05.05.10 · symplectic / lagrangian

Lie's classification of second-order ODEs and the symmetry algorithm for ODEs

shipped3 tiersLean: none

Anchor (Master): Olver §3.5; Lie *Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen* (Lie-Scheffers 1891); Ibragimov *Elementary Lie Group Analysis and Ordinary Differential Equations* (1999)

Intuition Beginner

A second-order differential equation tells you how the curvature of an unknown curve is fixed by its position and slope. Such an equation can have symmetries: motions of the plane that turn every solution curve into another solution curve. Lie discovered something surprising about how many such symmetries a second-order equation can have. The number of independent symmetries is never just anything you like. It is always exactly zero, one, two, three, or eight, and never four, five, six, or seven. There is a gap.

The top of that list, eight, is special. An equation reaches the maximum of eight symmetries only when it is, in disguise, the simplest second-order equation of all: the one whose solutions are straight lines. So a rich symmetry count is a signal. If you can find eight independent symmetries, you have learned that a clever change of variables turns your equation into the straight-line equation, and then you can solve it completely.

The other big idea is a recipe. Finding the symmetries is not guesswork. You write down the most general motion, demand that it carry solutions to solutions, and that demand collapses into a system of simpler equations you can actually solve. Once you have even two symmetries arranged the right way, you can integrate the original equation step by step, peeling off one order at a time.

Visual Beginner

Picture the straight-line equation, whose solutions are all the straight lines in the plane. You can shift the plane, rotate it, stretch it, shear it, and project it, and every one of those motions sends straight lines to straight lines. There are eight independent ways to do this, and together they form the largest possible symmetry collection a second-order equation can have.

The picture makes the gap believable. A generic curved equation breaks most of these motions, so it keeps only a few symmetries, usually none, one, two, or three. To keep all eight you must keep the whole rich family of plane motions intact, and that forces the solution curves to be exactly the straight lines. There is no middle ground that keeps four through seven of them, because the surviving motions always close up into one of the small sizes or jump all the way to the full eight.

Worked example Beginner

Take the straight-line equation: the second rate of change of the height is zero. Its solutions are all functions whose graph is a straight line, that is, height equals a slope times the input plus a starting value.

Now test two simple symmetries. First, slide every point up by a fixed amount, say add seven to every height. A straight line slid up by seven is still a straight line, so this motion carries solutions to solutions. Second, shift every input to the right by a fixed amount, say replace the input by the input minus three. A straight line shifted sideways is still a straight line, so this is a symmetry too.

Try a third, a stretch: multiply every height by two. The line with slope one becomes the line with slope two, still a straight line. Symmetry again. You can keep going: tilt, rotate, and even a projective squeeze all preserve straight lines. Counting carefully, you find eight independent motions, and not one more.

What this tells us: the simplest second-order equation already carries the maximum number of symmetries, eight. This is the fingerprint of the straight-line equation, and any equation that secretly equals it after a change of variables will show the same fingerprint of eight symmetries.

Check your understanding Beginner

Formal definition Intermediate+

Work on the jet bundle $J^{2} (R, R)$ with coordinates $(x, u, u^{'}, u^{''})$ of 05.05.05, where $x$ is the independent variable and $u$ the dependent one. A scalar second-order ODE is the submanifold $$ \mathcal{S} = {, \Delta(x, u, u', u'') = u'' - F(x, u, u') = 0 ,} \subset J^2(\mathbb{R}, \mathbb{R}). $$ A point symmetry is a generator $v = ξ (x, u) \partial_{x} + η (x, u) \partial_{u}$ whose second prolongation $pr^{(2)} v = ξ \partial_{x} + η \partial_{u} + η^{x} \partial_{u^{'}} + η^{xx} \partial_{u^{''}}$ satisfies Lie's infinitesimal criterion of 05.05.06, namely $pr^{(2)} v [Δ] = 0$ on $S$ . The development follows Olver ^{[Olver §3.5]} and Ibragimov ^{[Ibragimov Ch. 7]}.

The prolongation coefficients, from the formula $η^{J} = D_{J} Q + ξ u^{(∣ J ∣ + 1)}$ with characteristic $Q = η - ξ u^{'}$ of 05.05.06, are $$ \eta^x = \eta_x + (\eta_u - \xi_x) u' - \xi_u (u')^2, $$ $$ \eta^{xx} = \eta_{xx} + (2\eta_{xu} - \xi_{xx}) u' + (\eta_{uu} - 2\xi_{xu})(u')^2 - \xi_{uu}(u')^3 + (\eta_u - 2\xi_x - 3\xi_u u'),u''. $$

Definition (the symmetry algorithm). The determining equations are obtained by the following procedure. Substitute $u^{''} = F$ into $pr^{(2)} v [Δ] = η^{xx} - ξ F_{x} - η F_{u} - η^{x} F_{u^{'}} = 0$ , then regard the result as a polynomial in the remaining free coordinate $u^{'}$ . Because $ξ, η$ depend only on $(x, u)$ , the coefficient of each power of $u^{'}$ must vanish separately. This splitting turns one equation into an overdetermined linear homogeneous system of partial differential equations for $ξ (x, u)$ and $η (x, u)$ . Solving the system yields the symmetry algebra $g$ .

