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Rocky Raccoon

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During the whole of my undergraduate study of physics, this one thing always bothered me. It concerns the interplay of conserved quantities, symmetries, Noether's theorem and initial conditions.

For a system of N degrees of freedom, governed by the usual Newton's laws, one has to give 2N initial conditions (e.g. position and momenta) to arrive at the complete solution of the problem. Given that one of the initial condition refers to the initial time, we are left with 2N-1 constant describing the trajectory of the system. We can now form 2N-1 independent functions of these 2N-1 constant which will be, by construction, also constant in time. Smartly chosen, these functions can represent energy, momentum... of the system. This shows that there are, in general, (at least!) 2N-1 constants of motion for a system od N degrees of freedom. The only thing that was used in this derivation was the assumption of Newtons law that the accelaration is uniquely given by some function of position and velocity. Note that the 2N-1 conserved quantities should(?) corespond to the space-time properties of the system.

On the other hand, Noether's theorem clearly states the connection between the conserved quantities and the symmetries of the physical system. Since we usually deal with Galilei invariance (in classical mechanics), it is usualy stated that any closed system has 10 conserved quantities - enegry (1), momentum (3), angular momentum (3) and vector of center of mass (3). Since clearly 2N-1 is not equal to 10, it is obvious that in some cases (N < 5), the before mentioned 10 Galileian invariants are not independent, and in other cases (N > 5), there are further space-time symmetries besides the Galileian.

Is this correct? If so, how do other symmetries (e.g. gauge invariance) fit into the "initial conditions" picture?