In QM only one quantum state holds info for position, momentum and Energy and Newtonic also should have such an allencompassing, which is Lagrangian:

We say the integral of L is the S[L) stationary point of the integral:

This principle of least action is not proper, should be the principle of stationary action. By all sorts of math manipulation, we get below significant equations:

for example, looking at the first equation, we can say to change the Lagragian in terms of position, we look at the change of momentum in time, or, put in other word, momentum is the generator of spatial change. Position is the generator of momentum change. by the same reasoning, energy is the generator of time change.

A natural question is that if this generator is identical as in the Lie Algebra:

Generators in Quantum Mechanics and Lie Algebra:

• In quantum mechanics, the commutator structure replaces the Poisson bracket structure, and the idea of a “generator” becomes even more pronounced. For instance, the momentum operator �^p^​ generates spatial translations in the sense that:����^/ℏ�^�−���^/ℏ=�^+�eiap^​/ℏx^e−iap^​/ℏ=x^+aThis is a manifestation of the Stone-von Neumann theorem.
• The above expression is a specific example of a more general principle: in quantum mechanics, the exponential of an operator generates a transformation. This concept is deeply connected to Lie groups and Lie algebras. The operators themselves (like momentum, angular momentum, etc.) are the generators of the associated Lie algebra.

is generator in QM or Lagranian eqution of motion same concept as the generator in lie algebra? to chatGPT

Yes, there is a deep connection between the concept of a generator in quantum mechanics (QM), the Lagrangian formalism, and the generators of Lie algebras. Let’s unpack this relationship:

Generators in Lagrangian Mechanics:

• In the Lagrangian formalism, symmetries of the action lead to conserved quantities via Noether’s theorem. The infinitesimal transformations associated with these symmetries can be thought of as being generated by certain functions on the phase space.
• For example, an infinitesimal time translation is generated by the Hamiltonian, and spatial translations are generated by the momentum.

Generators in Lie Algebras:

• Lie algebras arise naturally when considering continuous symmetries and their associated conserved quantities. The elements (or basis elements) of a Lie algebra are called generators.
• The commutation relations between these generators define the structure of the Lie algebra and the associated Lie group.
• In quantum mechanics, the commutation relations between operators (like position and momentum) are reminiscent of the relations in a Lie algebra.

an example of generator in lie algebra:

In all, one can see the Lagragian concept is an elevated/high level same concept as quantum state Sai but in Newtonic world.