Geometrical framework Lie point symmetry

1 geometrical framework

1.1 infinitesimal approach
1.2 lie groups , lie algebras of infinitesimal generators
1.3 continuous dynamical systems
1.4 invariants

geometrical framework
infinitesimal approach

lie s fundamental theorems underline lie groups can characterized infinitesimal generators. these mathematical objects form lie algebra of infinitesimal generators. deduced infinitesimal symmetry conditions (defining equations of symmetry group) can explicitly solved in order find closed form of symmetry groups, , associated infinitesimal generators.

let



z
=
(

z

1


,
…
,

z

n


)


{\displaystyle z=(z_{1},\dots ,z_{n})}

set of coordinates on system defined



n


{\displaystyle n}

cardinal of



z


{\displaystyle z}

. infinitesimal generator



δ


{\displaystyle \delta }

in field




r

(
z
)


{\displaystyle \mathbb {r} (z)}

linear operator



δ
:

r

(
z
)
→

r

(
z
)


{\displaystyle \delta :\mathbb {r} (z)\rightarrow \mathbb {r} (z)}

has




r



{\displaystyle \mathbb {r} }

in kernel , satisfies leibniz rule:

∀
(

f

1


,

f

2


)
∈

r

(
z

)

2


,
δ

f

1



f

2


=

f

1


δ

f

2


+

f

2


δ

f

1




{\displaystyle \forall (f_{1},f_{2})\in \mathbb {r} (z)^{2},\delta f_{1}f_{2}=f_{1}\delta f_{2}+f_{2}\delta f_{1}}

.

in canonical basis of elementary derivations




{


∂

∂

z

1





,
…
,


∂

∂

z

n





}



{\displaystyle \left\{{\frac {\partial }{\partial z_{1}}},\dots ,{\frac {\partial }{\partial z_{n}}}\right\}}

, written as:

δ
=

∑

i
=
1


n



ξ


z

i




(
z
)


∂

∂

z

i







{\displaystyle \delta =\sum _{i=1}^{n}\xi _{z_{i}}(z){\frac {\partial }{\partial z_{i}}}}

where




ξ


z

i






{\displaystyle \xi _{z_{i}}}

in




r

(
z
)


{\displaystyle \mathbb {r} (z)}





i


{\displaystyle i}

in




{
1
,
…
,
n
}



{\displaystyle \left\{1,\dots ,n\right\}}

.

lie groups , lie algebras of infinitesimal generators

lie algebras can generated generating set of infinitesimal generators. every lie group, 1 can associate lie algebra. roughly, lie algebra





g




{\displaystyle {\mathfrak {g}}}

algebra constituted vector space equipped lie bracket additional operation. base field of lie algebra depends on concept of invariant. here finite-dimensional lie algebras considered.

continuous dynamical systems

a dynamical system (or flow) one-parameter group action. let denote





d




{\displaystyle {\mathcal {d}}}

such dynamical system, more precisely, (left-)action of group



(
g
,
+
)


{\displaystyle (g,+)}

on manifold



m


{\displaystyle m}

:

d


:


g
×
m


→


m





ν
×
z


→




d


(
ν
,
z
)






{\displaystyle {\begin{array}{rccc}{\mathcal {d}}:&g\times m&\rightarrow &m\\&\nu \times z&\rightarrow &{\mathcal {d}}(\nu ,z)\end{array}}}

such point



z


{\displaystyle z}

in



m


{\displaystyle m}

:

d


(
e
,
z
)
=
z


{\displaystyle {\mathcal {d}}(e,z)=z}





e


{\displaystyle e}

neutral element of



g


{\displaystyle g}

;
for



(
ν
,



ν
^



)


{\displaystyle (\nu ,{\hat {\nu }})}

in




g

2




{\displaystyle g^{2}}

,





d


(
ν
,


d


(



ν
^



,
z
)
)
=


d


(
ν
+



ν
^



,
z
)


{\displaystyle {\mathcal {d}}(\nu ,{\mathcal {d}}({\hat {\nu }},z))={\mathcal {d}}(\nu +{\hat {\nu }},z)}

.

a continuous dynamical system defined on group



g


{\displaystyle g}

can identified




r



{\displaystyle \mathbb {r} }

i.e. group elements continuous.

invariants

an invariant, speaking, element not change under transformation.