Operator (physics) - Operators in Classical Mechanics

Operators in Classical Mechanics

In classical mechanics, the dynamics of a particle (or system of particles) are completely determined by the Lagrangian L(q, q̇, t) or equivalently the Hamiltonian H(q, p, t), a function of the generalized coordinates q, generalized velocities q̇ = dq/dt and its conjugate momenta:

If either L or H are independent of a generalized coordinate q, meaning the L and H so not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry). Operators in classical mechanics are related to these symmetries.

More technically, when H is invariant under the action of a certain group of transformations G:

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the elements of G are physical operators, which map physical states among themselves.