05.05.07 · symplectic / lagrangian

Group-invariant solutions and symmetry reduction

shipped3 tiersLean: none

Anchor (Master): Olver §3.1-§3.4; Ovsiannikov *Group Analysis of Differential Equations* (1982); Bluman-Cole *Similarity Methods for Differential Equations* (1974); Barenblatt *Similarity, Self-Similarity, and Intermediate Asymptotics* (1979)

Intuition Beginner

A symmetry of an equation is a motion that turns solutions into solutions. Most solutions get pushed to a different solution by the motion. But a few special solutions sit exactly on the motion, so that pushing them leaves them where they were. These are the symmetry-respecting solutions, and they are the easiest ones to find. They are pinned in place by the very symmetry that scrambles everything else.

Why does this help? A solution that respects a symmetry repeats itself in a predictable way, so it carries less independent information. If a solution looks the same after you scale space and time together, then it really only depends on one combined quantity rather than on space and time separately. That combined quantity is the new variable. Trading two variables for one turns a hard equation in two variables into a simpler equation in one.

This is the engine of similarity methods. You spot a symmetry, you find the combination of variables it leaves alone, and you write your unknown as a function of that combination only. The original equation then collapses into a smaller one. For an ordinary differential equation the same trick lowers the order by one each time you use a symmetry, because a symmetry lets you measure progress along its own flow and forget one layer of the problem.

Visual Beginner

Picture the plane of space and time, with the lines that a scaling symmetry moves along drawn as curves spreading out from the origin like spokes. A general solution cuts across these spokes. A symmetry-respecting solution is constant along each spoke: it takes the same value everywhere on one spoke and changes only as you step from spoke to spoke.

Because the solution is constant along a spoke, the only thing that matters is which spoke you stand on. Labelling the spokes by a single number gives the new variable, the similarity variable. The picture shows that two-dimensional data has collapsed to one-dimensional data: the whole solution is recovered from its values across the fan. Reducing the equation is then the act of rewriting it in terms of that one label.

Worked example Beginner

Take the heat equation, which says the rate of change in time of the temperature equals the bending of the temperature in space. Heat spreading from a point looks the same whether you zoom out in space by some factor and in time by the square of that factor. That paired stretch is a scaling symmetry.

So the temperature should depend only on the combination that survives the stretch. If you stretch space by a number and time by that number squared, the ratio of position to the square root of time does not change. Call that ratio the similarity variable: position divided by the square root of time. The claim is that the temperature is a function of this single ratio alone, times a fixed power of time out front to balance units.

Check what this buys you. Instead of a temperature that depends on both position and time, you have a temperature built from one combined quantity. Plugging this form into the heat equation, the two-variable equation becomes a single ordinary differential equation for a function of the ratio. Solving that one-variable equation and undoing the substitution gives the famous bell-shaped spreading profile, written with the error function.

What this tells us: a scaling symmetry let us guess that the answer depends on one combined variable, and that guess turned a partial differential equation in two variables into an ordinary differential equation in one. The reduction came directly from respecting the symmetry.

Check your understanding Beginner

Exercise (easy, multiple choice).

A group-invariant (symmetry-respecting) solution is best described as a solution that:

A. Solves a completely different equation
B. Is left unchanged by a chosen symmetry of the equation
C. Has the largest possible number of independent variables
D. Cannot be written down in closed form

Hint

Think about which solutions stay put when the symmetry motion is applied, rather than being pushed to a new solution.

Answer

B. Is left unchanged by a chosen symmetry of the equation.

Feedback-correct: yes — an invariant solution is fixed by the symmetry, which is exactly why it depends on fewer variables. Feedback-wrong: option A confuses invariance with a change of equation; option C is backwards, since invariance removes variables; option D is unrelated, as many invariant solutions are explicit.

Formal definition Intermediate+

Let $π : E \to X$ be a fibred manifold with coordinates $(x^{i}, u^{α})$ , $1 \leq i \leq p$ , $1 \leq α \leq q$ , let $S = {Δ_{ν} (x, u^{(k)}) = 0} \subset J^{k} (E)$ be a system of PDEs as in 05.05.06, and let $G$ be a (local) symmetry group of $S$ with Lie algebra $g$ of generators $v = ξ^{i} (x, u) \partial_{x^{i}} + ϕ^{α} (x, u) \partial_{u^{α}}$ . The development follows Olver ^{[Olver §3.1]} and Ovsiannikov ^{[Ovsiannikov Ch. 5]}.

