One parameter-groups and coordinate transformations in phase-space

Question

I have given a function

$$G=p_1q_1 - p_2q_2$$

on a 4-dimensional phase-space. This function $G$ commutes with the Hamiltonian

$$H= \frac{p_1p_2}{m} + m\omega^2q_1q_2.$$

It generates a flow

$$(\vec{q},\vec{p}) \rightarrow (\vec{Q}(\vec{q},\vec{p},\alpha), \vec{P}(\vec{q},\vec{p},\alpha)) $$

defined by

$$\frac{dQ_i}{d\alpha} = \frac{\partial G}{\partial P_i}; \quad \vec{Q}(\vec{q},\vec{p},0) = \vec{q}$$

$$\frac{dP_i}{d\alpha} = -\frac{\partial G}{\partial Q_i}; \quad \vec{P}(\vec{q},\vec{p},0) = \vec{p}.$$

The calculation is no problem, but I do not understand its background. How does $G$ generate the transformation and why is it first defined in old coordinates $\vec{q}, \vec{p}$ but then used with the new ones $\vec{Q}, \vec{P}$? What is the connection to (Lie-) group theory? And can the transformation be understood in terms of vector fields on a manifold?

Since the whole concept is not really clear to me I would be very grateful for a comprehensive explanation. Book-suggestion are also highly welcome.

Answer 1

In terms of standard differential geometry, the phase space of physics is a symplectic manifold with symplectic structure $\omega = \mathrm{d}q^i \wedge \mathrm{d}p_i$ (with summation over repeated indices).

For any observable $f(q,p)$, the associated Hamiltonian vector field $X_f$ is defined abstractly by $\omega(X_f,V) = \mathrm{d}f(V)$ and in local coordinates this is the vector field $\left(\frac{\partial f}{\partial p_i}, -\frac{\partial f}{\partial q^i}\right)$. The flows of Hamiltonian vector fields are symplectomorphisms of the phase space manifold, or what physicists know as "canonical transformation" in some contexts (but not always, see this answer by Qmechanic).

So the one-parameter family of symplectomorphisms $\phi_t : (q^i, p_i)\mapsto (Q^i(t),P_i(t))$ generated as the flow of the Hamiltonian vector field associated with your function $G$ is the solution of the differential equations \begin{align} \partial_t Q^i & = \frac{\partial G}{\partial p_i} \\ \partial_t P_i & = -\frac{\partial G}{\partial q^i} \end{align} and the usage of the capital coordinates on the r.h.s. in your equations is extremely likely to be a typo. Note that this can also be written as $\partial_t \phi_t = \{\phi_t , G\}$, which is more directly how one might say that $G$ generates the transformation - the infinitesimal change under the transformation is given by the Poisson bracket with $G$.

The connection to Lie theory is that the map $f \mapsto X_f$ from phase space functions to vector fields is a Lie algebra homomorphism with the Poisson bracket and the standard Lie bracket of vector fields as the Lie brackets on the two sides, and the flow $\phi_t$ associated with $X_f$ is the one-parameter group of elements of the diffeomorphism Lie group generated by $X_f$ (the latter is difficult to make precise in full generality because the group of diffeomorphisms is not finite-dimensional, but you can always do it for finite-dimensional subgroups and subalgebras).