Definition (symmetry algebra and its dimension). The set of point symmetries is a finite-dimensional Lie algebra $g$ under the bracket of vector fields, since prolongation is a Lie-algebra homomorphism (proved in 05.05.06). Its dimension $r = dim g$ is the number of independent generators. An equation is linearisable by a point transformation when a change of variables $(x, u) \mapsto (\tilde{x} (x, u), \tilde{u} (x, u))$ carries $S$ to the straight-line equation $\tilde{u}^{''} = 0$ .

Definition (the four canonical 2D types). A two-dimensional symmetry algebra $g = span {v_{1}, v_{2}}$ is classified by two binary data: whether it is abelian ( $[v_{1}, v_{2}] = 0$ ) or non-abelian (a basis exists with $[v_{1}, v_{2}] = v_{1}$ ), and whether the action is transversal ( $v_{1}, v_{2}$ span the $(x, u)$ plane at a generic point) or non-transversal ( $v_{1} \land v_{2} = 0$ generically). Lie's normal forms place each of the four resulting types into canonical coordinates, listed in the Key theorem.

A non-example fixes the meaning. A generator $v = ξ \partial_{x} + η \partial_{u}$ that satisfies the first-prolongation criterion $η^{x} - (first-order part) = 0$ but not the full second-prolongation criterion is a symmetry of a related first-order equation, not of $S$ . The prolongation order must match the equation order, here two.

Counterexamples to common slips

The dimension count $0, 1, 2, 3, 8$ is for point symmetries of a scalar second-order ODE. Contact symmetries, higher-order generalised symmetries, or systems all obey different counts; the gap between $3$ and $8$ is specific to this case.
"Linearisable" means linearisable by a point transformation $(x, u) \mapsto (\tilde{x}, \tilde{u})$ , not merely "becomes linear after some substitution of the unknown". The equation $u^{''} = u^{- 3}$ is not point-linearisable despite being integrable, and indeed it does not carry an eight-dimensional algebra.
For a non-abelian 2D algebra the integration order matters. You must reduce first by the generator $v_{1}$ with $[v_{1}, v_{2}] = v_{1}$ (the one that is an ideal), or the reduced equation fails to inherit the second symmetry. Choosing the wrong order leaves a first-order equation with no usable symmetry.

Key theorem with proof Intermediate+

Theorem (Lie's dimension bound for scalar second-order ODEs). The point-symmetry algebra $g$ of a scalar second-order ODE $u^{''} = F (x, u, u^{'})$ has dimension $r \in {0, 1, 2, 3, 8}$ . The values $r = 4, 5, 6, 7$ do not occur. The maximal value $r = 8$ is attained if and only if the equation is linearisable by a point transformation, equivalently equivalent to $u^{''} = 0$ , in which case $g ≅ sl (3, R)$ .

The statement follows Olver ^{[Olver §3.5]} and Lie ^{[Lie-Scheffers 1891]}.

Proof. Run the symmetry algorithm. Substituting $η^{xx}$ and $η^{x}$ into $pr^{(2)} v [Δ] = 0$ on $u^{''} = F$ and collecting powers of $u^{'}$ gives the determining system; in the most-symmetric case $F = 0$ it reads, after splitting the polynomial in $u^{'}$ of degree three, $$ \xi_{uu} = 0, \quad \eta_{uu} - 2\xi_{xu} = 0, \quad 2\eta_{xu} - \xi_{xx} = 0, \quad \eta_{xx} = 0. $$ These four linear PDEs for $ξ (x, u), η (x, u)$ have, as their general solution, the space of polynomials $$ \xi = c_1 + c_3 x + c_5 u + c_7 x^2 + c_8 x u, \qquad \eta = c_2 + c_4 u + c_6 x + c_7 x u + c_8 u^2, $$ an eight-parameter family $c_{1}, \dots, c_{8}$ . (Direct check: $ξ_{uu} = 0$ and $η_{xx} = 0$ kill the $u^{2}$ -in- $ξ$ and $x^{2}$ -in- $η$ terms; the cross conditions force the shared coefficients $c_{7}, c_{8}$ .) Hence $dim g = 8$ for $u^{''} = 0$ . The eight generators $$ \partial_x,\ \partial_u,\ x\partial_x,\ u\partial_u,\ u\partial_x,\ x\partial_u,\ x(x\partial_x + u\partial_u),\ u(x\partial_x + u\partial_u) $$ are the infinitesimal projective transformations of the plane; their brackets reproduce the structure constants of $sl (3, R)$ , the Lie algebra of the projective group $PGL (3, R)$ acting on $RP^{2}$ and preserving the family of lines.

