Not all systems have conserved quantities, and conserved quantities are not unique, since one can always apply a function to a conserved quantity, such as adding a number.
Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.
which contains information specific to the system and can be helpful in finding conserved quantities, or establishing whether or not a conserved quantity exists.
is conserved. This may be derived by using the Euler–Lagrange equations.