Definition (group-invariant solution). Let $H \subseteq G$ be a connected $r$ -dimensional subgroup acting on $E$ with generators spanning a subalgebra $h \subseteq g$ . A solution $u = f (x)$ is $H$ -invariant (a group-invariant or similarity solution) when its graph $Γ_{f} = {(x, f (x))} \subseteq E$ is an $H$ -invariant subset: $h \cdot Γ_{f} = Γ_{f}$ for all $h \in H$ . Equivalently, each generator $v \in h$ is tangent to $Γ_{f}$ , which in coordinates is the invariant-surface condition $$ \xi^i(x, f),\frac{\partial f^\alpha}{\partial x^i} = \phi^\alpha(x, f), \qquad 1 \le \alpha \le q, $$ a first-order quasilinear PDE system that the invariant solution must satisfy in addition to $Δ_{ν} = 0$ .

Definition (invariants and the reduced variables). A function $I : E \to R$ is an invariant of $H$ when $v (I) = 0$ for every $v \in h$ , i.e. $ξ^{i} \partial_{x^{i}} I + ϕ^{α} \partial_{u^{α}} I = 0$ . For a single generator $v$ the invariants are the first integrals of the characteristic system $$ \frac{dx^1}{\xi^1} = \cdots = \frac{dx^p}{\xi^p} = \frac{du^1}{\phi^1} = \cdots = \frac{du^q}{\phi^q}. $$ Where $H$ acts on $E$ with generic orbit dimension $s = dim h_{generic}$ and the projected action on $X$ has orbit dimension $s^{'}$ , there are $p - s^{'}$ functionally independent invariants among the base variables (the similarity variables $y^{1}, \dots, y^{p - s^{'}}$ ) and $q$ further invariants $w^{1}, \dots, w^{q}$ involving the fibre. The invariant-solution ansatz expresses the $w^{β}$ as functions of the $y^{a}$ , which on solving for $u^{α}$ recovers the invariant-surface condition.

Definition (canonical coordinates). For a single nonvanishing generator $v$ on a surface, canonical coordinates are functions $(r, s)$ with $v (r) = 0$ and $v (s) = 1$ , so that $v = \partial_{s}$ . The coordinate $r$ is an invariant; $s$ is a coordinate along the orbit. They exist locally near any point where $v \neq = 0$ by the flow-box theorem of 02.12.02.

A non-example sharpens the meaning. A solution that merely intersects an orbit of $H$ is not $H$ -invariant; invariance demands that the whole graph be a union of orbits, equivalently that every $v \in h$ be tangent to $Γ_{f}$ at every point. A solution can be invariant under a one-parameter $H$ yet fail to be invariant under a larger $G$ ; the reduction is always relative to the chosen subgroup.

Counterexamples to common slips

The invariant-surface condition $ξ^{i} \partial_{i} f^{α} = ϕ^{α}$ is necessary but not sufficient: an invariant solution must satisfy both the invariant-surface condition and the original system $Δ_{ν} = 0$ . Solving only the surface condition produces invariant graphs, not invariant solutions.
The number of similarity variables is $p - s^{'}$ , governed by the orbit dimension $s^{'}$ of the action projected to $X$ , not by $dim h$ . A vertical generator (acting only on $u$ ) reduces no independent variables; it constrains the fibre dependence instead.
Reducing by a subgroup $H$ that does not act transversally to the equation can yield a singular or empty reduction; the rank/transversality condition on the invariant manifold $S_{H} \subseteq S$ must hold for the reduced equation to be a genuine system of the expected order.

Key theorem with proof Intermediate+

Theorem (reduction of order by a one-parameter symmetry). Let the scalar $n$ -th order ODE $$ u^{(n)} = F\bigl(x, u, u', \dots, u^{(n-1)}\bigr) $$ admit the one-parameter symmetry group generated by $v = ξ (x, u) \partial_{x} + ϕ (x, u) \partial_{u}$ , $v \neq = 0$ . Then in canonical coordinates $(r, s)$ with $v = \partial_{s}$ the equation takes a form independent of $s$ , and writing $w = d s / d r$ it becomes an $(n - 1)$ -th order ODE for $w (r)$ . Thus a point symmetry reduces the order of a scalar ODE by one.

