Continuous symmetry

<h2 id="formalization">Formalization</h2>
<p>The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of a <a href="/facts/Topological_group/cAgNMMRU">topological group</a>, <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> and <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a>. For most practical purposes, continuous symmetry is modelled by a <i>group action</i> of a topological group that preserves some structure. Particularly, let 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 be a function, and 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is a group that acts on 
  
    
      
        X
      
    
    {\displaystyle X}
  
; then a subgroup 
  
    
      
        H
        ⊆
        G
      
    
    {\displaystyle H\subseteq G}
  
 is a symmetry of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 if 
  
    
      
        f
        (
        h
        ⋅
        x
        )
        =
        f
        (
        x
        )
      
    
    {\displaystyle f(h\cdot x)=f(x)}
  
 for all 
  
    
      
        h
        ∈
        H
      
    
    {\displaystyle h\in H}
  
.
</p>
<h3>One-parameter subgroups</h3>
<p>The simplest motions follow a <a href="/facts/One-parameter_subgroup/Ep7zASA0">one-parameter subgroup</a> of a Lie group, such as the <a href="/facts/Euclidean_group/VdTSdwzl">Euclidean group</a> of <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a>. For example <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a> parallel to the <i>x</i>-axis by <i>u</i> units, as <i>u</i> varies, is a one-parameter group of motions. Rotation around the <i>z</i>-axis is also a one-parameter group.
</p>
<h2 id="noethers-theorem">Noether's theorem</h2>
<p>Continuous symmetry has a basic role in <a href="/facts/Noether%2527s_theorem/VFjqrEvz">Noether's theorem</a> in <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, in the derivation of <a href="/facts/Conservation_law_(physics)/6g1Ay2FR">conservation laws</a> from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Goldstone%2527s_theorem/zGBpWRYU">Goldstone's theorem</a></li>
<li><a href="/facts/Infinitesimal_transformation/XCN1WpGS">Infinitesimal transformation</a></li>
<li><a href="/facts/Noether%2527s_theorem/VFjqrEvz">Noether's theorem</a></li>
<li><a href="/facts/Sophus_Lie/CplQVa8S">Sophus Lie</a></li>
<li><a href="/facts/Motion_(geometry)/x5zUzg7l">Motion (geometry)</a></li>
<li><a href="/facts/Circular_symmetry/gEgSEFxy">Circular symmetry</a></li></ul>

<ul><li>Barker, William H.; Howe, Roger (2007). <a href="https://books.google.com/books?id=88EjDwAAQBAJ&pg=PT7"><i>Continuous Symmetry: from Euclid to Klein</i></a>. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3900-3.</li></ul>

Continuous symmetry open-in-new

Continuous symmetry