr/480a/480ants/ageq&max/ageq&max.html   (1288 words)

  
  Untitled</u>   <i>(Site not responding. Last check: 2007-11-01)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A chain rule which unifies discrete and continuous settings is presented for our so-called alpha <a href="/topics/Derivative" title="Derivative" class=fl>derivatives</a> on generalized time scale s. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> This chain rule allows <b>transformation</b> of linear Hamiltonian systems on time scal es under simultaneous change of independent and dependent variables, thus extending the change of dependent variables recently obtained by Dosly and Hilscher. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> We also give the <b>Legendre</b> <b>transformation</b> for nonlinear Euler-Lagrange equations on time scales to Hamiltonian systems on time scales.</td></tr> <tr><td></td><td colspan=2><font color=gray>web.umr.edu /~bohner/papers/hsots.html</font>   (95 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.wit.edu/academics/em/courses/birkett/Lolita.htm">Classical Physics Graphics</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Φn(x) are the orthogonal <b>Legendre</b> polynomials on [-1,1] </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The discrete <b>Legendre</b> <b>transformation</b> of an array is </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> and is equivalent to the familiar <b>Legendre</b> <b>transformation</b> in</td></tr> <tr><td></td><td colspan=2><font color=gray>www.wit.edu /academics/em/courses/birkett/Lolita.htm</font>   (685 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://pruffle.mit.edu/3.00/Lecture_21_web/node2.html">The Other Energy Functionals: The Legendre transformations</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> To change to another set of natural variables, a new function is defined by subtracting off a particular conjugate pair: </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> ``The <b>Legendre</b> <b>transformation</b>'' is defined as the procedure of subtracting off conjugate pair to change that particular variable. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> It is fairly easy to show that the inverse <b>Legendre</b> <b>transformation</b> is simply</td></tr> <tr><td></td><td colspan=2><font color=gray>pruffle.mit.edu /3.00/Lecture_21_web/node2.html</font>   (88 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.math.tamu.edu/~fnarc/m642/s06/m642s06_hw.html">Math 642-600 Assignments</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Find the Fourier <b>transform</b> of T this way. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Then, find the Fourier <b>transform</b> of this tempered distribution. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Finally, use Fourier <b>transform</b> properties to obtain the desired <b>transform</b>.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.math.tamu.edu /~fnarc/m642/s06/m642s06_hw.html</font>   (355 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://diffiety.org/students/its2000/v1.htm">Forino IV, 4</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> M be 1-jet bundle of smooth functions of manifold M and let L </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> is a nowhere wanishing smooth function on M. Using the <b>Legendre's</b> <b>transformation</b>, constract a multivalue solutions of some 1-order scalar PDEs. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Constract an example to show that this statement is not true globally.</td></tr> <tr><td></td><td colspan=2><font color=gray>diffiety.org /students/its2000/v1.htm</font>   (249 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.emory.edu/PHYSICS/Faculty/Benson/380-96/notes/4/4.html">No Title</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> We noted why the Lagrangian is unchanged by this <b>transformation</b>, to first order in </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Expressed as a coordinate <b>transformation</b>, time translation causes </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> is an example of a <b>Legendre</b> <b>transformation</b>, which you will frequently run into in <a href="/topics/Thermodynamics" title="Thermodynamics" class=fl>thermodynamics</a>).</td></tr> <tr><td></td><td colspan=2><font color=gray>www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/4/4.html</font>   (389 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.loc.gov/catdir/toc/dover031/85029168.html">Table of contents for Library of Congress control number 85029168</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Coordinate <b>transformations</b> as a method of solving mechanical problems </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The motion of the phase fluid as a continuous succession of canonical <b>transformations</b> </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Hamilton's principal function and the motion of the phase fluid</td></tr> <tr><td></td><td colspan=2><font color=gray>www.loc.gov /catdir/toc/dover031/85029168.html</font>   (281 