Prolongation of vector fields and the infinitesimal symmetry criterion

Intuition Beginner

A symmetry of an equation is a way of moving the picture that leaves the equation looking the same. Rotating a circle does not change the fact that it is a circle. In the same spirit, a symmetry of a differential equation is a transformation of the variables that turns every solution into another solution. If you know one solution and you know a symmetry, you get a second solution for free.

The catch is that a differential equation talks about derivatives, not just about points. So a transformation that mixes the input and output variables also reshuffles the slopes and curvatures. To check whether a candidate motion is a symmetry, you have to know how it acts on those derivatives too. The rule that extends a motion of the plain variables to a motion of the variables-with-their-derivatives is called prolongation. It promotes a transformation of points into a transformation of the whole stack of derivative data.

Once you can prolong, testing a symmetry becomes a calculation rather than a guess. You ask: does the prolonged motion keep you on the surface where the equation holds? Lie's insight was to do this one infinitesimal step at a time. Instead of a whole transformation, you study its velocity at the identity, a vector field. Asking that the prolonged velocity stay tangent to the equation gives a clean, mechanical test, and solving that test produces every continuous symmetry the equation has.

Visual Beginner

Imagine the graph of a solution drawn as a curve. A symmetry slides and bends the whole plane, dragging the curve to a new curve. For the motion to be a symmetry, the new curve must again be a solution graph. The arrows of the vector field show, at each point, the direction the plane is being nudged.

The picture also shows a small slope marker riding along with one point of the curve. As the point is nudged, the slope marker is nudged too, and prolongation is the rule that says exactly how. The plain arrows move the points; the extended arrows, which prolongation supplies, move the points together with their attached slopes and curvatures. A motion is a symmetry when the extended arrows never push you off the family of solution graphs.

Worked example Beginner

Take the simplest growth law: the rate of change of a quantity equals the quantity itself. A function solves it when its slope at every input equals its height there. The exponential function is one solution.

Now try a symmetry: multiply every output value by a fixed positive number, say three, while leaving the input alone. If a function had height ten at some input, the new function has height thirty there. What happens to the slope? Multiplying every height by three multiplies every slope by three as well, because stretching a graph vertically by three stretches all its slopes by three.

So the new slope equals three times the old slope, and the new height equals three times the old height. The old function satisfied slope equals height. Multiply both sides by three: three times the slope equals three times the height, which says the new slope equals the new height. The stretched function solves the same law.

What this tells us: scaling the output is a symmetry of this growth law, and to confirm it we had to track how the rule moves the slope, not only the height. That tracking of the slope is prolongation in miniature, and the test "the moved function still satisfies the law" is the symmetry criterion in miniature.

Check your understanding Beginner

Formal definition Intermediate+

Let $\pi :E\to X$ be a fibred manifold with coordinates $(x^i,u^{\alpha })$, $1\le i\le p$, $1\le \alpha \le q$, and let $J^k(E)$ be its $k$-jet bundle with coordinates $(x^i,u^{\alpha },u_J^{\alpha })$ for $1\le |J|\le k$, as in 05.05.05. A system of $k$-th order PDEs is the vanishing locus $$ \mathcal{S} = {, \Delta_\nu(x, u^{(k)}) = 0,\ \nu = 1, \dots, m ,} \subset J^k(E), $$ where $u^{(k)}$ abbreviates all jet coordinates up to order $k$; a section $f$ of $\pi$ solves the system when its prolongation ${\mathrm{pr}}^{(k)}f$ lands in $\mathcal{S}$. The development follows Olver ^{[Olver §2.3]} and Ovsiannikov ^{[Ovsiannikov Ch. 2]}.

A one-parameter symmetry group of $\mathcal{S}$ is a local one-parameter group $g_{\epsilon }:E\to E$ of fibre-preserving diffeomorphisms (acting on $(x,u)$) such that for every solution $f$ and small $\epsilon$, the transformed section $g_{\epsilon }\cdot f$ is again a solution. The associated infinitesimal generator is the vector field on $E$ whose phase flow (in the sense of 02.12.02) is $g_{\epsilon }$: $$ v = \xi^i(x, u),\frac{\partial}{\partial x^i} + \phi^\alpha(x, u),\frac{\partial}{\partial u^\alpha}. $$

