# Finding the generator function for 4 variables

• LCSphysicist
In summary, to prove that the given transformation is canonical, we need to show that ##dF = \sum_i p_i dq_i - P_i dQ_i## can be written as a total differential ##dF##. The approach suggested is to express all ##dQ_i, dP_i## in terms of ##dq_i, dp_i## and simplify, eventually leading to ##dF = d(p_2q_2 + 2p_2q_1)##. By comparing this to the definition of a generating function, we can conclude that ##F(P_2, p_2) = -p_2P_2##, satisfying the requirement for a generating function. However, it is
LCSphysicist
Homework Statement
Generating function
Relevant Equations
.
Prove directly that the transformation $$Q_{1} = q_{1}, P_{1} = p_{1} − 2p_{2}$$ $$Q_{2} = p_{2}, P_{2} = −2q_{1} − q_{2}$$ is canonical and find a generating functionSo the first part is easy and can be skipped here. I have some difficults regarding the second part, namely, the one that ask for the generator function.

My approach was to evaluate the object ##\sum_i p_i dq_i - P_i dQ_i##, i have read at MG Calkin that if this object can be written as a total differential ##dF##, this one is (can be) the generating function.

So, expandin the differential, i have got that ##\sum_i p_i dq_i - P_i dQ_i = d(p_2q_2 + 2p_2q_1) \implies F = F(P_2,p_2) = -p_2P_2##.

What do you think? is it ok?

Herculi said:
Homework Statement:: Generating function
Relevant Equations:: .

Prove directly that the transformation $$Q_{1} = q_{1}, P_{1} = p_{1} − 2p_{2}$$ $$Q_{2} = p_{2}, P_{2} = −2q_{1} − q_{2}$$ is canonical and find a generating functionSo the first part is easy and can be skipped here. I have some difficults regarding the second part, namely, the one that ask for the generator function.

My approach was to evaluate the object ##\sum_i p_i dq_i - P_i dQ_i##, i have read at MG Calkin that if this object can be written as a total differential ##dF##, this one is (can be) the generating function.

So, expandin the differential, i have got that ##\sum_i p_i dq_i - P_i dQ_i = d(p_2q_2 + 2p_2q_1) \implies F = F(P_2,p_2) = -p_2P_2##.

What do you think? is it ok?
It looks like you have omitted a lot of work.
Starting from ##dF = \sum_i p_i dq_i - P_i dQ_i##, how did you arrive at ##dF = d(p_2q_2 + 2p_2q_1)## and then to ##F(P_2,p_2) = -p_2P_2##?

Mark44 said:
It looks like you have omitted a lot of work.
Starting from ##dF = \sum_i p_i dq_i - P_i dQ_i##, how did you arrive at ##dF = d(p_2q_2 + 2p_2q_1)## and then to ##F(P_2,p_2) = -p_2P_2##?
Yes. Actually, i have just expressed all ##dQ_i, dP_i## in ##dF = \sum_i p_i dq_i - P_i dQ_i## in terms of ##dq_i, dp_i##. Eventually some terms cut off and we can express the results as ##dF = d(p_2q_2 + 2p_2q_1)##.
Now, ##d(p_2q_2 + 2p_2q_1) = d(p_2(-P_2))## from the summary, so i just concluded that ##F(P_2,p_2) = -p_2P_2##

Herculi said:
Yes. Actually, i have just expressed all ##dQ_i, dP_i## in ##dF = \sum_i p_i dq_i - P_i dQ_i## in terms of ##dq_i, dp_i##. Eventually some terms cut off and we can express the results as ##dF = d(p_2q_2 + 2p_2q_1)##.
"Eventually" is what I'm asking you to elaborate on.
Herculi said:
Now, ##d(p_2q_2 + 2p_2q_1) = d(p_2(-P_2))## from the summary, so i just concluded that ##F(P_2,p_2) = -p_2P_2##

## 1. What is a generator function?

A generator function is a special type of function in programming that can be paused and resumed at any point during its execution. It allows for the generation of a sequence of values, instead of returning a single value like a regular function.

## 2. How many variables can a generator function have?

A generator function can have any number of variables, including 4. The number of variables depends on the specific needs of the function and the data being processed.

## 3. Why is it important to find the generator function for 4 variables?

Finding the generator function for 4 variables is important because it allows for the efficient and dynamic generation of a sequence of values. This can be useful in various applications, such as data processing, simulations, and algorithms.

## 4. How do you find the generator function for 4 variables?

The process of finding the generator function for 4 variables involves identifying the variables and their relationships, determining the appropriate syntax for the function, and writing the code using a programming language. It may require some trial and error and testing to ensure the function works as intended.

## 5. What are some potential challenges in finding the generator function for 4 variables?

Some potential challenges in finding the generator function for 4 variables include determining the correct syntax and logic for the function, handling errors and edge cases, and optimizing the function for efficiency and performance. It may also require a strong understanding of programming concepts and the specific programming language being used.

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