For the upper bound and the gap, take any $F$ admitting a symmetry $v$ , which by $v \neq = 0$ can be straightened to $v = \partial_{u}$ in canonical coordinates of 05.05.07. The criterion forces $F$ to be independent of $u$ , so the equation is $u^{''} = F (x, u^{'})$ . The remaining symmetries form a subalgebra fixing this normal form; analysing their determining equations shows the residual freedom is governed by the same four split equations but with $F$ -dependent inhomogeneities, whose solution space has dimension at most that of the homogeneous $F = 0$ case minus the constraints imposed by a nonzero $F$ . A counting argument on the prolonged action — the symmetry group of a second-order ODE acts on the three-dimensional manifold $J^{1}$ with an action whose isotropy and orbit structure permit stabiliser dimensions only $0, 1, 2, 3$ off the maximal stratum, and the full $8$ on it — closes the list at ${0, 1, 2, 3, 8}$ . The intermediate dimensions $4, 5, 6, 7$ would require a transitive-enough action that already forces $F = 0$ , collapsing to the maximal case; this is the source of the gap.

For the equivalence at the top, $r = 8$ forces the four homogeneous split equations above (any inhomogeneity from $F \neq = 0$ cuts the solution dimension below $8$ ), and those equations hold precisely when $F_{u^{'} u^{'}} = 0$ together with the structural relations that characterise $F = 0$ up to a point change of variables. Thus $r = 8$ iff a point transformation carries the equation to $u^{''} = 0$ , which is the assertion of point-linearisability. $□$

Bridge. This dimension theorem is the structural backbone of the symmetry method for ODEs: the count of symmetries is a discrete invariant that already decides integrability questions. It builds toward the integration-by-quadrature results below, where a 2D or 3D algebra is enough to solve the equation in closed form, and the foundational reason the count matters is that each independent symmetry buys one reduction of order via 05.05.07. It appears again in 03.03.01, because the maximal algebra $sl (3, R)$ is the Lie algebra of a concrete projective group, so the most symmetric ODE is governed by a specific simple Lie group. This is exactly the Galois-theoretic analogy Lie pursued: the discrete symmetry count and the algebra's structure decide solvability, just as a Galois group decides solvability by radicals. The bridge is that prolongation, proved a Lie-algebra homomorphism in 05.05.06, makes the symmetry set a finite-dimensional algebra whose dimension is the invariant classified here.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that $v_{1} = \partial_{x}$ and $v_{2} = x \partial_{x} + u \partial_{u}$ generate a non-abelian 2D algebra, compute $[v_{1}, v_{2}]$ , and state which generator must be used for the first reduction of order.

Hint

Compute the bracket as a vector field. For a non-abelian algebra with $[v_{1}, v_{2}] = v_{1}$ , reduce first by the generator that spans the derived ideal.

Answer

$[v_{1}, v_{2}] = [\partial_{x}, x \partial_{x} + u \partial_{u}]$ . Acting on coordinates, $\partial_{x} (x) = 1$ and $\partial_{x} (u) = 0$ , so $[\partial_{x}, x \partial_{x} + u \partial_{u}] = \partial_{x} = v_{1}$ . The algebra is non-abelian with $[v_{1}, v_{2}] = v_{1}$ , so $span {v_{1}}$ is the derived ideal. The first reduction must use $v_{1} = \partial_{x}$ , since reducing by the ideal leaves a quotient equation that still admits the projected $v_{2}$ .

Rubric: full credit for $[v_{1}, v_{2}] = v_{1}$ and the choice of $v_{1}$ for the first reduction; partial credit for the bracket alone.

Exercise 4 (medium, symbolic).

Use the symmetry algorithm to verify that $v = x \partial_{x}$ is a point symmetry of the equation $u^{''} = (u^{'})^{2} / u^{'}$ ... instead take the equidimensional equation $u^{''} = (u^{'})^{2} / u$ and verify $v = u \partial_{u}$ is a symmetry by computing $pr^{(2)} v [u^{''} - (u^{'})^{2} / u]$ on the equation.

Hint

For $v = u \partial_{u}$ : $ξ = 0$ , $η = u$ , so $η^{x} = u^{'}$ and $η^{xx} = u^{''}$ . Substitute into $pr^{(2)} v [Δ]$ with $Δ = u^{''} - (u^{'})^{2} / u$ .

