What is a generating function (GF) in physics?

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Discussion Overview

The discussion centers on the concept of generating functions (GF) in physics, particularly in the context of classical mechanics (CM) and electromagnetism (E&M). Participants explore the nature and role of GFs, seeking to clarify their definitions and applications in different areas of physics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the role of generating functions in physics, particularly in relation to Hamiltonian mechanics and their function in canonical transformations.
  • Another participant explains that generating functions in CM serve to link old canonical variables with new ones during transformations, suggesting a specific application of GFs in this context.
  • A participant mentions the use of delta functions in E&M as generating function solutions for equations of motion, but questions the relationship between GFs in E&M and CM.
  • There is a correction from one participant acknowledging a misunderstanding about the relationship between GFs in CM and E&M, reiterating that GFs in CM primarily relate old and new canonical coordinates.

Areas of Agreement / Disagreement

Participants generally agree that generating functions in CM are used to relate canonical variables, but there is disagreement regarding the connection between GFs in CM and E&M, with some participants asserting that they are not related.

Contextual Notes

There are unresolved questions about the broader implications and definitions of generating functions across different areas of physics, as well as the specific nature of their relationships in various contexts.

Living_Dog
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I am trying to understand CM wrt QFT and found out that I need to understand the HJE. This brought me to reading about all related subjects. The history lesson alone has been awesome. However, now I am reading about the HJE and found the Wikipedia pages lacking as to exactly what is the nature/idea behind/role of GF's in physics.

[1] GF's in CM
Wikipedia.com says: "Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In the case of physics, generating functions act as a bridge between two sets of canonical variables when performing canonical transformation."

So F(q,P,t) => Q and p can be determined from this F? So F "generates" Q and p??

[2] GF's in E&M:
I recall that there is an equation of motion in E&M which has a delta function as a "generating function" solution.

NOTE: I don't need the physics - I need the _idea_ of what a GF is ... and now that I think about it - are they related?

Thanks in advance.

-LD
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To get the solution of a partial differential equation with a inhomogeneous term ,we introduce the delta function on the right side of the equation.The corresponding solution is called Green function.I don,t think GF's in E&M has some relations with GF's in CM.The GF's in CM only helps us link the old canonical variables with the new canonical variables during the transformation.
 
Dyson said:
To get the solution of a partial differential equation with a inhomogeneous term ,we introduce the delta function on the right side of the equation.The corresponding solution is called Green function.I don,t think GF's in E&M has some relations with GF's in CM.

D'OH! You are right sir! - my bad ... and apologies.

Dyson said:
The GF's in CM only helps us link the old canonical variables with the new canonical variables during the transformation.

So that's all the four types of generating functions do - is relate the old canonical coordinates to the new ones!

Thanks for the awesome help! I can now move on in my studies.

-LD
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