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--- advanced_tools:group_theory [2018/04/08 17:15]
georgefarr ↷ Links adapted because of a move operation
+++ advanced_tools:group_theory [2020/04/02 15:01]
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 A square is defined mathematically as a set of points. A symmetry of the square is a transformation that maps this set of points into itself. This means concretely that by the transformation, no point is mapped to a point outside of the set that defines the square.
-Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$, $270^\circ$, and of course $0^\circ$.
+Obvious examples of such transformations are rotations, by $90^{\circ}$, $180^{\circ}$, $270^{\circ}$, and of course $0^{\circ}$.
 {{ :advanced_tools:einheitsquadrat-gedreht22-150x150.png?nolink|}}
-A counter-example is a rotation by, say $5^\circ$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^\circ$ is no symmetry of the square.
+A counter-example is a rotation by, say $5^{\circ}$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^{\circ}$ is no symmetry of the square.
-A characteristic property of the symmetries of the square is that the combination of two transformations that leave the square invariant is again a symmetry. For example, combining a rotation by $90^\circ$ and $180^\circ$ is equivalent to a rotation of $270^\circ$, which is again a symmetry of the square. We will elaborate on this in the next post. In fact, the basic axioms of group theory can be derived from such an easy example.
+A characteristic property of the symmetries of the square is that the combination of two transformations that leave the square invariant is again a symmetry. For example, combining a rotation by $90^{\circ}$ and $180^{\circ}$ is equivalent to a rotation of $270^{\circ}$, which is again a symmetry of the square. We will elaborate on this in the next post. In fact, the basic axioms of group theory can be derived from such an easy example.
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 <blockquote>A man who is tired of group theory is a man who is tired of life. <cite> Sidney Coleman</cite></blockquote>
+<blockquote>Group theory is, in short, the mathematics of symmetries. You already know that
+symmetries can be very important in understanding or simplifying physics problems.
+When you study classical mechanics, you learn that symmetries of a system
+are intimately related to the existence of conserved charges. Their existence often
+makes solving for the dynamics a lot simpler. Even if a symmetry is not present
+exactly (for instance, when a system is almost-but-not-quite spherically symmetric),
+we can often describe the system as a small perturbation of a system that does
+exhibit symmetry. A surprisingly large number of physics problems is built around
+that idea; in fact, practically all systems for which we can solve the dynamics exactly
+exhibit some sort of symmetry that allow us to reduce the often horrible secondorder
+equations of motion to much simpler first-order conservation equations.<cite>http://maths.dur.ac.uk/users/kasper.peeters/pdf/groups.pdf</cite></blockquote>
 <tabbox Overview>

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