Answer

With $ξ = 0$ , $η = u$ : from the formulas $η^{x} = η_{x} + (η_{u} - ξ_{x}) u^{'} - ξ_{u} (u^{'})^{2} = u^{'}$ and $η^{xx} = u^{''}$ . Then $$ \mathrm{pr}^{(2)}v[\Delta] = \eta^{xx} - \eta,\partial_u!\Big(!-\tfrac{(u')^2}{u}\Big) - \eta^x,\partial_{u'}!\Big(!-\tfrac{(u')^2}{u}\Big) = u'' - u\cdot\tfrac{(u')^2}{u^2} - u'\cdot\big(!-\tfrac{2u'}{u}\big). $$ This is $u^{''} - (u^{'})^{2} / u + 2 (u^{'})^{2} / u = u^{''} + (u^{'})^{2} / u$ . On the equation $u^{''} = (u^{'})^{2} / u$ this equals $(u^{'})^{2} / u + (u^{'})^{2} / u = 2 (u^{'})^{2} / u$ , which does not vanish, so $v = u \partial_{u}$ alone is not a symmetry; the scaling that works is $v = x \partial_{x}$ (with $η = 0$ , $ξ = x$ ), for which $η^{x} = - u^{'}$ , $η^{xx} = - 2 u^{''}$ and $pr^{(2)} v [Δ] = - 2 u^{''} + 2 (u^{'})^{2} / u = - 2Δ$ vanishes on $S$ .

Rubric: full credit for the correct on-shell computation identifying $v = x \partial_{x}$ as the genuine symmetry; partial credit for a correct prolongation of either candidate.

Exercise 5 (medium, symbolic).

For the equation $u^{''} = 0$ , write down the two generators $x (x \partial_{x} + u \partial_{u})$ and $u (x \partial_{x} + u \partial_{u})$ and explain in one or two sentences why they are the "projective" part of $sl (3, R)$ .

Hint

These two have quadratic coefficients; their flows are fractional-linear (projective) maps of the plane that still send lines to lines.

Answer

The generators are $v_{7} = x^{2} \partial_{x} + xu \partial_{u}$ and $v_{8} = xu \partial_{x} + u^{2} \partial_{u}$ . Their coefficients are quadratic in $(x, u)$ , so their one-parameter flows are projective (fractional-linear) transformations of $RP^{2}$ rather than affine ones. Projective transformations carry straight lines to straight lines, hence preserve the solution family of $u^{''} = 0$ ; together with the six affine generators ( $\partial_{x}, \partial_{u}, x \partial_{x}, u \partial_{u}, u \partial_{x}, x \partial_{u}$ ) they exhaust the eight-dimensional $sl (3, R) = Lie (PGL (3, R))$ acting on the plane.

Rubric: full credit for naming both quadratic generators and the projective / line-preserving explanation; partial credit for the generators alone.

Exercise 6 (hard, symbolic).

Integrate the equation $u^{''} = \frac{( u ^{'} ) ^{2}}{u} - \frac{u ^{'}}{x}$ given that it admits the abelian transversal 2D algebra with generators $v_{1} = x \partial_{x}$ and $v_{2} = u \partial_{u}$ (check $[v_{1}, v_{2}] = 0$ ). Use canonical coordinates to reduce twice.

Hint

The joint invariants of $v_{1} = x \partial_{x}$ , $v_{2} = u \partial_{u}$ suggest logarithmic variables. Set $t = ln x$ , $w = ln u$ ; both generators become translations, so the equation in $(t, w)$ should be autonomous, which drops the order by one, and a further reduction by the remaining translation finishes it.

Answer

With $t = ln x$ , $w = ln u$ one has $v_{1} = \partial_{t}$ , $v_{2} = \partial_{w}$ , commuting. Compute $u^{'} = d u / d x = (u / x) \overset{w}{˙}$ where $\overset{w}{˙} = d w / d t$ , and a second differentiation gives $u^{''} = (u / x^{2}) (\overset{w}{¨} + \overset{w}{˙}^{2} - \overset{w}{˙})$ . Also $(u^{'})^{2} / u = (u / x^{2}) \overset{w}{˙}^{2}$ and $u^{'} / x = (u / x^{2}) \overset{w}{˙}$ . Substituting, the equation becomes $\overset{w}{¨} + \overset{w}{˙}^{2} - \overset{w}{˙} = \overset{w}{˙}^{2} - \overset{w}{˙}$ , i.e. $\overset{w}{¨} = 0$ . This is the straight-line equation in $(t, w)$ , so $w = a t + b$ , giving $ln u = a ln x + b$ , hence $u = C x^{a}$ . The two commuting symmetries reduced the equation to $\overset{w}{¨} = 0$ , integrated by two quadratures.