Definition (prolongation of a vector field). The $k$-th prolongation of $v$ is the vector field on $J^k(E)$ $$ \mathrm{pr}^{(k)} v = \xi^i,\frac{\partial}{\partial x^i} + \phi^\alpha,\frac{\partial}{\partial u^\alpha} + \sum_{1 \le |J| \le k} \phi^\alpha_J,\frac{\partial}{\partial u^\alpha_J}, $$ whose flow is the prolongation ${\mathrm{pr}}^{(k)}{g}_{\epsilon }$ of the flow of $v$. Its coefficients are given by the prolongation formula ^{[Olver §2.3]} $$ \phi^\alpha_J = D_J!\left(\phi^\alpha - \xi^i u^\alpha_i\right) + \xi^i, u^\alpha_{J,i}, \qquad Q^\alpha := \phi^\alpha - \xi^i u^\alpha_i, $$ where $D_J=D_1^{j_1}\cdots D_p^{j_p}$ is the iterated total derivative of 05.05.05 and $Q^{\alpha }$ is the characteristic of $v$. The vector field $v_Q=Q^{\alpha }{\partial }_{u^{\alpha }}$ on $E$, vertical with respect to $\pi$, is the evolutionary (or characteristic) form of $v$; its prolongation has the simpler coefficients ${\varphi }_J^{\alpha }(v_Q)=D_J{Q}^{\alpha }$.

Definition (infinitesimal symmetry). The field $v$ is an infinitesimal symmetry of $\mathcal{S}$ when ${\mathrm{pr}}^{(k)}v$ is tangent to $\mathcal{S}$, equivalently $$ \mathrm{pr}^{(k)} v,[\Delta_\nu] = 0 \quad\text{whenever}\quad \Delta_1 = \dots = \Delta_m = 0, $$ that is, on the submanifold $\mathcal{S}$. This is Lie's infinitesimal symmetry criterion. Splitting the criterion in the free jet coordinates $u_J^{\alpha }$ that survive after eliminating the equations ${\Delta }_{\nu }=0$ produces the determining equations: an overdetermined linear homogeneous PDE system for the unknown coefficient functions ${\xi }^i(x,u)$ and ${\varphi }^{\alpha }(x,u)$.

A non-example fixes the meaning. The bare field ${\xi }^i{\partial }_{x^i}+{\varphi }^{\alpha }{\partial }_{u^{\alpha }}$, applied to ${\Delta }_{\nu }$ without prolongation, ignores how the derivative coordinates $u_J^{\alpha }$ transform; it is not tangent to $\mathcal{S}$ in general and gives a wrong criterion. The prolongation is precisely the correction that makes tangency on $J^k(E)$ the right condition.

Counterexamples to common slips

• Tangency is required only on $\mathcal{S}$, not on all of $J^k(E)$. Demanding ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }]=0$ identically is strictly stronger and rejects genuine symmetries; the correct statement allows ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }]={\lambda }_{\nu }^{\mu }{\Delta }_{\mu }$ for smooth multipliers ${\lambda }_{\nu }^{\mu }$.
• The characteristic $Q^{\alpha }={\varphi }^{\alpha }-{\xi }^i{u}_i^{\alpha }$ is not ${\varphi }^{\alpha }$; dropping the ${\xi }^i{u}_i^{\alpha }$ term collapses the evolutionary representative and corrupts every ${\varphi }_J^{\alpha }$. The two fields $v$ and $v_Q$ differ by the total field ${\xi }^i{D}_i$, which is tangent to all prolongations and so does not affect the criterion, but the bookkeeping must use $Q$.
• The prolongation order must match the equation order $k$: prolonging to order $k-1$ leaves $\mathrm{pr}v[{\Delta }_{\nu }]$ undefined on the top derivatives, while prolonging past $k$ adds vanishing terms but wastes computation.

Key theorem with proof Intermediate+

Theorem (prolongation formula). Let $v={\xi }^i{\partial }_{x^i}+{\varphi }^{\alpha }{\partial }_{u^{\alpha }}$ generate the one-parameter group $g_{\epsilon }$ on $E$, with characteristic $Q^{\alpha }={\varphi }^{\alpha }-{\xi }^i{u}_i^{\alpha }$. Then the prolonged generator ${\mathrm{pr}}^{(k)}v$, defined as the infinitesimal generator of the prolonged flow ${\mathrm{pr}}^{(k)}{g}_{\epsilon }$ on $J^k(E)$, has coefficient of ${\partial }_{u_J^{\alpha }}$ equal to $$ \phi^\alpha_J = D_J Q^\alpha + \xi^i, u^\alpha_{J,i}, \qquad 1 \le |J| \le k. $$

