Generating function of infinitesimal translation - classical mechanics

Question

I am reading Sakurai's Modern quantum mechanics and at some point it's trying to draw a parallel between classical and quantum mechanics.

It says

An infinitesimal translation in classical mechanics can be regarded as a canonical transformation, $$ \mathbf{x}_{\mathrm{new}} \equiv \mathbf{X} = \mathbf{x} + d\mathbf{x}, \quad \mathbf{p}_{\mathrm{new}} \equiv \mathbf{P} = \mathbf{p}, \tag{1.6.28} $$ obtainable from the generating function $$ F_2(\mathbf{x}, \mathbf{P}) = \mathbf{x}\cdot \mathbf{P} + \mathbf{p}\cdot d\mathbf{x}. \tag{1.6.29} $$

From the wikipedia page it seems that a generating function is something that one can differentiate to obtain the equation of motion of the system.

I mean I assume I need to differentiate $F$ with respect to $\mathbf{X}$ and $\mathbf{P}$? What equation of motion am I supposed to get from here?

Answer 1

Hints:

1. The infinitesimal change in position $d{\bf x}=\varepsilon~ {\bf f}({\bf x})$ can be thought of as an infinitesimal parameter $\varepsilon$ times a vector-valued function of the old position ${\bf x}$.

2. A type-2 generating function $F_2({\bf x},{\bf P},t)$ for a canonical transformation depends on the old position ${\bf x}$, the new momentum ${\bf P}$, and possibly time $t$.

3. If the above does not make sense, then you should study Hamiltonian mechanics and canonical transformations, as Cosmas Zachos suggests in above comment, e.g. Ref. 1.

References:

1. H. Goldstein, Classical Mechanics; Section 9.