Rubric: full credit for reaching $\overset{w}{¨} = 0$ and the power-law solution $u = C x^{a}$ ; partial credit for the logarithmic change of variables and a correctly autonomous reduced equation.

Exercise 7 (hard, short-answer).

Explain why the dimension of the point-symmetry algebra of a scalar second-order ODE cannot equal $4, 5, 6$ , or $7$ , referring to what happens to the equation as the symmetry count grows.

Hint

A large symmetry group acts transitively enough on $J^{1}$ that the equation is forced into the maximal normal form $u^{''} = 0$ before the count can land in the forbidden range.

Answer

The symmetry group acts on the three-dimensional first-jet space $J^{1}$ , and the equation $u^{''} = F$ is recovered as a section assigning $u^{''}$ over each point of $J^{1}$ . A symmetry algebra of dimension $r$ produces an orbit/stabiliser structure on $J^{1}$ : for $r \leq 3$ the action leaves enough invariant functions for a genuinely curved $F$ to survive, giving the cases $0, 1, 2, 3$ . Once $r \geq 4$ the prolonged action becomes transitive on an open set with small isotropy, and the only invariant second-order field compatible with that much symmetry is $F = 0$ up to a point transformation, which jumps the dimension straight to the maximal $8 = dim sl (3, R)$ . There is no second-order ODE whose symmetry algebra has the intermediate transitivity needed for $4 \leq r \leq 7$ ; demanding it collapses the equation to $u^{''} = 0$ . Hence the count skips from $3$ to $8$ .

Rubric: full credit for the transitivity / forced-normal-form argument explaining the collapse to $F = 0$ ; partial credit for stating the gap and naming $sl (3, R)$ as the maximal case.

Lean formalization Intermediate+

Mathlib has Lie algebras and $sl_{n}$ but no link from an ODE to its symmetry algebra; the block fixes intended signatures, with the gap detailed in Mathlib gap analysis.

import Mathlib.Algebra.Lie.Classical
import Mathlib.Geometry.Manifold.ContMDiff.Basic

/-- A point generator v = ξ ∂_x + η ∂_u on J^0(ℝ,ℝ). -/
structure PointGenerator where
  xi  : ℝ → ℝ → ℝ
  eta : ℝ → ℝ → ℝ

/-- A scalar second-order ODE u'' = F(x,u,u'). -/
structure SecondOrderODE where
  F : ℝ → ℝ → ℝ → ℝ

/-- The point-symmetry algebra (its dimension is the invariant classified by Lie).
    Stated against the absent jet/prolongation apparatus of 05.05.05–05.05.06. -/
noncomputable def symmetryDim (_ : SecondOrderODE) : ℕ :=
  sorry  -- dimension of {v : PointGenerator // pr² v [Δ] = 0 on Δ = 0}

/-- Lie's dimension theorem: the symmetry dimension lies in {0,1,2,3,8}. -/
theorem lie_dimension_bound (E : SecondOrderODE) :
    symmetryDim E ∈ ({0, 1, 2, 3, 8} : Set ℕ) := by
  sorry  -- the 4–7 gap; rests on the prolongation/determining-equation machinery

/-- The maximal case is sl(3,ℝ): u'' = 0 has an 8-dimensional algebra. -/
theorem max_case_is_sl3 :
    symmetryDim ⟨fun _ _ _ => 0⟩ = 8 := by
  sorry  -- isomorphic to sl(3,ℝ) = Mathlib's `sl 3 ℝ`

The sorrys rest on the jet/prolongation gap of 05.05.05 and 05.05.06 plus the determining-equation splitting and the dimension-gap argument. Mathlib supplies $sl_{n}$ via Mathlib.Algebra.Lie.Classical, so the target structure exists; the missing piece is the symmetry-algebra functor and its dimension, each a Mathlib-contribution-sized target named in Mathlib gap analysis.

Advanced results Master

The dimension theorem is the coarse invariant; the finer classification of the small-dimensional cases and the integration procedures are the working content of Lie's theory for ODEs.

Lie's table of two-dimensional algebras. A two-dimensional Lie algebra of vector fields in the plane falls into exactly four canonical types, by the abelian/non-abelian and transversal/non-transversal dichotomies. In suitable canonical coordinates $(x, u)$ the representatives are $$ \begin{array}{llll} \text{Type I} & \mathfrak{g}_1: & [v_1, v_2] = 0, & v_1 = \partial_x,\ v_2 = \partial_u \quad(\text{transversal}); \ \text{Type II} & \mathfrak{g}_2: & [v_1, v_2] = 0, & v_1 = \partial_x,\ v_2 = x\partial_x \quad(\text{non-transversal, } v_1 \wedge v_2 = 0); \ \text{Type III} & \mathfrak{g}_3: & [v_1, v_2] = v_1, & v_1 = \partial_x,\ v_2 = x\partial_x + u\partial_u \quad(\text{transversal}); \ \text{Type IV} & \mathfrak{g}_4: & [v_1, v_2] = v_1, & v_1 = \partial_u,\ v_2 = u\partial_u \quad(\text{non-transversal}). \end{array} $$ Each type forces a canonical form on any second-order ODE admitting it ^{[Lie-Scheffers 1891]}. For Type I the most general invariant equation is $u^{''} = f (u^{'})$ ; for Type III it is $u^{''} = u^{'} f (u^{'} / \dots)$ scaled appropriately. The canonical form is what makes the equation integrable by quadratures: in each case the two symmetries reduce the order twice.

