# Confirm my reasoning on a generating function proof

• Coffee_
In summary, the conversation discusses the equation (6.5) and the reasoning behind it. The participant proposes a way of looking at the equation and discusses the conditions under which the extra term with the ##F## vanishes. It is determined that the proposed method is correct and that ##F## must depend only on ##q##, ##Q##, and perhaps explicitly on ##t##. The conversation also mentions the difference between Hamilton's principle in the Lagrangian and Hamiltonian forms, with the latter being more general due to its consideration of variations in phase space. The concept of canonical transformations is also briefly discussed. In the end, the participant requests confirmation or refutation of a summary written about yesterday's thread.
Coffee_

It's about equation (6.5) I'm not entirely getting the reasoning explained by the author so I came up with the following, can anyone confirm or refute. One way to look at equation (6.5) would be:

We create variations on the ##q## variables, in the form of ##\delta q(t)##. Since ##Q=Q(q,p,t)## the former variation induces a unique variation ##\delta Q(t)##. Since I want both of the action integrals to make sense, both ##\delta q(t) = \delta Q(t) ## have to be ##0## on the end points. This is not so subtle because technically a variation in ##q(t)## could change ##p(t)## in a non trivial way and ##\delta Q(t)## wouldn't be zero on the end points. However I have to make thsi a condition for both integrals over the Lagrangians to be identifiable as the action.

Under these conditions it is clear that the extra term with the ##F## vanishes.

Is this totally wrong, is this partially correct?

It's correct. That's why ##F## must depend only on ##q##, ##Q## (and perhaps explicitly on ##t##). Note that there are no canonical momenta involved here since the proof deals with Hamilton's principle in the Lagrangian form. I'd also not call these transformations "canonical", because in the Lagrangian formalism you can only do point transformations, i.e., transformations between old and new configuration-space variables, since in the Lagrangian form the variation is with respect to curves in configuration space.

It's important to note that the Hamilton principle in the Hamiltonian form is more general than in the Lagrangian form, because it's about variations in phase space (unrestricted for the momenta and at fixed boundary values for the position variables). Then you have more freedom and thus also more symmetry transformations, namely the canonical ones of the Hamilton description of the dynamics (see the other thread about canonical transformations, last discussed yesterday).

Thanks so much for helping me out. About yesterdays thread, I wrote some kind of a summary at the end. I would appreciate it if you could confirm or refute

## 1. How do I know if my reasoning for a generating function proof is correct?

To confirm your reasoning on a generating function proof, you can follow these steps:

- Start by clearly stating your assumptions and defining all variables used in the proof.

- Use mathematical equations and logical reasoning to show how each step in the proof follows from the previous one.

- Check for any errors or mistakes in your calculations or reasoning.

- Consider different cases or examples to test the validity of your proof.

- If possible, seek feedback from peers or experts in the field to verify your proof.

## 2. Can I use any generating function for my proof?

No, it is important to carefully select the generating function for your proof based on the problem at hand. The generating function should have the same coefficients as the sequence you are trying to prove, and it should also have a simple form that allows for easy manipulation and analysis.

## 3. Do I need to use complex numbers in my generating function proof?

Not necessarily. While some generating functions may involve complex numbers, there are also many generating functions that only use real numbers. The choice of using complex numbers depends on the problem and the most efficient way to prove it.

## 4. How can I make my generating function proof more concise?

To make your generating function proof more concise, you can try to reduce unnecessary steps or simplify complicated equations. You can also use properties and identities of generating functions to streamline your proof. Additionally, using clear and concise language can also make your proof easier to understand.

## 5. What are some common mistakes to avoid in a generating function proof?

Some common mistakes to avoid in a generating function proof include:

- Not clearly stating assumptions or defining variables.

- Skipping steps or not showing the logical reasoning behind each step.

- Using an incorrect generating function that does not match the sequence being proven.

- Making mistakes in calculations or algebraic manipulations.

- Not considering all cases or examples to test the validity of the proof.

• Classical Physics
Replies
5
Views
1K
• Classical Physics
Replies
5
Views
1K
• Classical Physics
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
616
• Biology and Chemistry Homework Help
Replies
1
Views
542
• Classical Physics
Replies
17
Views
4K
• Classical Physics
Replies
11
Views
1K
• Biology and Chemistry Homework Help
Replies
1
Views
218
• Classical Physics
Replies
6
Views
1K
• Biology and Chemistry Homework Help
Replies
2
Views
282