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For two sets of canonical coordinates $(q,p)$ and $(Q,P)$, the following relation holds true:

$$p_i \dot q_i - H = P_i \dot Q_i -K + \frac{dF}{dt}$$

I have three questions regarding the last term:

  1. Why do we add $\frac{dF}{dt}$ and not just $F$?
  2. Why is the variation of this term at endpoints of the action integral is 0?
  3. In my book (Goldstein), it says it's useful if $F$ is expressed as half the function of old coordinates and half of new. Why?
Qmechanic
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Prasoon
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2 Answers2

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The requirement that must be satisfied is Hamilton’s Principle $$ \delta S = \delta\int_{t_1}^{t_2} L \ dt = 0 $$ which states that the variation of the action must vanish. We choose, by definition, that the variation $\delta$ keeps the endpoints fixed. In Hamiltonian mechanics, we can express Hamilton’s principle by rewriting the Lagrangian in terms of the Hamiltonian $$ \delta S = \delta\int _{t_1}^{t_2} \left( p_i \dot{q}_i - H\ \right) dt = 0 $$ So any alternative choice of coordinates $p_i, q_i \rightarrow P_i, Q_i$ and “Kamiltonian” $H \rightarrow K$ has to satisfy the same variational equation. $$ \delta\int_{t_1}^{t_2} \left( P_i \dot{Q}_i - K\ \right) dt = 0 $$ So now the question is: what can we do to the action $S$ that keeps the variation $\delta S $ zero? We can either multiply it by a constant $\lambda$ (which Goldstein discusses) or add a new function $F$ which only changes $S$ at the endpoints where there is no variation. To only affect the endpoints, we can add a term within the action integral which is a total derivative. This will only impact the endpoints, which will leave the variation procedure implied by $\delta$ unaffected. $$ \int_{t_1}^{t_2} \frac{dF}{dt}dt = F(p(t_2), q(t_2), t_2) - F(p(t_1), q(t_1), t_1) $$ Note that $F$ is a function of the phase space variables and time. As to your third question, consider what the utility of the generating function would be if it only depends on one half of the coordinates. When there is one from each (i.e. $F(q,Q,t)$), you can find relations like those given in table 9.1 in Goldstein, which allow you to solve for the new coordinates given the old ones. The procedure to find these relations is laid out there as well.

Da Monster
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  1. Adding a term to eq. (1) that is not a total derivative would typically not ensure that the the Hamilton's equations are form invariant under a canonical transformation (CT).

  2. Boundary conditions are e.g. discussed in this Phys.SE post.

  3. Concerning the usefulness of CTs of type 1-4, see e.g. this Phys.SE post.

Qmechanic
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