Integration by a 2D algebra. When $g$ is two-dimensional and solvable (every 2D Lie algebra is solvable), the equation integrates by two quadratures, with one structural caveat. For an abelian algebra the two reductions commute and may be done in either order. For a non-abelian algebra with $[v_{1}, v_{2}] = v_{1}$ the reduction must use $v_{1}$ first: $v_{1}$ spans the derived ideal $[g, g]$ , and only reduction by an ideal leaves a quotient on which the second generator descends to a symmetry. Reducing by $v_{2}$ first leaves a first-order equation on which the image of $v_{1}$ is not a symmetry, and the method stalls. This ordering rule is the ODE shadow of the solvable-series structure exploited in Lie's analogue of Galois theory ^{[Olver §3.5]}.

The maximal case and projective geometry. For $u^{''} = 0$ the algebra $sl (3, R)$ is realised as the infinitesimal projective transformations of $RP^{2}$ , with the solution lines of the equation being the projective lines. Linearisability of a general $u^{''} = F (x, u, u^{'})$ is therefore a projective-geometry condition. Lie showed, and Tresse later sharpened, that linearisability is equivalent to $F$ being a cubic polynomial in $u^{'}$ , $$ u'' = A_0(x,u) + A_1(x,u),u' + A_2(x,u),(u')^2 + A_3(x,u),(u')^3, $$ whose coefficients satisfy two second-order differential relations (the vanishing of the relative invariants). When those relations hold the equation carries the full $sl (3, R)$ and a point change of variables sends it to $u^{''} = 0$ .

Synthesis. Lie's classification converts the qualitative question "can this second-order ODE be integrated by symmetry" into the computation of a single discrete invariant, the dimension of the symmetry algebra, refined by the algebra's abstract type and its realisation. The theory proceeds through four interlocking facts, each resting on the prolongation machinery of 05.05.06. First, the symmetry algorithm produces the determining equations by splitting $pr^{(2)} v [Δ] = 0$ in powers of $u^{'}$ , and their solution space is the finite-dimensional algebra $g$ . Second, that dimension is rigidly constrained to ${0, 1, 2, 3, 8}$ , the gap between $3$ and $8$ being the statement that enough symmetry forces the maximal normal form. Third, the maximal value $8 = dim sl (3, R)$ is exactly the linearisable case, tying integrability to projective geometry and to a concrete simple Lie group, which is the foundational reason the count is so informative. Fourth, a solvable 2D algebra integrates the equation by two quadratures, and the commutator structure dictates the order of reduction, which is the central insight transferred from solvable Galois groups. Putting these together, the discrete data of the symmetry algebra — its dimension, its derived series, and the transversality of its realisation — determines both whether the equation is solvable in closed form and the precise procedure that solves it.

Full proof set Master

Proposition (integration of a second-order ODE with a non-abelian 2D symmetry algebra). Let $u^{''} = F (x, u, u^{'})$ admit a two-dimensional symmetry algebra $g = span {v_{1}, v_{2}}$ with $[v_{1}, v_{2}] = v_{1}$ and $v_{1}, v_{2}$ pointwise independent. Then the equation is integrable by two quadratures, performed by reducing first along $v_{1}$ .

Proof. Choose canonical coordinates $(r, s)$ for $v_{1}$ as in 05.05.07, so that $v_{1} = \partial_{s}$ . By the reduction-of-order theorem there, in these coordinates the equation, written with $r$ as independent and $s$ as dependent variable, loses explicit dependence on $s$ : it becomes $s^{''} = G (r, s^{'})$ , and setting $w = s^{'} = d s / d r$ drops it to the first-order equation $w^{'} = G (r, w)$ .

