Chapter 1
Symmetries of a Classical System
The realization of symmetries in physical systems has proven to be of help in the
description of physical phenomena: it makes it possible to relate the behaviour of
similar systems and therefore it leads to a great simplification of the mathematical
description of Nature.
The simplest concept of symmetry occurs at the geometrical or kinematical level
when the shape of an object or the configuration of a physical system is invariant
or symmetric under geometric transformations like rotations, reflections, etc. At the
dynamical level, a system is symmetric under a transformation of the coordinates or
of the parameters which identify its configurations, if correspondingly its dynamical
behaviour is symmetric in the sense that the action of the symmetry transformation
and of time evolution commute.
To formalize the concept of dynamical symmetry, we first recall that the description of a classical physical system consists in
i) the identification of all its possible configurations {S γ }, with γ running over an
index set of coordinates or parameters which identify the configuration S γ ;
ii) the determination of their time evolution
α
t
: S γ → α
t S γ ≡ S γ(t) .
(1.1)
A symmetry g of a physical system is a transformation of the coordinates (or of
the parameters) γ, g: γ → gγ, which
1) induces an invertible mapping of configurations
g : S γ → gS γ ≡ S gγ
(1.2)
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_1
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