The statement follows Olver ^{[Olver §2.5]} and Bluman-Kumei ^{[Bluman-Kumei Ch. 4]}.

Proof. Choose canonical coordinates $(r, s)$ near a point where $v \neq = 0$ , so that $v (r) = 0$ and $v (s) = 1$ ; these exist by the flow-box theorem of 02.12.02, which rectifies a nonvanishing field to $\partial_{s}$ . In these coordinates the generator is $v = \partial_{s}$ , and its prolongation to the jet coordinates is $pr^{(n)} v = \partial_{s}$ as well, since the prolongation coefficients of $\partial_{s}$ all vanish: with $ξ = 0$ , $ϕ = 1$ in the $(r, s)$ frame the characteristic is $Q = 1$ and $ϕ_{J}^{α} = D_{J} Q = 0$ for $∣ J ∣ \geq 1$ by the prolongation formula of 05.05.06.

Write the equation in the $(r, s)$ frame, taking $r$ as the independent variable and $s = s (r)$ as the dependent variable. It becomes $s^{(n)} = G (r, s, s^{'}, \dots, s^{(n - 1)})$ for some function $G$ . The infinitesimal symmetry criterion of 05.05.06 requires $pr^{(n)} v [s^{(n)} - G] = 0$ on the equation. Since $pr^{(n)} v = \partial_{s}$ , this reads $$ \partial_s\bigl(s^{(n)} - G\bigr) = -,\partial_s G = 0, $$ so $G$ does not depend on $s$ : $G = G (r, s^{'}, \dots, s^{(n - 1)})$ . The equation in the canonical frame contains $s$ only through its derivatives.

Introduce the new dependent variable $w = s^{'} = d s / d r$ . Then $s^{''} = w^{'}$ , and inductively $s^{(j)} = w^{(j - 1)}$ for $j \geq 1$ , so the $n$ -th order equation $s^{(n)} = G (r, s^{'}, \dots, s^{(n - 1)})$ becomes $$ w^{(n-1)} = G\bigl(r, w, w', \dots, w^{(n-2)}\bigr), $$ an ODE of order $n - 1$ for $w (r)$ . Solving it for $w (r)$ and integrating $s^{'} = w$ once recovers $s$ up to a constant, the constant being the orbit parameter along $v$ ; undoing the canonical change of coordinates returns the original solution. The order has dropped from $n$ to $n - 1$ , with one quadrature restoring the lost order. $□$

Bridge. This is the elementary core of Lie's program: each symmetry buys one reduction of order, and a solvable $n$ -dimensional symmetry algebra integrates a scalar ODE completely by successive reductions. It builds toward the PDE case below, where canonical coordinates are replaced by joint invariants and reduction of order becomes reduction of the number of independent variables. It appears again in the optimal-system classification, where one selects which one-parameter subgroup to reduce by, and in 02.12.02, whose flow-box rectification is the geometric content of canonical coordinates. The construction reuses the prolongation formula of 05.05.06: straightening $v$ to $\partial_{s}$ kills every prolongation coefficient, so the criterion degenerates to the single statement that $s$ is absent, which is exactly the structural fact that one variable has become ignorable.

Exercises Intermediate+

Exercise 6 (hard, symbolic).

Burgers' equation $u_{t} + u u_{x} = κ u_{xx}$ admits the traveling-wave symmetry generated by $v = \partial_{t} + c \partial_{x}$ (a Galilean-free space-time translation). Find the invariants, write the traveling-wave ansatz $u = U (z)$ , and reduce Burgers' equation to an ODE for $U$ .

Hint

The characteristic system is $d t /1 = d x / c = d u /0$ . The invariants are $z = x - c t$ and $u$ . Substitute $u = U (x - c t)$ .

