How Lagrangian Drive Modern Theory
How Lagrangians drive modern theory
Lagrangian theory (as well as Hamiltonian theory) has a highly influential role in modern physics, there being many remarkable uses to which ti can be put. For example, there is an important theorem, known as Nther’s theorem, which tells us that, if an ordinary Lagrangian possesses some continuous (smooth) symmetry, then there will be a conservation law associated with that symmetry. In particular, if there is invariance of the Lagrangian under time translation (i.e. independent of time), then there is a conserved energy; if it is invariant under some spatial translation, then a momentum is conserved. Furthermore, if there is invariance under angular rotation about some axis, then there is a conserved angular momentum about same axis. For an isolated system in flat space-time, these symmetries are to be expected.
It is probable that most physicists would regard the changes in our picture of the world that quantum mechanics has wrought as being fare more revolutionary even than the extraordinary curved space time of Einstein’s general relativity. In fact, what quantum theory actually tells us to believe about reality at the submicroscopic levels of atoms or of fundamental particles is.
Yet, despite all this, it is very remarkable how much of the Lagrangian|Hamiltonian of procedures that comprehensive but entirely classical scheme that grew out of 17th century Newtonian mechanics provides the essential background of quantum-mechanical theory. Of course, there had to be changes in the mathematical formalism.
Otherwise the new theory would be just a copy of the old. But it is as waiting for quantum mechanics to come, with parts of its machinery of just the right shape and size, so that the new quantum ingredients could simply be inserted in their place.