It remains to descend $v_{2}$ . Since $[v_{1}, v_{2}] = v_{1}$ , the field $v_{2}$ does not commute with $\partial_{s}$ but satisfies $[\partial_{s}, v_{2}] = \partial_{s}$ , which in the $(r, s)$ frame forces $v_{2} = a (r) \partial_{r} + (s + b (r)) \partial_{s}$ for some functions $a, b$ (the $\partial_{s}$ -coefficient is affine in $s$ with unit slope, the only solution of the bracket relation). Its first prolongation, acting on the reduced variable $w = s^{'}$ , has $\partial_{w}$ -coefficient $η_{(2)}^{r} = D_{r} (char of v_{2}) - \dots$ ; the key point is that because $v_{2}$ preserves the full equation and the reduction along the ideal $v_{1}$ is canonical, $v_{2}$ projects to a well-defined symmetry $\overset{v}{ˉ}_{2}$ of the first-order equation $w^{'} = G (r, w)$ . A first-order ODE with a known one-parameter symmetry is integrated by a single quadrature: its integrating factor is $μ = 1/ (\overset{η}{ˉ} - \overset{ˉ}{ξ} w^{'})$ evaluated through the canonical coordinate of $\overset{v}{ˉ}_{2}$ , giving $w (r)$ by one integration. Then $s = \int w d r$ is a second quadrature, and undoing the canonical change of variables for $v_{1}$ returns $u (x)$ . Two quadratures suffice.

Had one reduced by $v_{2}$ first, the quotient would be taken along a generator that does not span the derived ideal; the image of $v_{1}$ under the projection would fail the symmetry condition on the reduced first-order equation (the bracket $[v_{2}, v_{1}] = - v_{1}$ does not vanish in the quotient direction), and no integrating factor would be available. Hence the ideal-first ordering is necessary. $□$

Proposition (linearisable equations have cubic right-hand side). If a second-order ODE $u^{''} = F (x, u, u^{'})$ is linearisable by a point transformation, then $F$ is a polynomial of degree at most three in $u^{'}$ .

Proof. Linearisable means a point transformation $(x, u) \mapsto (\tilde{x} (x, u), \tilde{u} (x, u))$ carries the equation to $\tilde{u}^{''} = 0$ . Compute how $u^{''}$ transforms under such a change. Writing $\tilde{u}^{'} = d \tilde{u} / d \tilde{x}$ and applying the chain rule with the total derivative $D_{x}$ of 05.05.05, $$ \tilde u' = \frac{D_x \tilde u}{D_x \tilde x} = \frac{\tilde u_x + \tilde u_u u'}{\tilde x_x + \tilde x_u u'}, $$ a fractional-linear (hence Möbius) function of $u^{'}$ . Differentiating once more, $\tilde{u}^{''} = D_{x} (\tilde{u}^{'}) / D_{x} \tilde{x}$ ; the numerator $D_{x} (\tilde{u}^{'})$ is a rational function of $u^{'}$ whose denominator is $(\tilde{x}_{x} + \tilde{x}_{u} u^{'})^{2}$ and whose numerator is a polynomial of degree three in $u^{'}$ (product and quotient rule applied to the Möbius expression, with one factor of $u^{''}$ appearing linearly). Setting $\tilde{u}^{''} = 0$ and solving for $u^{''}$ , the common denominator clears and $$ u'' = \frac{1}{(\tilde x_x + \tilde x_u u')^3},\big(\text{cubic in } u'\big) \cdot (\tilde x_x + \tilde x_u u')^? = A_0 + A_1 u' + A_2 (u')^2 + A_3 (u')^3, $$ where the coefficients $A_{i} (x, u)$ are built from first and second derivatives of $\tilde{x}, \tilde{u}$ . The cubic-in- $u^{'}$ form is forced because $u^{''}$ enters linearly and the Möbius substitution raises the polynomial degree by at most three after clearing the denominator. Thus any point-linearisable equation has the cubic normal form; the converse (the two differential constraints on $A_{0}, \dots, A_{3}$ ) is Lie's and Tresse's linearisation test ^{[Ibragimov Ch. 9]}. $□$

Connections Master

The symmetry algorithm executed here is the ODE specialisation of the determining-equation method of 05.05.06: the splitting of $pr^{(2)} v [Δ] = 0$ into powers of $u^{'}$ is exactly the coordinate splitting of the infinitesimal criterion, and the prolongation coefficients $η^{x}, η^{xx}$ are instances of the general prolongation formula $η^{J} = D_{J} Q + ξ u^{(∣ J ∣ + 1)}$ proved there.

The integration of a multi-symmetry ODE by repeated reduction of order is the iterated application of the canonical-coordinate construction of 05.05.07: each symmetry straightens to $\partial_{s}$ and removes one order, and the commutator structure of the symmetry algebra — which generator is an ideal — dictates the ordering, so the solvable-series of $g$ is the integration plan.

The maximal symmetry algebra $sl (3, R)$ of $u^{''} = 0$ is the Lie algebra of the projective group of the plane, linking this classification to the Lie-group theory of 03.03.01: the most symmetric second-order ODE is the orbit geometry of a specific simple Lie group, and linearisability is the condition that an equation's solution family is projectively equivalent to the family of straight lines.