Answer

From $d t /1 = d x / c$ , $z = x - c t$ is invariant, and $d u /0$ forces $u$ invariant, so the ansatz is $u = U (z)$ , $z = x - c t$ . Then $u_{t} = - c U^{'}$ , $u_{x} = U^{'}$ , $u_{xx} = U^{''}$ , and Burgers' equation becomes $$ -c,U' + U,U' = \kappa,U'', \qquad\text{i.e.}\qquad \kappa,U'' = (U - c),U'. $$ Integrating once: $κ U^{'} = \frac{1}{2} U^{2} - c U + a$ for a constant $a$ , a first-order separable ODE whose bounded monotone solution is the $tanh$ shock profile $U (z) = c - b tanh (\frac{b}{2 κ} (z - z_{0}))$ joining the two roots of the right-hand quadratic.

Rubric: full credit for $κ U^{''} = (U - c) U^{'}$ and the first integral; partial credit for the reduced second-order ODE alone.

Exercise 7 (hard, short-answer).

Explain why two invariant solutions built from conjugate subalgebras $h$ and $g_{*} h$ (related by the adjoint action of some $g \in G$ ) are equivalent, and why this motivates classifying reductions by an optimal system of subalgebras.

Hint

If $f$ is $h$ -invariant and $g \in G$ is a symmetry, what can you say about $g \cdot f$ and the subalgebra $Ad (g) h$ ?

Answer

Let $f$ be $H$ -invariant and let $g \in G$ be a symmetry of the equation. Then $g \cdot f$ is again a solution, and it is invariant under the conjugate subgroup $g H g^{- 1}$ , whose algebra is $Ad (g) h$ : if $v \in h$ is tangent to $Γ_{f}$ then $g_{*} v = Ad (g) v$ is tangent to $g \cdot Γ_{f} = Γ_{g \cdot f}$ . Hence the reduction by $h$ and the reduction by $Ad (g) h$ produce solutions carried to each other by the known symmetry $g$ , so they contain no new information relative to one another. To enumerate genuinely distinct invariant solutions one therefore lists subalgebras up to conjugacy under the adjoint action — an optimal system of subalgebras — picking one representative per conjugacy class. Reducing by every subalgebra would redundantly recompute group-equivalent solutions.

Rubric: full credit for the $Ad (g)$ -conjugacy argument and the optimal-system conclusion; partial credit for noting $g \cdot f$ is invariant under the conjugate.

Lean formalization Intermediate+

Mathlib has flows and Lie groups but no symmetry-reduction apparatus for differential equations; the block fixes intended signatures, with the gap detailed in Mathlib gap analysis.

import Mathlib.Geometry.Manifold.ContMDiff.Basic
import Mathlib.Analysis.ODE.Gronwall

variable {p q : ℕ}

/-- A generator on E: horizontal part ξ and vertical part φ. -/
structure Generator (p q : ℕ) where
  xi  : (Fin p → ℝ) → (Fin q → ℝ) → (Fin p → ℝ)
  phi : (Fin p → ℝ) → (Fin q → ℝ) → (Fin q → ℝ)

/-- An invariant of v: a function annihilated by v = ξ^i ∂_{x^i} + φ^α ∂_{u^α}. -/
def IsInvariant
    (v : Generator p q)
    (I : (Fin p → ℝ) → (Fin q → ℝ) → ℝ) : Prop :=
  ∀ x u, (∑ i, v.xi x u i * (deriv (fun t => I (Function.update x i t) u) (x i)))
       + (∑ α, v.phi x u α * (deriv (fun t => I x (Function.update u α t)) (u α))) = 0

/-- Invariant-surface condition: ξ^i ∂_i f^α = φ^α along the graph of f. -/
def InvariantSurface
    (v : Generator p q) (f : (Fin p → ℝ) → (Fin q → ℝ)) : Prop :=
  ∀ x α, (∑ i, v.xi x (f x) i * (deriv (fun t => f (Function.update x i t) α) (x i)))
       = v.phi x (f x) α

/-- Canonical coordinates (r,s) straighten v to ∂_s: v(r)=0, v(s)=1.
    Reduction of order then follows; statement against the absent jet apparatus. -/
theorem reduction_of_order
    (v : Generator 1 1) : True := by
  trivial  -- placeholder: in (r,s) the n-th order ODE loses s and drops to order n-1

The by-block placeholder and the deriv-based stand-ins rest on the jet/prolongation gap of 05.05.05 and 05.05.06 plus the reduced-equation construction. Each sits on the existing flow and Lie-group libraries and is a Mathlib-contribution-sized target named in Mathlib gap analysis.

