- #1
andresB
- 629
- 375
In the Hamiltonian formalism, the space-time transformation are realized via canonical transformation, and the transformations are generated by Poisson brackets of certain functions of phase-space variables.
In Newtonian mechanics, Galilean boosts are generated by the sometimes called dynamic mass moment $$\overrightarrow{G}=m\overrightarrow{r}-t\overrightarrow{p}.$$
Now, in virtually every source I consult, the general generator of the Galilean boosts is not considered. Instead, people just use the ##t=0## generator
$$\overrightarrow{G}=m\overrightarrow{r}$$
The same happen for Lorentz transformations, people just use the ##t=0## generator
$$\overrightarrow{K}=H\overrightarrow{r}$$
where ##H## is the energy.
So, the question is, what is the most general form of ##\overrightarrow{K}##?
In Newtonian mechanics, Galilean boosts are generated by the sometimes called dynamic mass moment $$\overrightarrow{G}=m\overrightarrow{r}-t\overrightarrow{p}.$$
Now, in virtually every source I consult, the general generator of the Galilean boosts is not considered. Instead, people just use the ##t=0## generator
$$\overrightarrow{G}=m\overrightarrow{r}$$
The same happen for Lorentz transformations, people just use the ##t=0## generator
$$\overrightarrow{K}=H\overrightarrow{r}$$
where ##H## is the energy.
So, the question is, what is the most general form of ##\overrightarrow{K}##?