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Each Conservation Law is accompanied by a symmetry. Or rather each conservatino law is due to a symmetry in the physical world. The following are certain conservation laws:

Conservation of charge
Conservatino of mass and energy
Conservation of momentum

what else is there? what is the symmetry that accopmanies these laws? and HOW does a symmetry lead to a conservatino law?

Classically, the symmetry referred to that of the Lagrangian L(t,q,q'), which describes the action of a physical system of particles. It is a function of time t, some generalizes coordinates {q_i}, and generalized velocities {q_i' = dq_i/dt}. In this formalism, the state of the system is a described as point in the {q_i}×{q'_i} phase space, and the evolution of the system is certain trajectory in this phase space described by the Euler-Lagrange equations (which is actually a geodesic if length is identified with action, as is natural there). If the Lagrangian is independent of some coordinate q_k, then ∂L/∂q_k = 0, and furthemore by the corresponding Euler-Lagrange equation reduces to [d/dt](∂L/∂q'_k) = 0, so that the generalized conjugate momentum p_k = ∂L/∂q'_k = constant. Thus, symmetry of the Lagrangian with respect to a generalized coordinate leads to conservation of something. Since I do not wish to get into a full treatise on Lagrangian mechanics, I'll simply end with the following summary...

Time-translation symmetry implies energy conservation, spatial-translation symmetry implies linear momentum conservation, rotation symmetry implies angular momentum conservation, and gauge symmetry implies charge conservation. Conservation of mass has historically been an empirical law rather than a theoretical one, at least until relativity. In GTR with the formalism of differential geometry, instead of symmetries of the Lagrangian, one deals with Killing vector [fields], which are generators of isometries of the manifold.

Try using & part without the space
∂

Forgive me for my descent but vorpal's description really seems to me way more mathematical than it needs to be to give a good answer to the question. To me the equivalency of symmetries and conservation laws seems a straightforward consequence of what each of them are. A symmetry is something that doesn't change under a certain type of transformation and a conservation law is nothing more than the statement that something doesn't change. So a conservation law is a symmetry itself in the sense that it expresses the idea that something doesn't change with respect to anything else be it time or what have you. There is actually a theorem (Noether's theorem) that states the one to one correspondence of conservation laws and symmetries but I am not familiar with the math behind it so I just wanted to mention its existence. Emmy Noether though is an interesting historical figure and if only I could find it I know she proved an excellent result involving energy conservation in gtr. In any case though noether's theorem operates only on continuous symmetries. That is to say that because you can move continuously from one frame of reference to annother via rotation or translation then conservation of linear momentum and angular momentum follow because if the laws of physics do not change with rotations or translations then even though the specific momentums as measured from different frames of reference will differ their sum will not change while inside that frame of reference.

yes I am.