Advanced results Master

Reduction turns an invariance into a smaller equation, and the precise bookkeeping of how much smaller is the content of the reduction theorem. Two structural results frame the method, and three worked reductions exhibit it producing closed-form solutions.

The reduction theorem. Let $G$ be an $r$ -dimensional symmetry group of $S = {Δ_{ν} = 0} \subseteq J^{k} (E)$ acting on $E$ so that its projection to $X$ has generic orbit dimension $s^{'}$ , and assume the action is transversal to the equation in the sense that the invariant jet manifold has maximal rank. Then the $G$ -invariant solutions are governed by a reduced system in $p - s^{'}$ independent variables, the similarity variables $y^{a}$ , obtained by restricting $Δ_{ν}$ to the invariant jets. The map from invariant solutions of $S$ to solutions of the reduced system is a bijection (Olver ^{[Olver §3.2]}, Ovsiannikov ^{[Ovsiannikov Ch. 5]}). For $p = 2$ , $s^{'} = 1$ this is the passage from a PDE in $(x, t)$ to an ODE in one similarity variable; the heat-equation reduction below is the canonical instance.

The heat-equation source solution. For $u_{t} = u_{xx}$ the scaling subalgebra is spanned by $v_{4} = x \partial_{x} + 2 t \partial_{t}$ and the dilation $v_{3} = u \partial_{u}$ . Reducing by the one-parameter group generated by $v = v_{4} - 2 β v_{3} = x \partial_{x} + 2 t \partial_{t} - 2 β u \partial_{u}$ gives base invariant $η = x / t$ and fibre invariant $w = u t^{β}$ , so the ansatz is $u = t^{- β} ψ (η)$ . Substitution yields the reduced ODE $$ \psi'' + \tfrac12\eta,\psi' + \beta,\psi = 0. $$ For $β = 1/2$ the equation integrates to $ψ (η) = C e^{- η^{2} /4}$ , and undoing the ansatz gives the fundamental (source) solution $$ u(x,t) = \frac{C}{\sqrt t},\exp!\Bigl(-\frac{x^2}{4t}\Bigr), $$ the heat kernel up to normalisation. For $β = 0$ the reduced ODE is $ψ^{''} + \frac{1}{2} η ψ^{'} = 0$ with solution the error-function profile $ψ (η) = A + B erf (η /2)$ , the spreading step. Both are pure outputs of the scaling reduction; the choice of $β$ selects the self-similar exponent, exactly the self-similar solution of the first kind in Barenblatt's terminology ^{[Barenblatt Ch. 1]}.

The porous-medium / Burgers traveling wave. The space-time translation $v = \partial_{t} + c \partial_{x}$ reduces any translation-invariant evolution PDE to a traveling-wave ODE in $z = x - c t$ . For Burgers $u_{t} + u u_{x} = κ u_{xx}$ this is $κ U^{''} = (U - c) U^{'}$ , integrating to the $tanh$ shock profile; for the porous-medium equation $u_{t} = (u^{m})_{xx}$ the same translation gives a degenerate ODE whose compactly supported traveling fronts are the Barenblatt-type waves with finite propagation speed. The traveling-wave reduction is the one-dimensional shadow of the two-dimensional translation subgroup.

Optimal systems. Distinct invariant solutions correspond to subalgebras of $g$ taken up to conjugacy under the adjoint action $Ad : G \to GL (g)$ . An optimal system of $θ$ -dimensional subalgebras is a list of representatives, one per conjugacy class, computed by exponentiating the adjoint action $Ad (exp ε v) w = w - ε [v, w] + \frac{ε ^{2}}{2} [v, [v, w]] - \dots$ and using it to simplify an arbitrary generator to canonical form. For the heat equation, the one-dimensional optimal system reduces the six-dimensional point algebra to a short list — translation $\partial_{x}$ , scaling $v_{4}$ , Galilean $v_{5}$ , and a few combinations — and every reduction not equivalent to these is redundant. Ovsiannikov introduced the systematic computation ^{[Ovsiannikov Ch. 6]}.

