Λ-symmetries for first-order ODE systems

See the works of Cicogna: @cicognareduction.

Overview

A $\mathrm{\Lambda }$-symmetry is a generalized notion of symmetry for an autonomous system (system of first order ODEs)
${\dot{u}}^a=f^a(u),A=f^a(u){\partial }_{u^a},$
generated by a vector field

1. Definition (bracket form)

Compute the Lie bracket
$[X,A{]}^a={\phi }^b{\partial }_b{f}^a-f^b{\partial }_b{\phi }^a.$
We say that $X$ is a Λ-symmetry of the system if there exists a matrix field ${\mathrm{\Lambda }}^a{}_b(u)$ such that
$[X,A{]}^a=(\mathrm{\Lambda }\phi {)}^a={\mathrm{\Lambda }}^a{}_b{\phi }^b.$
Equivalently, in vector notation,
$[X,A]=\mathrm{\Lambda }X.$

Remarks:

• This equation always defines a matrix $\mathrm{\Lambda }$ pointwise (by solving linear equations) whenever the components ${\phi }^b$ are known. The nontrivial content is that $\mathrm{\Lambda }(u)$ is a smooth matrix field of a form that is useful for reduction.
• Special cases:
  • $\mathrm{\Lambda }\equiv 0$ ⇢ Lie (exact) symmetry: $[X,A]=0$.
  • $\mathrm{\Lambda }=\lambda (u),I$ ⇢ scalar lambda-symmetry: $[X,A]=\lambda X$.

2. Geometric meaning: why reduction may fail or be weaker

Classical symmetry reduction uses Frobenius: if $[X,A]$ lies in the distribution generated by $X$ and $A$ (in particular if $[X,A]$ is a linear combination of $X$ and $A$) then the distribution is involutive and projects to a well-defined reduced vector field on the quotient by the flow of $X$.

For a general $\mathrm{\Lambda }$-symmetry, $[X,A]=\mathrm{\Lambda }X$ is not necessarily a linear combination of $X$ and $A$. Consequently:

• The distribution $\mathrm{span}\{X,A\}$ need not be involutive.
• Collapsing (quotienting) along $X$ is not a geometric quotient that produces a closed reduced vector field automatically.

Nonetheless, $\mathrm{\Lambda }$ controls precisely how the invariants of $X$ evolve under $A$, so an algebraic reduction is possible: write the system in coordinates adapted to $X$ (invariants + orbit coordinate) and use the information encoded in $\mathrm{\Lambda }$ to understand the dependence on the orbit coordinate.

3. Adapted coordinates and the reduction recipe (practical)

Step 0 — choose adapted coordinates. Locally choose coordinates $(w^i,z)$ such that
$X(w^i)=0(i=1,\dots ,n-1),X(z)=1.$
Thus the $w^i$ are invariants of $X$ and $z$ parametrizes the orbits.

Write the dynamical vector field in these coords:
$A=F^i(w,z){\partial }_{w^i}+G(w,z){\partial }_z.$

Step 1 — bracket in adapted coordinates. Because $X={\partial }_z$, the bracket is
$[X,A]=[{\partial }_z,F^i{\partial }_{w^i}+G{\partial }_z]=({\partial }_z{F}^i){\partial }_{w^i}+({\partial }_{z}G){\partial }_z.$

Step 2 — impose the Λ-condition. The Λ-equation becomes
${\partial }_z{F}^i=(\mathrm{\Lambda }\phi {)}^{w^i},{\partial }_{z}G=(\mathrm{\Lambda }\phi {)}^z.$
But in the adapted frame $\phi =(0,\dots ,0,1{)}^T$ (last component =1), so the right-hand sides are simply the last column of $\mathrm{\Lambda }$ in these coordinates:
${\partial }_z{F}^i={\mathrm{\Lambda }}^{w^i}{}_z,{\partial }_{z}G={\mathrm{\Lambda }}^z{}_z.$

Interpretation / reduction criterion.

• If the last column entries ${\mathrm{\Lambda }}^{w^i}{}_z$ vanish identically, then ${\partial }_z{F}^i=0$ and the equations for $w^i$ do not depend on $z$: the $w$-subsystem is closed and we have a true reduction (as in Lie/λ cases).
• If the last column is simple (e.g. functions of $w$ only, or constants), the $z$-dependence is controlled and may be integrable.
• In general the last column measures the mixing of orbit-direction and invariants; the more structured/sparse it is, the more effective the reduction.

Practical recipe to attempt reduction:

1. Pick a candidate generator $X$.
2. Compute adapted coordinates $(w,z)$ (solve $X(w)=0$, $X(z)=1$ locally).
3. Compute $F(w,z),G(w,z)$ from $f$.
4. Compute ${\partial }_{z}F$ and read off the last column of $\mathrm{\Lambda }$. If these vanish (or are simple), the $w$-subsystem reduces.

4. Formula for Λ in general coordinates

Given $X={\phi }^a{\partial }_{u^a}$ and $A=f^a{\partial }_{u^a}$, compute $B^a:=[X,A{]}^a$. Then the defining linear system for ${\mathrm{\Lambda }}^a{}_b$ is
$B^a={\mathrm{\Lambda }}^a{}_b{\phi }^b.$
This is a system of $n$ linear equations (index $a$) for the $n^2$ unknown entries of $\mathrm{\Lambda }$. It is underdetermined in general.

Two remarks:

• If $\phi$ has no zero components and one wants a scalar form $\mathrm{\Lambda }=\lambda I$ one must check that $B^a$ is proportional to ${\phi }^a$ (this yields $\lambda$).
• In adapted coordinates the system simplifies dramatically (last column equals $B^a$ because $\phi =(0,\dots ,1)$).

5. Small 2D example (illustrative)

System: harmonic oscillator
$\dot{x}=y,\dot{y}=-x.$
Take $X=x{\partial }_x+y{\partial }_y$ (radial scaling). Then
$[X,A]=-2(x{\partial }_y+y{\partial }_x).$
Expressed as $[X,A]=\mathrm{\Lambda }X$ with $\phi =(x,y{)}^T$, one choice is
$\mathrm{\Lambda }=\left(\begin{array}{cc}0& -2\\ -2& 0\end{array}\right).$

Adapted invariants: $w=y/x$. Compute
$\dot{w}=-(1+w^2),$
so $w$ evolves autonomously in this example. That autonomy is a special feature of the example (the last column of $\mathrm{\Lambda }$ produces a closed $w$-equation).

6. When is a Λ-symmetry useful?

• If the relevant entries of $\mathrm{\Lambda }$ (the last column in adapted coords) vanish or depend only on the invariants $w$, the invariant subsystem closes or becomes simple.
• If $\mathrm{\Lambda }$ is sparse, constant, triangular, or depends on fewer variables than the full phase space, reduction will often be effective.
• If $\mathrm{\Lambda }$ is fully general and complicated, the $\mathrm{\Lambda }$-symmetry statement is true but not practically helpful.

7. Short summary

• A $\mathrm{\Lambda }$-symmetry is the condition $[X,A]=\mathrm{\Lambda }X$. A matrix $\mathrm{\Lambda }$ always exists formally, but usefulness depends on its structure.
• In adapted coordinates the last column of $\mathrm{\Lambda }$ measures exactly how invariants $w$ depend on the orbit coordinate $z$.
• Reduction is weaker than for Lie symmetries, but still very effective whenever the last column is simple/structured.