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.mathtable.com/hode/hode_toc.html">Table of Contents: Handbook of Differential Equations</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Transformations</b> of Second Order Linear ODEs - 1 </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Transformations</b> of Second Order Linear ODEs - 2 </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Transformation</b> of an ODE to an Integral Equation</td></tr> <tr><td></td><td colspan=2><font color=gray>www.mathtable.com /hode/hode_toc.html</font>   (41 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>CiteULike: Legendre transformation and lifting of multi-vectors</u>   <i>(Site not responding. Last check: 2007-11-01)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> First to recall the link between the classical Legendre-Fenschel <b>transformation</b> and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> We also show that, in the antisymmetric case, this lifting respects the Schouten bracket. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Finally we give an application to the study of the stability of singular points of Poisson manifold and Lie algebroids.}, author = {Dufour, Jean-Paul }, citeulike-article-id = {87155}, eprint = {math.DG/0502068}, keywords = {fenchel grassmannian <b>legendre</b> tensor}, month = {February}, title = <b>{Legendre</b> <b>transformation</b> and lifting of multi-vectors}, url = {http://arxiv.org/abs/math.DG/0502068}, year = {2005} }</td></tr> <tr><td></td><td colspan=2><font color=gray>www.citeulike.org /article/87155</font>   (234 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.uconn.edu/~cdavid/latex2html/thermo1/node3.html">y[x] as an example</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Up: <b>Legendre</b> <b>Transformations</b> Previous: dE = dq + </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> For the rest of this discussion we will write y[x] rather than y(x) to emphasize the functionality. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Anyway, y will be analogous to E, and x will be analogous to V, so that we are doing the constant entropy part of the <b>transformation</b> from dE to dH.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.uconn.edu /~cdavid/latex2html/thermo1/node3.html</font>   (146 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.phy.olemiss.edu/~luca/Topics/l.html">Topics: L</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Local Pseudogroup of <b>Transformations</b> of a Manifold > see <a href="/topics/Derivative" title="Derivative" class=fl>differentiable</a> maps. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Logarithm > see asymptotic flatness [logarithmic <b>transformations];</b> entropy [defomed log's]. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Idea: The name given to an approach to quantum gravity that originated with loop-based solutions of the connection representation of canonical quantum gravity and the of loop representation of qg; the name has stuck despite the fact that states of quantum gravity in this approach are now based on spin networks or spin foam models.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.phy.olemiss.edu /~luca/Topics/l.html</font>   (1574 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>h32a-06 in fm96</u>   <i>(Site not responding. Last check: 2007-11-01)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> ------------------------------ AN: H32A-06 TI: The Topographic-Coordinate <b>Transformation</b> --- A New Analytic Solution Method for Nonlinear Landform PDEs AU: Scott D. Peckham AF: USGS-WRD, 3215 Marine Street, Boulder, CO 80303-1066 EM: peckham@usgs.gov AB: There is an increasing interest in nonlinear partial <a href="/topics/Derivative" title="Derivative" class=fl>differential</a> equations (PDEs) that arise in the context of "equilibrium" or ideal landforms. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A new technique for solving such equations will be presented, one that is based on the simple idea of a topographic map, and is yet very powerful, in that it reduces the PDE to a set of often-separable ordinary <a href="/topics/Derivative" title="Derivative" class=fl>differential</a> equations. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Unlike other coordinate-transformation methods, such as the <b>Legendre</b> <b>transformation</b>, this one yields useful information such as elevation contours and flow lines, even when a closed-form inversion is not possible.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.agu.org /cgi-bin/SFgate/SFgate?&listenv=table&multiple=1&range=1&directget=1&application=fm96&database=/data/epubs/wais/indexes/fm96/fm96&maxhits=200&="H32A-06"</font>   (5482 words)</td></tr> </table> </td> </tr> </table><script language="JavaScript"> <!-- // This function displays the ad results. // It must be defined above the script that calls show_ads.js // to guarantee that it is defined when show_ads.js makes the call-back. function google_ad_request_done(google_ads) { // Proceed only if we have ads to display! if (google_ads.length < 1 ) return; 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