Timeline for Derive the generating function for canonical transformation of type $F_3$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 14, 2014 at 12:41 | comment | added | Deliang Zhong | Please read carefully what I have done in the dE example. I think you can understand this by calculating by yourself. | |
| Oct 13, 2014 at 23:22 | comment | added | JackReacher | I guess I am not so much worried about the geometric meaning at the moment. I'm just confused with how we form the different generating functions from $F_1$. For example, I get $F_2$ by doing $F_1 + QP$, and I get $F_3$ by doing $F_1 - qp$, how would i get $F_4$? And how do we know that $F_2, F_3$ could be formed by adding/subtracting those conjugate coordinates? | |
| Oct 13, 2014 at 21:47 | comment | added | Deliang Zhong | $df=udx+wdy$, $df = udx + d(wy) - ydw$. So we need to compute the differential $d(wy)$... If you really want to know legedre transformation, you can read Arnold's book to find its geometric meanings. But in practice, what you need to remember is that legedre transformation is just "integration by parts". | |
| Oct 13, 2014 at 21:40 | comment | added | JackReacher | Sorry, still not really getting it, I may be thinking about this in the opposite direction. Here is my thought process: We start with a function $f(x,y)$ and wish to perform a Legendre transform. if we take the differential we get: $df = udx + wdy$, then my notes say, compute the differential $d(wy)$ - I guess my question is why $wy$? When I compute this, I'm having trouble seeing why its the same as integration by parts? | |
| Oct 13, 2014 at 21:20 | comment | added | Deliang Zhong | The sign is very important. As you can see in my deriviation, the sign can be derived by "integration by parts". For example, $dE = TdS - pdV$, $d(E+PV) = TdS + vdp, d(E-TS) = -SdT - pdv$... The point is the independent variable appears as the $d ..$ | |
| Oct 13, 2014 at 20:57 | comment | added | JackReacher | So for a Legendre transform, does the sign not matter? In that sense, how is one supposed to know how to derive the other generating functions? Just by guess/memorization? For example, if I wanted to derive $F_4$, I have already done $F_2 = F_1 + PQ$, $F_3 = F_1 - qp$, aren't these the only conjugate variables? Can you do also $pP$? But then aren't these not conjugate to each other? Thanks so much for your help. | |
| Oct 13, 2014 at 20:08 | comment | added | Deliang Zhong | $F_2 = F_1 +PQ$ rather than minus... | |
| Oct 13, 2014 at 12:09 | comment | added | JackReacher | Thanks, I think I have gotten a little further with my understanding, could you look at my edit in my question, if you are able to help me? | |
| Oct 13, 2014 at 11:46 | comment | added | Deliang Zhong | we have $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy \equiv u dx + wdy$ | |
| Oct 13, 2014 at 11:24 | comment | added | JackReacher | Thanks for your help, but I'm still confused, and to be honest still confused about the definition of a legendre transform. My understanding so far is that the definition of the Legendre transform of a function $f(x,y)$ is $G = f-wy$ where $w= \frac{\partial f}{\partial y}$ - and then this can be done with a few combinations of variables. I'm a bit confused as to why differentials come into play? | |
| Oct 13, 2014 at 9:58 | review | First posts | |||
| Oct 13, 2014 at 11:07 | |||||
| Oct 13, 2014 at 9:54 | history | answered | Deliang Zhong | CC BY-SA 3.0 |