# Number of conservation laws

I saw a discussion about the relation of symmetries of Lagrangian and conservation laws on a textbook of analytical mechanics. A part that was counterintuitive to me was that all the discussion was done in a generalized coordinate system.

Does it imply that the number of conservation laws does not change after a coordinate transformation?

(This question might not be an accurate because you can make another conservation law by simply adding a constant. But I want to know the case up to such a trivial change.)

• Jun 18 at 9:57

Along with adding constants, an arbitrary function of existing conserved quantities will also be conserved. If $$X_1, X_2, \dots X_n$$ are all conserved, then $$f(X_1, X_2, \dots X_n)$$ will also be conserved for any function $$f$$. So really the conserved quantities should form a space, with some number of dimensions. The number of dimensions is the number of "nontrivial conserved quantities".
If the system has $$N$$ degrees of freedom (i.e. its phase space is $$2N$$ dimensional) then the dimension of the conserved space is also $$2N$$. Why? Conserved quantities are allowed to depend on the $$t$$ coordinate in general, so given a particular state at a particular time, we can just run time backwards to $$t=0$$, and we get the $$N$$ initial position coordinates, and the $$N$$ initial momentum coordinates. At any point in the trajectory, running time back to $$t=0$$ will give the same initial condition, so these $$2N$$ variables must all be conserved. (And the initial condition can vary any of these variables independently, so the space must have a full $$2N$$ dimensions.)
A coordinate transformation can't change the number of dimensions in a space, so there will always be $$2N$$ conserved quantities, though some of them may end up looking very complicated.
• I think for integrable systems, the conserved quantites are not allowed to depend on $t$, just the position and momentum coordinates. Jun 19 at 3:17