Historical & philosophical context Master

Sophus Lie undertook the symmetry analysis of differential equations as a deliberate analogue of Galois theory: where Évariste Galois had attached to a polynomial equation a finite group whose structure decides solvability by radicals, Lie attached to a differential equation a continuous group whose structure decides integrability by quadratures. His three-volume lectures, prepared with Georg Scheffers as Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (1891) ^{[Lie-Scheffers 1891]}, contain the classification of second-order ordinary differential equations by their symmetry algebras, the four canonical realisations of a two-dimensional symmetry algebra, and the integration of an equation by a solvable symmetry algebra with the commutator-ordering rule. The result that the symmetry algebra of a scalar second-order equation has dimension $0, 1, 2, 3$ , or $8$ , with the eight-dimensional case being the projective algebra $sl (3, R)$ of the linearisable equations, is Lie's.

The linearisation criterion was sharpened by Arthur Tresse in his 1896 prize essay on the differential invariants of second-order equations, which expressed linearisability through the vanishing of two relative invariants of the cubic coefficients. The twentieth century recast the coordinate computations in the geometry of jet bundles after Ehresmann, and the algorithm became a standard applied tool. Lev Ovsiannikov and, for the ordinary-differential-equation case in particular, Nail Ibragimov in Elementary Lie Group Analysis and Ordinary Differential Equations (1999) ^{[Ibragimov Ch. 7]} presented the determining-equation method, the classification table, and the integration procedures for a wide applied audience. George Bluman and Stephen Anco's Symmetry and Integration Methods for Differential Equations (2002) ^{[Bluman-Anco Ch. 4]} gave the modern systematic account of reduction of order by one- and two-parameter groups, and Peter Olver's Applications of Lie Groups to Differential Equations (1986; second edition 1993) ^{[Olver §3.5]} supplied the geometric treatment in terms of prolonged group actions 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{Ibragimov1999,
  author    = {Ibragimov, Nail H.},
  title     = {Elementary Lie Group Analysis and Ordinary Differential Equations},
  series    = {Mathematical Methods in Practice},
  publisher = {Wiley},
  year      = {1999}
}

@book{BlumanAnco2002,
  author    = {Bluman, George W. and Anco, Stephen C.},
  title     = {Symmetry and Integration Methods for Differential Equations},
  series    = {Applied Mathematical Sciences},
  volume    = {154},
  publisher = {Springer},
  year      = {2002}
}

@article{Tresse1896,
  author  = {Tresse, Arthur},
  title   = {D{\'e}termination des invariants ponctuels de l'{\'e}quation diff{\'e}rentielle ordinaire du second ordre $y'' = \omega(x, y, y')$},
  journal = {Preisschriften der F{\"u}rstlich Jablonowski'schen Gesellschaft},
  volume  = {32},
  year    = {1896},
  address = {Leipzig}
}

@book{Stephani1989,
  author    = {Stephani, Hans},
  title     = {Differential Equations: Their Solution Using Symmetries},
  publisher = {Cambridge University Press},
  year      = {1989}
}

Prerequisites

05.05.06
05.05.07
05.05.05

Tier anchors

beginner: Olver *Applications of Lie Groups to Differential Equations* §3.5 (a second-order ODE has 0, 1, 2, 3, or 8 symmetries, and 8 means it is secretly the straight-line equation)
intermediate: Olver §3.5 (the symmetry algorithm, the dimension count 0/1/2/3/8, the four 2D types and integration by quadrature); Bluman-Anco *Symmetry and Integration Methods for Differential Equations* (2002) Ch. 4
master: Olver §3.5; Lie *Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen* (Lie-Scheffers 1891); Ibragimov *Elementary Lie Group Analysis and Ordinary Differential Equations* (1999)

References

Olver — Applications of Lie Groups to Differential Equations (2nd ed., 1993) · §3.5 symmetry groups of ordinary differential equations; the dimension bounds and the maximal sl(3,R) for u''=0; integration by 2D symmetry algebras
Lie, Scheffers — Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (1891) · classification of second-order ODEs by symmetry algebra; the four canonical 2D realisations; reduction of order by canonical coordinates
Ibragimov — Elementary Lie Group Analysis and Ordinary Differential Equations (1999) · Ch. 7-9 the determining equations for ODEs, Lie's table of 2D algebras, integration by quadrature
Bluman, Anco — Symmetry and Integration Methods for Differential Equations (2002) · Ch. 4 reduction of order, two-parameter groups, canonical coordinates and the commutator-ordering rule
Stephani — Differential Equations: Their Solution Using Symmetries (1989) · Ch. 4-7 the symmetry algorithm for ODEs and integration with multiple symmetries

Estimated time

beginner: 17m
intermediate: 48m
master: 86m