Time is not special regarding symmetries and conserved quantities. In general you can consider any family of continuous transformations parametrised by some rea

Time is not special regarding symmetries and conserved quantities. In general you can consider any family of continuous transformations parametrised by some real variable s: be it translations by a distance x, rotations by an angle φ, etc. These are technically one-parameter subgroups of a Lie group. Then, if your dynamical system is symmetrical under these transformations you can construct a quantity whose derivative wrt s is zero.