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In physics, a symmetry of a physical system is a physical or mathematical feature of the system
(observed or intrinsic) that is preserved or remains fixed under some transformation.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete
(e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and
discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be explained
using Lie groups while discrete symmetries are described by finite groups (see Symmetry group).
These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.
Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is known in mathematical terms as the Poincaré group, the symmetry group of special relativity. Another crucial example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important concept in general relativity.
Symmetries may be broadly classified as global or local. A global symmetry is one that holds at all points of spacetime, whereas a local symmetry is one that has a different symmetry transformation at different points of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates. Local symmetries play an important role in physics as they form the basis for gauge theories.
Certain type of transformation when applied provides invariance for the lagrangian in field theory. Gauge transformations
are transformations in which the Lagrangian is invariant under a continuous group of local transformations. An invariant is
a model that holds true no matter the
mathematical procedure applied to it. This continuous group shows some symmertry. This kind of symmetry is
called gauge symmetry :
The symmetries are in the transfromations of the generalized coordinates q(k) and in turns , in Lagrangian L. The transformation can be a time reversal or other change , which will leave the lagrangian invariant. We are varying the langrangian and generalized coordinates but the quantity in the parenthesis does not vary over time. This is deduced from the principles of the extrema values of functions. The quantity in the parenthesis may be called "constant of motion". The constant of motion, in general" is the quantity which is conserved. Angular momentum is a constant of motion.