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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.

Qmechanic
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Fatho
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1 Answers1

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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.

AV23
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  • The problem occuring for me here, is that the construction is really a guess guided by historical developments. I'm searching for a way to do the construction for a mostly arbitrary symmetry group or a construction which doesn't use the foliation of spacetime in the Galilean case as the given one and thereby might lead to a generalisation not diretly using the spevial symmetry group. – Fatho Apr 29 '15 at 07:24
  • There is certainly a difference between the two constructions presented here: one is explicitly invariant under its symmetry transformations, the other isn't. While it is possible to actually construct a Galilean invariant action (http://physics.stackexchange.com/a/15315/77679), it requires introducing auxiliary variables. Once again, we can have a different Lorentz invariant action with auxiliary variables, generally used to treat the massless case (http://physics.stackexchange.com/q/167296/77679). But they both look different; if there is a general method, it is, at least, hidden. – AV23 Apr 29 '15 at 10:24