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:
- .
the elements of G are physical operators, which map physical states among themselves.
Read more about this topic: Operator (physics)
Famous quotes containing the words classical and/or mechanics:
“Culture is a sham if it is only a sort of Gothic front put on an iron building—like Tower Bridge—or a classical front put on a steel frame—like the Daily Telegraph building in Fleet Street. Culture, if it is to be a real thing and a holy thing, must be the product of what we actually do for a living—not something added, like sugar on a pill.”
—Eric Gill (1882–1940)
“the moderate Aristotelian city
Of darning and the Eight-Fifteen, where Euclid’s geometry
And Newton’s mechanics would account for our experience,
And the kitchen table exists because I scrub it.”
—W.H. (Wystan Hugh)