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Noether's Theorem

For every continuous symmetry in a physical system, there exists a corresponding conserved quantity. This is fundamental to Hamiltonian mechanics.
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The statement of the theorem

Let L(q,dqdt,t)L(q, \frac{\text{d}q}{\text{d}t}, t) be the Lagrangian of a system, and let q∈Rn\mathbf{q} \in \mathbb{R}^n be the generalized coordinates. If the action S=∫LdtS = \int L \text{d}t is invariant under a continuous transformation parameterized by ϵ\epsilon, such that δL=ddt(ϵ⋅dF/dt)\delta L = \frac{\text{d}}{\text{d}t} (\epsilon \cdot \text{d}F/\text{d}t), then there exists a conserved quantity GG (the Noether charge) defined by the generalized momentum associated with the symmetry, such that dGdt=0\frac{\text{d}G}{\text{d}t} = 0.
Source: Wikipedia