theorems:noethers_theorems
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theorems:noethers_theorems [2018/06/08 15:19] jakobadmin [Why is it interesting?] |
theorems:noethers_theorems [2021/03/31 18:43] (current) edi [Concrete] |
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| When an object is at rest it has no momentum. When it has a lot of momentum it changes its location quickly. Hence there is a connection between momentum and the change of location. Similarly, if an object doesn't rotate it has no angular momentum. Hence there is a connection between rotations and angular momentum. In a similar spirit we can say that an object that does not change at all over time has no energy. | When an object is at rest it has no momentum. When it has a lot of momentum it changes its location quickly. Hence there is a connection between momentum and the change of location. Similarly, if an object doesn't rotate it has no angular momentum. Hence there is a connection between rotations and angular momentum. In a similar spirit we can say that an object that does not change at all over time has no energy. | ||
| - | We say momentum generates translations (= changes of the position), angular momentum generates rotations and energy generates temporal translations (= movements forward in time). Momentum is responsible that an object changes its location, angular momentum that it rotates and energy that it changes as time passes on. | + | In a somewhat more formal way we say that momentum generates translations (= changes of the position), angular momentum generates rotations and energy generates temporal translations (= movements forward in time). Momentum is responsible that an object changes its location, angular momentum that it rotates and energy that it changes as time passes on. |
| Noether's theorem tells us that there is also a connection the other way round. Namely starting from the transformations: change of location, rotations and movements forward in time; we can //derive// the quantities; momentum, angular momentum and energy. | Noether's theorem tells us that there is also a connection the other way round. Namely starting from the transformations: change of location, rotations and movements forward in time; we can //derive// the quantities; momentum, angular momentum and energy. | ||
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| Therefore, if $G$ is a symmetry then $G$ is conserved. Equally, if $G$ is conserved we automatically know that it generates a canonical transformation. | Therefore, if $G$ is a symmetry then $G$ is conserved. Equally, if $G$ is conserved we automatically know that it generates a canonical transformation. | ||
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| + | ---- | ||
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| + | **Graphical Summary** | ||
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| + | The diagram below shows the relationship between symmetry and conservation in the Lagrangian formalism for the case of a linear point transformation, such as a rotation. For a more detailed explanation see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||
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| + | {{:theorems:sym_and_cons_lagrange.jpg?nolink}} | ||
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| + | The diagram below shows the relationship between symmetry and conservation in the Hamiltonian formalism. For a more detailed explanation see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||
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| + | {{:theorems:sym_and_cons_hamilton.jpg?nolink}} | ||
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| - | See Chapter 1 in The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century | + | * [[https://arxiv.org/abs/1902.01989|Colloquium: A Century of Noether's Theorem]] by Chris Quigg |
| + | * See also Chapter 1 in The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century | ||
| by Yvette Kosmann-Schwarzbach | by Yvette Kosmann-Schwarzbach | ||
| </tabbox> | </tabbox> | ||
theorems/noethers_theorems.1528463945.txt.gz · Last modified: 2018/06/08 13:19 (external edit)
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