3

So I apologize if this vague, but I am just looking for steps to figure this out.

I have the following CT $$Q_1(q_1)$$ $$Q_2(q_2,p_2)$$ $$P_1(p_1,p_2,q_1,q_2)$$ $$P_2(p_2,q_1)$$

Where I am just indicating what old coordinates are used in the new coordinate functions. So when looking through Goldstein (and any source online), there are 4 different generating functions, as given in the table.

I am really unsure of how to determine which to use, let alone going about finding it.

Qmechanic
  • 201,751
yankeefan11
  • 1,778
  • 3
  • 18
  • 32
  • An example: https://physics.stackexchange.com/q/329427/2451 – Qmechanic May 14 '17 at 12:58
  • If you already have the Q,P as explicit functions of q,p preserving PBs, what more do you want? You already know the transformation is canonical. Is there some further clarification due? – Cosmas Zachos May 14 '17 at 13:51

1 Answers1

1

Here is how I understand it: let's say you have some coordinates $p_i$ and some momenta $q_i$. You want to find transformations

$$Q_i\equiv Q_i(q_i,\,...,\,q_n,\,p_i,...,\,p_n,\,t)\qquad P_i\equiv P_i(q_i,\,...,\,q_n,\,p_i,...,\,p_n,\,t) $$

These variables $p_i$, $q_i$, $Q_i$, and $P_i$ must satisfy the Hamilton's equations of motion:

$$\dot{q}_i=\frac{\partial H}{dp_i},\quad \dot{p}_i=-\frac{\partial H}{dq_i},\quad \dot{Q}_i=\frac{\partial K}{dP_i},\quad \dot{P_i} = -\frac{\partial K}{dQ_i}$$

where $K$ is the transformed Hamiltonian $K\equiv K(Q,\,P,\,t)$.

To perform the canonical transformation, we may be able to introduce the function $F=F_1$, where

$$P_i \dot{Q}_i-K+\frac{dF_1}{dt}=p_i \dot{q}_i-H$$

or we could introduce $F=F_2-Q_iP_i$, and we thus have

$$-\dot{P}_i Q_i-K+\frac{dF_2}{dt}=p_i \dot{q}_i-H $$

Similarly for the generating functions for $F_3$ and $F_4$.

Now that we understand their differences, we have to ask how we use them. What you use is based upon what you know. If I want to find $F_1$, I integrate $p_i$ with respect to $q_i$ and $-P_i$ with respect to $Q_i$, and combine the result to find the whole value for $F_1$. A similar process follows for the identities for $F_2$, $F_3$, and $F_4$. Now, if I have the opposite problem--namely, I have $F_1$, and I want to find the coordinates and/or momenta--then I take the partial with respect to $q_i$ to find $p_i$ and the partial with respect to $Q_i$, take the negative, and I find $P_i$.

If I understand your question correctly, you have a set of $Q$'s and $P$'s, and you want to find $F$. You can pick whatever $F$ you want--you just need to change the definition of the generating function as given above.

Hope this helps. These slides might also be of assistance: http://users.physics.harvard.edu/~morii/phys151/lectures/Lecture20.pdf

Y2H
  • 700
Joshuah Heath
  • 2,605