Synthesis. Symmetry reduction is the conversion of an invariance into a quotient equation, and the orbit structure of the chosen subgroup is the conversion ratio. The method proceeds along four linked stages, each carried by the generators produced in 05.05.06. First, a subgroup $H \subseteq G$ is selected and its joint invariants are computed as first integrals of the characteristic system $d x^{i} / ξ^{i} = d u^{α} / ϕ^{α}$ , replacing the original coordinates by similarity variables. Second, the invariant-surface condition $ξ^{i} \partial_{i} f^{α} = ϕ^{α}$ is imposed, expressing the dependent variables through those invariants, which is the invariant-solution ansatz. Third, substitution into $Δ_{ν} = 0$ produces the reduced system in $p - s^{'}$ independent variables, an ODE when the count reaches one, integrated in closed form for the heat, Burgers, and porous-medium examples. Fourth, the adjoint action of $G$ on $g$ collapses the redundant reductions into an optimal system of subalgebras, so that finitely many representatives exhaust all group-inequivalent invariant solutions. The reduction of order of an ODE and the reduction of independent variables of a PDE are the same operation seen at $s^{'} = 0$ and $s^{'} \geq 1$ , and the canonical coordinates that straighten a generator to $\partial_{s}$ are the local model that both specialise.

Full proof set Master

Proposition (similarity reduction of a scaling-invariant evolution equation). Let an evolution equation $u_{t} = N [u]$ in one space dimension admit the scaling group $(x, t, u) \mapsto (λ x, λ^{a} t, λ^{b} u)$ with generator $v = x \partial_{x} + a t \partial_{t} + b u \partial_{u}$ , $a \neq = 0$ . Then every $v$ -invariant solution has the self-similar form $u (x, t) = t^{b / a} ψ (η)$ with $η = x t^{- 1/ a}$ , and $ψ$ satisfies an ODE obtained by substitution; conversely each solution of that ODE yields a $v$ -invariant solution.

Proof. The invariants of $v$ are the first integrals of the characteristic system $d x / x = d t / (a t) = d u / (b u)$ . From $d x / x = d t / (a t)$ , $ln x - \frac{1}{a} ln t = const$ , giving the base invariant $η = x t^{- 1/ a}$ . From $d t / (a t) = d u / (b u)$ , $\frac{1}{a} ln t \cdot b - ln u$ ... more directly $d u / u = (b / a) d t / t$ , so $ln u - \frac{b}{a} ln t = const$ , giving the fibre invariant $w = u t^{- b / a}$ . The invariant-surface condition $v$ -tangency forces $w$ to be a function of $η$ alone: $w = ψ (η)$ , i.e. $$ u(x, t) = t^{b/a},\psi(\eta), \qquad \eta = x,t^{-1/a}. $$ Differentiating, $u_{t} = t^{b / a - 1} (\frac{b}{a} ψ - \frac{1}{a} η ψ^{'})$ and each spatial derivative $\partial_{x}^{j} u = t^{b / a - j / a} ψ^{(j)} (η)$ , since $\partial_{x} η = t^{- 1/ a}$ . Substituting into $u_{t} = N [u]$ and dividing by the common power $t^{b / a - 1}$ — which is consistent precisely because $v$ is a symmetry, so every term carries the same scaling weight — eliminates $t$ and leaves an ODE $N [ψ] (η) = \frac{b}{a} ψ - \frac{1}{a} η ψ^{'}$ purely in $η$ . Conversely, given a solution $ψ$ of that ODE, the function $u = t^{b / a} ψ (x t^{- 1/ a})$ satisfies $u_{t} = N [u]$ by reversing the substitution, and it is $v$ -invariant because $v (u t^{- b / a}) = v (ψ (η)) = 0$ as $η$ is an invariant. Hence the correspondence is a bijection between $v$ -invariant solutions and solutions of the reduced ODE. $□$

Proposition (commuting reductions and successive order drop). Let a scalar ODE of order $n$ admit a two-dimensional abelian symmetry algebra $h = span {v_{1}, v_{2}}$ with $[v_{1}, v_{2}] = 0$ and $v_{1}, v_{2}$ pointwise independent. Then the equation can be reduced to order $n - 2$ by two successive canonical-coordinate reductions, and the reductions can be performed in either order.

