I've got a question regarding the different Symmetrie-Lie-Groups of Newtonian Mechanics and special realtivity. Is there a canonical way to obtain the equations of motion for a free particle only by the symmetrie-group?
I just found that according to J.M. Souriau the Lie-Group should act transitively so there is no further structure.
There is, in a sense, a way to 'guide' oneself to the equations of motion based on the symmetries. The form of mechanics most suitable for this purpose is Hamilton's principle - the system takes a path for which the action has a stationary value for variations with fixed endpoints: $$\delta S=0$$
$S$ is generally expressed as (under some parametrization of paths with parameter $t$): $$S = \int_{path} L(x, \dot{x})dt$$
Leading to the equations of motion: $$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} = \frac{\partial L}{\partial x}$$
Now, in general, one expects the action respect the symmetries of the problem. Consider the following examples:
Newtonian mechanics
Here, we have particles where the position is given as a function of time: $x(t)$. Due to the translation and rotation invariance respectively, we require that the action $S$ does not depend on any preferred position or direction. The form we effectively have is then $L = L(|\dot{x}|)$. We further require it to respect Galilean symmetry. For this, we require that $\delta S$ is invariant under $\dot{x} \rightarrow \dot{x}+u$ for constant $u$. This is possible when $L = \frac{1}{2}m\dot{x}^2$, so that under this transformation $$S = \int\frac{1}{2}m\dot{x}^2dt \rightarrow \int\frac{1}{2}m\dot{x}^2dt + \int mu\ dx + \int\frac{1}{2}mu^2dt$$ The last two terms are only functions of the endpoint of the path, and therefore their variation vanishes i.e. $\delta\int dx = \delta\int dt = 0$. This leads to the equation of motion: $$m\ddot{x} = 0$$ or $$\ddot{x} = 0$$
This is precisely the equation of motion one would obtain for a free particle according to Newton's second law. (Note: Mechanics is presented this way, for example, in Landau and Lifshitz, Mechanics).
Special Relativity
Here, we require that the action is invariant under Lorentz transformations, in addition to being translation invariant. Thus, for the action, we choose the integral over paths of the spacetime interval itself: $$S = m\int \sqrt{\eta_{\mu\nu} dx^{\mu} dx^{\nu}}$$
Setting $\delta S=0$ leads to the geodesic equation of motion for a free particle i.e. the motion is a straight line in spacetime.
I'm not sure if this qualifies as a 'canonical' way; it is more like making educated guesses.