Proof. It suffices to treat the evolutionary field $v_Q=Q^{\alpha }{\partial }_{u^{\alpha }}$ and then restore the horizontal part. Write $v=v_Q+v_H$ with $v_H={\xi }^i({\partial }_{x^i}+u_i^{\alpha }{\partial }_{u^{\alpha }}+\dots )={\xi }^i{D}_i$ the total field, the truncation of ${\xi }^i{D}_i$ to $J^k(E)$. The total field is tangent to every prolongation: along ${\mathrm{pr}}^{(k)}f$ the flow of ${\xi }^i{D}_i$ acts by moving the base point, and the chain-rule identity of 05.05.05 gives $D_i(\mathrm{pr}f)=\mathrm{pr}({\partial }_{i}f)$, so ${\xi }^i{D}_i$ prolongs to itself, with ${\partial }_{u_J^{\alpha }}$-coefficient ${\xi }^i{u}_{J,i}^{\alpha }$. Thus $$ \phi^\alpha_J = \phi^\alpha_J(v_Q) + \xi^i u^\alpha_{J,i}, $$ and the claim reduces to ${\varphi }_J^{\alpha }(v_Q)=D_J{Q}^{\alpha }$.

For the evolutionary field, its flow $g_{\epsilon }^Q$ fixes the base coordinate $x$ and evolves the fibre by ${\partial }_{\epsilon }{u}^{\alpha }=Q^{\alpha }$. Prolonging means differentiating the transformed section. Let $f_{\epsilon }=g_{\epsilon }^Q\cdot f$, a one-parameter family of sections with $f_0=f$ and ${\partial }_{\epsilon }{f}^{\alpha }{|}_0=Q^{\alpha }(x,{\mathrm{pr}}^{(k)}f)$. The coordinate $u_J^{\alpha }$ of the prolongation evaluates to ${\partial }_J{f}_{\epsilon }^{\alpha }$, so the ${\partial }_{u_J^{\alpha }}$-component of the prolonged generator is $$ \phi^\alpha_J(v_Q) = \frac{\partial}{\partial \varepsilon}\bigg|{0} \bigl(\partial_J f\varepsilon^\alpha\bigr) = \partial_J!\left(\frac{\partial}{\partial \varepsilon}\bigg|{0} f\varepsilon^\alpha\right) = \partial_J\bigl(Q^\alpha(x, \mathrm{pr}^{(k)} f)\bigr), $$ the interchange of ${\partial }_{\epsilon }$ and the spatial ${\partial }_J$ being legitimate by smoothness of the joint dependence. The key identity of 05.05.05 states that ${\partial }_i$ of a jet function evaluated along a prolongation equals the total derivative $D_i$ of that function evaluated along the next prolongation. Applying it $|J|$ times, $$ \partial_J\bigl(Q^\alpha(x, \mathrm{pr}^{(k)} f)\bigr) = \bigl(D_J Q^\alpha\bigr)(x, \mathrm{pr}^{(k+|J|)} f). $$ Reading off the coefficient as a function on the jet bundle gives ${\varphi }_J^{\alpha }(v_Q)=D_J{Q}^{\alpha }$. Combining with the horizontal contribution yields ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^i{u}_{J,i}^{\alpha }$. $\square$

Bridge. This formula is the computational heart of Lie's method. It builds toward the determining equations, where each ${\varphi }_J^{\alpha }$ is expanded in $\xi ,\varphi$ and the criterion is split coordinate by coordinate. It appears again in 05.00.04 (Noether's theorem), where the characteristic $Q^{\alpha }$ of a variational symmetry is exactly the datum paired with the conserved current, so that the evolutionary form computed here is the bridge from symmetry to conservation law. The construction reuses the total derivative of 05.05.05 in a new guise: there $D_i$ advanced a jet coordinate, here $D_J$ propagates the characteristic up the derivative stack. The unifying content is that prolongation of a symmetry, prolongation of an equation, and the total derivative are one operator seen from three sides, and the determining system is what that operator produces when tangency is imposed coordinate-wise.

Exercises Intermediate+

Lean formalization Intermediate+

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

import Mathlib.Geometry.Manifold.ContMDiff.Basic
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs

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)

/-- The characteristic Q^α = φ^α − ξ^i u^α_i (depends on first jets u^α_i). -/
noncomputable def characteristic
    (v : Generator p q) (x : Fin p → ℝ) (u : Fin q → ℝ)
    (ux : Fin q → Fin p → ℝ) : Fin q → ℝ :=
  fun α => v.phi x u α - ∑ i, v.xi x u i * ux α i

/-- Prolongation coefficient φ^α_J = D_J Q^α + ξ^i u^α_{J,i}.
    Stated against the (absent) jet/total-derivative apparatus. -/
noncomputable def prolongCoeff
    (v : Generator p q) : ℕ → ℝ :=
  fun _ => sorry  -- D_J(characteristic) + ξ^i u^α_{J,i}

/-- Lie's infinitesimal criterion: pr^(k) v is tangent to Δ = 0. -/
theorem infinitesimal_symmetry_criterion
    (v : Generator p q) : True := by
  trivial  -- placeholder: pr^(k) v [Δ_ν] = 0 on {Δ = 0}

The sorry and placeholders rest on the jet-bundle gap of 05.05.05 plus the prolongation-of-a-vector-field construction and the tangency criterion. Each sits on the existing flow and manifold libraries and is a Mathlib-contribution-sized target named in Mathlib gap analysis.

