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My understanding is that there are 5 implicit assumptions when introducing the Lagrangian formulation.

Assumptions:

  1. We appear to know that the principle of stationary action is true for the universe
  2. We define Kinetic energy of the system to be to be $T = \sum f(\mathcal{P}_n(\dot{q}))$ where $\mathcal{P}_n$ is a polynomial of some degree
  3. We define $V(q)$ to be the potential energy of the system
  4. We assume the system is time-translation invariant
  5. We define Lagrangian to be $L(\dot{q}, q) = T(\dot{q})-V(q)$

Question:

If our system is the universe, how do we know the universe is time-translation invariant? I am not fully convinced with the "Hamiltonian is a generator of time transformation", because it reasoning appears a little circular to me, given that's how the Hamiltonian is defined in the first place. Or maybe I have a fundamental mis-understanding of the Hamiltonian?

Also, how do we know that the duration of 1 second, say 50 years ago, is the same as the duration of 1 second today? Can we guarantee that it'll remain the same 50 years in the future from now?

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Partially answering your question, but the Universe is not time-translation invariant.

From Noether's theorem, we know that we can connect symmetries with conserved quantities, and as you said energy conservation is connected to time-translation invariance. However, universal expansion breaks time-translation invariance, and hence energy is not conserved. (At least in the standard cosmological models).

For the second part, I think you have to be more specific when you think about time measurement and specify who observes and what exactly. For example, a comoving observer in his clock would measure a constant duration of 1 sec always.

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The lagrangian formalism doesn't require that the state of the system be time-translation invariant. It only requires that the differential equations governing the system be so.

The universe is not time-translation invariant, and nobody has ever imagined that it was.

Also, how do we know that the duration of 1 second, say 50 years ago, is the same as the duration of 1 second today? Can we guarantee that it'll remain the same 50 years in the future from now?

Relativity says that time is what a clock measures. Therefore this question is meaningless. Relativity expresses gravity as curvature of spacetime, not as a time-dependent dilation of time.

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  • $\begingroup$ Thank you for the answer. I accepted @MariosN3's answer because his answer was first, even though it's very similar to yours. $\endgroup$ – skittish Apr 30 at 12:52

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