Proof. By the reduction-of-order theorem, choose canonical coordinates $(r, s)$ for $v_{1}$ , so $v_{1} = \partial_{s}$ and the equation, written for $s (r)$ , loses explicit dependence on $s$ ; setting $w = s^{'}$ drops the order to $n - 1$ in the variable $w (r)$ . It remains to show $v_{2}$ descends to a symmetry of the reduced equation. Since $[v_{1}, v_{2}] = 0$ , the field $v_{2}$ commutes with $\partial_{s}$ , so in the $(r, s)$ frame $v_{2} = ρ (r) \partial_{r} + σ (r) \partial_{s}$ has coefficients independent of $s$ (commuting with $\partial_{s}$ forbids $s$ -dependence). Its action on the $s$ -free reduced equation is well defined and is again a symmetry, because prolongation commutes with the projection that forgets $s$ : the reduced equation is the restriction of the original to the $s$ -independent jets, and $pr v_{2}$ preserves that restriction as it preserved the full equation. Hence the order- $(n - 1)$ equation for $w$ admits the projected one-parameter symmetry $\overset{v}{ˉ}_{2}$ , and a second canonical-coordinate reduction drops the order to $n - 2$ . Commutativity $[v_{1}, v_{2}] = 0$ makes the two flows independent, so reducing first by $v_{2}$ and then by $v_{1}$ yields the same order- $(n - 2)$ equation up to a change of variables, by symmetry of the argument. $□$

Connections Master

The generators and characteristics computed in 05.05.06 are the raw material of this unit: the invariant-surface condition $ξ^{i} \partial_{i} f^{α} = ϕ^{α}$ is the vanishing of the characteristic $Q^{α} = ϕ^{α} - ξ^{i} u_{i}^{α}$ along the solution graph, so an invariant solution is one on which every generator's characteristic vanishes, and the determining equations there select which generators are available to reduce by here.

The jet bundle and total derivative of 05.05.05 persist as the substrate: the reduced equation is the restriction of $Δ_{ν}$ to the invariant jet submanifold, computed by the same total-derivative prolongation, and the similarity variables are coordinates on the quotient of the jet bundle by the prolonged group action.

Canonical coordinates straightening a generator to $\partial_{s}$ are the flow-box rectification of 02.12.02: reduction of order is the statement that an ignorable coordinate along a symmetry flow can be integrated out, the differential-equation analogue of passing to a quotient by a one-parameter group, and the orbit parameter restored by quadrature is the flow time of that group.

Historical & philosophical context Master

Sophus Lie developed the reduction of differential equations by their symmetry groups as the centerpiece of his theory of continuous groups, explicitly modeling it on Galois theory: where Galois resolved the solvability of a polynomial through the structure of its permutation group, Lie sought to resolve the integrability of a differential equation through the structure of its continuous symmetry group. His lectures, edited by Georg Scheffers as Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (1891) ^{[Lie 1895]}, laid out reduction of order by canonical coordinates and the integration of an ODE by a solvable symmetry algebra. The central theorem — that a one-parameter symmetry lowers the order of a scalar ODE by exactly one, and a solvable $n$ -dimensional algebra integrates an $n$ -th order equation completely — is Lie's.

The method was carried into mathematical physics through dimensional analysis and similarity solutions before the group-theoretic origin was widely recognised. Garrett Birkhoff's Hydrodynamics (1950) ^{[Birkhoff 1950]} framed similarity solutions of fluid equations explicitly as invariance under groups, connecting Lie's reduction to the physicists' dimensional method. George Barenblatt's Similarity, Self-Similarity, and Intermediate Asymptotics (1979) ^{[Barenblatt Ch. 1]} distinguished self-similar solutions of the first kind, whose exponents are fixed by dimensional analysis, from those of the second kind, whose exponents are determined by a nonlinear eigenvalue problem and lie outside naive scaling. Lev Ovsiannikov's Group Analysis of Differential Equations (1982) ^{[Ovsiannikov Ch. 5]} systematised invariant and partially invariant solutions and the optimal-system classification for continuum mechanics, and George Bluman and Julian Cole's Similarity Methods for Differential Equations (1974) and their 1969 nonclassical method ^{[Bluman-Cole Ch. 2]} extended the construction to ansätze invariant under symmetries of the invariant-surface condition rather than of the equation itself. Peter Olver's Applications of Lie Groups to Differential Equations (1986; second edition 1993) ^{[Olver §3.1]} gave the modern geometric account in terms of group invariants, the reduced equation, and optimal systems used in this unit.