Advanced results Master

The prolongation formula converts symmetry analysis into linear algebra over a function space. Two worked systems show the algorithm producing a Lie algebra of symmetries, and the structural results frame the general theory.

The heat equation. For $u_t=u_{xx}$ on $J^2({\mathbb{R}}^2,\mathbb{R})$ take $v=\xi {\partial }_x+\tau {\partial }_t+\varphi {\partial }_u$ with $\xi ,\tau ,\varphi$ functions of $(x,t,u)$. The criterion is ${\varphi }^t={\varphi }^{xx}$ on the variety $u_t=u_{xx}$. Expanding the prolonged coefficients through the formula ${\varphi }_J^{\alpha }=D_{J}Q+{\xi }^i{u}_{J,i}^{\alpha }$ and collecting powers of the free coordinates $u_x,u_{xx},u_{xt}$ splits the criterion into the determining equations. Solving them (the standard computation in Olver ^{[Olver §2.4]} and Bluman-Kumei ^{[Bluman-Kumei Ch. 4]}) yields a six-dimensional point-symmetry algebra plus an infinite-dimensional piece. The finite generators are $$ \begin{aligned} &v_1 = \partial_x, \quad v_2 = \partial_t, \quad v_3 = u,\partial_u, \quad v_4 = x,\partial_x + 2t,\partial_t, \ &v_5 = 2t,\partial_x - x u,\partial_u \quad(\text{Galilean boost}), \ &v_6 = 4tx,\partial_x + 4t^2,\partial_t - (x^2 + 2t)u,\partial_u \quad(\text{projective / dilation}), \end{aligned} $$ together with $v_{\beta }=\beta (x,t){\partial }_u$ for every solution $\beta$ of $u_t=u_{xx}$ (the linearity / superposition symmetry).

Here $v_4$ is the scaling $x\mapsto \lambda x$, $t\mapsto {\lambda }^{2}t$ that reflects the heat equation's parabolic weight, and $v_5$ is the Galilean symmetry that drags a solution along a moving frame while compensating by the Gaussian factor $e^{-(xv+v^{2}t)/4}$ — applied to the fundamental solution it generates the full Gaussian family. The commutators close: $[v_1,v_4]=v_1$, $[v_1,v_5]=-v_3-\frac{1}{2}(\dots )$ in the standard normalisation, and the finite part is isomorphic to a central extension of $\mathfrak{sl}(2,\mathbb{R})⋉\mathfrak{h}$ with the Heisenberg algebra $\mathfrak{h}$ generated by $v_1,v_5,v_3$.

The harmonic oscillator. For the ODE $\ddot{u}+u=0$ on $J^2(\mathbb{R},\mathbb{R})$ (time $t$, state $u$) the determining equations admit an eight-dimensional symmetry algebra $\mathfrak{sl}(3,\mathbb{R})$, the maximal dimension for a scalar second-order ODE ^{[Olver §2.4]}. It contains the two solution-translation symmetries $u\mapsto u+\cos t$ and $u\mapsto u+\sin t$, the scaling $u\mapsto \lambda u$, the time-translation ${\partial }_t$, and a copy of $\mathfrak{sl}(2,\mathbb{R})$ acting projectively. That every linear second-order ODE carries $\mathfrak{sl}(3,\mathbb{R})$ is Lie's classification result; the symmetry algebra detects linearisability.

Structure of the symmetry algebra. For a non-degenerate system the infinitesimal symmetries form a finite-dimensional Lie algebra $\mathfrak{g}$ under the bracket of vector fields, because prolongation is a Lie-algebra homomorphism: ${\mathrm{pr}}^{(k)}[v,w]=[{\mathrm{pr}}^{(k)}v,{\mathrm{pr}}^{(k)}w]$. The symmetry group is the corresponding (local) Lie group of point transformations, recovering a one-parameter group from each generator by integrating its flow as in 02.12.02. The characteristics $Q^{\alpha }$ of the generators feed directly into Noether's theorem 05.00.04 when the system is variational: a symmetry of the action produces a conservation law $D_i{P}^i=0$ whose density is built from $Q^{\alpha }$ and the Lagrangian.

