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by Aardwolf
1325 days ago
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I would love to get intuition about this, but everytime I try to read about it I get lost in the math behind Lagrangians etc... and never get the intuition. Specifically, I'd love the intuition behind why it must be so that if the laws of physics are time/position invariant, it must be impossible to create or destroy energy/momentum. This because I can perfectly imagine a universe where you can create/destroy energy or momentum at will, yet have this work with the same physics at any time or position (e.g. a battlemage in some RPG able to cast fireball spells with heat and momentum out of nothing, where it works the same no matter at what position or point in time the battlemage is). So what about Lagrangians is preventing this, in an intuitively imaginable way? |
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The geometrical intuition is that the momentum operators are the generator of spatial translations (i.e. acting with the momentum operator P_x on a system is the same as applying an infinitesimal displacement in the direction x). https://en.wikipedia.org/wiki/Translation_operator_(quantum_...
And the Hamiltonian is the generator of time evolution (i.e. acting with the Hamiltonian on a system shifts it in time an infinitesimal amount). This is quite literally what the Schrödinger equation says, btw. H |Psi> ~ d/dt |Psi>
If the physics of a system (which are given by its Hamiltonian) are invariant under translations then it must be the case that a shift in time (Hamiltonian generates shifts in time) of the momentum (generates shifts in space) of a system is 0.
As the Hamiltonian gives us the energy of the system, if the system is invariant under time translations then its energy is conserved. Using the previous argument.
Rinse and repeat for any symmetry of the system. For instance, angular momentum operators are the generators of rotations. If your system is invariant under rotations, then it conserves angular momentum. Invariance under relative movement (relativistic invariance) gives conservation of center of mass. Etc... https://en.wikipedia.org/wiki/Symmetry_(physics)#Conservatio...