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peterliepa's profile
Peter Liepa
Peter Liepa
Peter Liepa
@peterliepa

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Peter Liepa

@peterliepa

Visual math, mainly geometry -- euclidean, hyperbolic, projective, conformal, computational.

Toronto
Joined May 2009

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    1. Bob (probably)‏ @knighton_bob Feb 12
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      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

      5 replies 99 retweets 474 likes
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    2. Bob (probably)‏ @knighton_bob Feb 12
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      Given a path p taken by a particle, we assign to it an *action* S[p]. Nature chooses paths which minimize (or maximize) this action. I can translate the path p to get a new path p', which has an action S[p']. If translation is a symmetry, then S[p]=S[p']. 2/npic.twitter.com/zMeIE1A871

      1 reply 0 retweets 49 likes
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    3. Bob (probably)‏ @knighton_bob Feb 12
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      If p happens to be the path that minimizes S[p], then "nearby" paths S[p+δp] have action very close to S[p] (that is, S[p+δp]=S[p] for δp very small). In this case, the action S[p] and the action along the red path should be very close (and will be equal as Δx gets smaller) 3/npic.twitter.com/EMYPJirxYu

      2 replies 0 retweets 39 likes
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    4. Bob (probably)‏ @knighton_bob Feb 12
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      Let us call the action of the blue path at the bottom P₁Δx and the action of the blue path at the top P₂Δx. Then the fact that S[p]=S[p'] and S[p+δp]=S[p], we immediately obtain P₁=P₂. That is, we have a conserved quantity, namely the momentum of the particle! 4/npic.twitter.com/h3oGBBG4iL

      1 reply 1 retweet 55 likes
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    5. Bob (probably)‏ @knighton_bob Feb 12
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      This is the essence of the proof of Noether's theorem (if one is a bit more careful). The evaluation of the action along the direction of symmetry turns out to be conserved. This even holds when our theory is more complicated than a particle moving on a trajectory! 5/5

      4 replies 0 retweets 46 likes
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      Peter Liepa‏ @peterliepa Feb 12
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      Replying to @knighton_bob

      I'm missing something. How does momentum, much less conservation of momentum, pop out of particles, paths and actions? I don't see anything that ties them together.

      12:18 PM - 12 Feb 2020
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      2 replies 0 retweets 1 like
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        2. Caleb Moses‏ @Caleb_Speak Feb 12
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          Replying to @peterliepa @knighton_bob

          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.

          1 reply 0 retweets 0 likes
        3. Caleb Moses‏ @Caleb_Speak Feb 12
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          Replying to @Caleb_Speak @peterliepa @knighton_bob

          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.

          1 reply 0 retweets 0 likes
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        1. Bob (probably)‏ @knighton_bob Feb 12
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          Replying to @peterliepa

          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.

          0 replies 0 retweets 1 like
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