- #1
Coffee_
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I am confused about the ''4 basic types'' of generating functions. I have searched for this a bit on google but haven't found anything that truly made the click for me on this concept so I'll try here:
What I do understand and need no elaboration on:
1) When considering the Hamiltonian and Kamiltonian for which minimal action still holds, their sum together with the sum of two other terms pq' and PQ', can differ up to a total derivative of a function ##F(Q,P,t)## or ##F(q,p,t)##. So I understand the reasoning behind introducing the concept ''generating function''.
2) Once you give me one of the four types, I can derive the canonical transformations that follow from this.
What I don't understand.
1) Why any generating function different than ##F(q,p,t)## or ##F(Q,P,t)## is introduced because when considering this total derivative as mentioned earlier - we are taking this total derivative inside an action integral that goes over either ##q,p## or ##Q,P## independently. Basically we want that F is the same in both end points of the action path under the integral, but the integral is taken either over variations of ##q,p## or variations of ##Q,P##
2) When 1 is answeren, and it would becoem clear why it makes sense to introduce different kinds of generating functions I still do not get how exactly they are related by the Legendre transforms.
I hope someone can enlighten me a bit on this subject.
What I do understand and need no elaboration on:
1) When considering the Hamiltonian and Kamiltonian for which minimal action still holds, their sum together with the sum of two other terms pq' and PQ', can differ up to a total derivative of a function ##F(Q,P,t)## or ##F(q,p,t)##. So I understand the reasoning behind introducing the concept ''generating function''.
2) Once you give me one of the four types, I can derive the canonical transformations that follow from this.
What I don't understand.
1) Why any generating function different than ##F(q,p,t)## or ##F(Q,P,t)## is introduced because when considering this total derivative as mentioned earlier - we are taking this total derivative inside an action integral that goes over either ##q,p## or ##Q,P## independently. Basically we want that F is the same in both end points of the action path under the integral, but the integral is taken either over variations of ##q,p## or variations of ##Q,P##
2) When 1 is answeren, and it would becoem clear why it makes sense to introduce different kinds of generating functions I still do not get how exactly they are related by the Legendre transforms.
I hope someone can enlighten me a bit on this subject.