Noether's theorem visualized -- a thread: If a physical system is symmetric under some continuous action (moving horizontally, time translation, rotation), then there is a corresponding "conserved charge" (momentum, energy, angular momentum). This is Noether's Theorem. 1/n
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Momentum is a consequence of the translation invariance of the laws of physics. Lagrangian mechanics is cool because it takes a global view, thinking in terms of trajectories instead of individual particles and the forces on them at small time scales.
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It’s easy to see that in a situation where the forces on a particle depend on location (like under gravity), momentum would *not* be conserved. It turns out that momentum is conserved if and only if the forces on a particle are invariant under translations.
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The standard action formula for a single particle (from which Newton’s laws are derived) evaluated along a path which moves in space but not in time evaluates to exactly P=mv.
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