Bibliography Master

@book{LieScheffers1891,
  author    = {Lie, Sophus and Scheffers, Georg},
  title     = {Vorlesungen {\"u}ber Differentialgleichungen mit bekannten infinitesimalen Transformationen},
  publisher = {B. G. Teubner},
  address   = {Leipzig},
  year      = {1891}
}

@book{OlverLieGroups,
  author    = {Olver, Peter J.},
  title     = {Applications of Lie Groups to Differential Equations},
  series    = {Graduate Texts in Mathematics},
  volume    = {107},
  publisher = {Springer},
  year      = {1993},
  edition   = {2nd}
}

@book{Ovsiannikov1982,
  author    = {Ovsiannikov, Lev V.},
  title     = {Group Analysis of Differential Equations},
  publisher = {Academic Press},
  year      = {1982},
  note      = {translated by W. F. Ames}
}

@book{BlumanCole1974,
  author    = {Bluman, George W. and Cole, Julian D.},
  title     = {Similarity Methods for Differential Equations},
  series    = {Applied Mathematical Sciences},
  volume    = {13},
  publisher = {Springer},
  year      = {1974}
}

@article{BlumanCole1969,
  author  = {Bluman, George W. and Cole, Julian D.},
  title   = {The general similarity solution of the heat equation},
  journal = {Journal of Mathematics and Mechanics},
  volume  = {18},
  year    = {1969},
  pages   = {1025--1042}
}

@book{Barenblatt1979,
  author    = {Barenblatt, Grigory I.},
  title     = {Similarity, Self-Similarity, and Intermediate Asymptotics},
  publisher = {Consultants Bureau},
  address   = {New York},
  year      = {1979}
}

@book{Birkhoff1950,
  author    = {Birkhoff, Garrett},
  title     = {Hydrodynamics: A Study in Logic, Fact, and Similitude},
  publisher = {Princeton University Press},
  year      = {1950}
}

@book{BlumanKumei1989,
  author    = {Bluman, George W. and Kumei, Sukeyuki},
  title     = {Symmetries and Differential Equations},
  series    = {Applied Mathematical Sciences},
  volume    = {81},
  publisher = {Springer},
  year      = {1989}
}

Prerequisites

05.05.06
05.05.05
02.12.02

Tier anchors

beginner: Olver *Applications of Lie Groups to Differential Equations* §3.1 (a symmetry-respecting solution depends on fewer variables)
intermediate: Olver §3.1-§3.4 (group invariants, the invariant-solution ansatz, reduction of order, optimal systems); Bluman-Kumei *Symmetries and Differential Equations* (1989) Ch. 4
master: Olver §3.1-§3.4; Ovsiannikov *Group Analysis of Differential Equations* (1982); Bluman-Cole *Similarity Methods for Differential Equations* (1974); Barenblatt *Similarity, Self-Similarity, and Intermediate Asymptotics* (1979)

References

Olver — Applications of Lie Groups to Differential Equations (2nd ed., 1993) · §3.1 group-invariant solutions; §3.2 invariants and the reduced equation; §3.3 optimal systems; §3.4 nonclassical and further reductions
Ovsiannikov — Group Analysis of Differential Equations (1982) · Ch. 5-6 invariant solutions, the rank of the invariant manifold, and partially invariant solutions
Bluman, Cole — Similarity Methods for Differential Equations (1974) · Ch. 2-4 similarity variables, reduction of PDEs, and the nonclassical method
Lie — Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (1891/1895) · reduction of order of an ODE by a one-parameter group via canonical coordinates
Barenblatt — Similarity, Self-Similarity, and Intermediate Asymptotics (1979) · Ch. 1-3 dimensional analysis, self-similar solutions of the first and second kind
Birkhoff — Hydrodynamics: A Study in Logic, Fact, and Similitude (1950) · Ch. 4 dimensional analysis, groups, and similarity solutions in fluid mechanics

Estimated time

beginner: 17m
intermediate: 47m
master: 84m