Synthesis. Lie's method is the conversion of an invariance question into a solvable linear system, and the prolongation formula is the conversion factor. The argument runs along four linked stages, each carried by the total derivative of 05.05.05. First, a one-parameter symmetry group is differentiated at the identity into an infinitesimal generator $v$, replacing a transformation by a vector field. Second, $v$ is prolonged by ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^i{u}_{J,i}^{\alpha }$, so that the action on derivative coordinates is computed mechanically from the single characteristic $Q^{\alpha }$. Third, tangency ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }]=0$ on $\mathcal{S}$ is imposed and split coordinate-wise into the overdetermined determining equations, which are linear in $\xi ,\varphi$ regardless of the nonlinearity of $\Delta$. Fourth, the solution space of the determining system is a finite-dimensional Lie algebra whose generators integrate back to the symmetry group and whose characteristics feed Noether's theorem. The heat equation and the harmonic oscillator instantiate every stage, and the same operator $D_i$ that defined the jet-bundle contact ideal here propagates the characteristic, splits the criterion, and closes the bracket.

Full proof set Master

Proposition (equivalence of the infinitesimal criterion and group invariance). Let $\mathcal{S}=\{{\Delta }_{\nu }=0\}\subset J^k(E)$ be a non-degenerate system (maximal rank: $d{\Delta }_1,\dots ,d{\Delta }_m$ independent on $\mathcal{S}$). A connected one-parameter group $g_{\epsilon }$ with generator $v$ maps solutions to solutions if and only if ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }]=0$ on $\mathcal{S}$ for each $\nu$.

Proof. The group $g_{\epsilon }$ maps solutions to solutions exactly when its prolongation ${\mathrm{pr}}^{(k)}{g}_{\epsilon }$ preserves $\mathcal{S}$ as a set: a section $f$ solves the system iff ${\mathrm{pr}}^{(k)}f(X)\subset \mathcal{S}$, and $g_{\epsilon }$ carries the prolongation of $f$ to the prolongation of $g_{\epsilon }\cdot f$ by functoriality of jets, so $g_{\epsilon }\cdot f$ solves iff ${\mathrm{pr}}^{(k)}{g}_{\epsilon }(\mathcal{S})\subset \mathcal{S}$.

($\Leftarrow$) Suppose ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }]=0$ on $\mathcal{S}$. Fix a point $z\in \mathcal{S}$ and let $z_{\epsilon }={\mathrm{pr}}^{(k)}{g}_{\epsilon }(z)$. The functions $h_{\nu }(\epsilon ):={\Delta }_{\nu }(z_{\epsilon })$ satisfy $h_{\nu }(0)=0$ and $$ \frac{d}{d\varepsilon} h_\nu(\varepsilon) = \mathrm{pr}^{(k)} v[\Delta_\nu]\big(z_\varepsilon\bigr). $$ By hypothesis the right side vanishes whenever $z_{\epsilon }\in \mathcal{S}$, i.e. whenever all $h_{\mu }(\epsilon )=0$. By maximal rank we may write ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }]={\sum }_{\mu }{\lambda }_{\nu }^{\mu }{\Delta }_{\mu }$ for smooth multipliers ${\lambda }_{\nu }^{\mu }$ in a neighbourhood of $\mathcal{S}$, so $h_{\nu }^{\prime }={\sum }_{\mu }{\lambda }_{\nu }^{\mu }(z_{\epsilon })h_{\mu }$. This is a linear homogeneous ODE system in $\epsilon$ with $h(0)=0$; by uniqueness $h\equiv 0$, hence $z_{\epsilon }\in \mathcal{S}$ for all $\epsilon$. Thus ${\mathrm{pr}}^{(k)}{g}_{\epsilon }$ preserves $\mathcal{S}$.

($\Rightarrow$) Suppose ${\mathrm{pr}}^{(k)}{g}_{\epsilon }(\mathcal{S})\subset \mathcal{S}$ for all small $\epsilon$. For $z\in \mathcal{S}$, ${\Delta }_{\nu }({\mathrm{pr}}^{(k)}{g}_{\epsilon }(z))=0$ for all $\epsilon$, so differentiating at $\epsilon =0$ gives ${\mathrm{pr}}^{(k)}v[{\Delta }_{\nu }](z)=0$. Since $z\in \mathcal{S}$ was arbitrary, the criterion holds on $\mathcal{S}$. The reduction from a finite transformation to the infinitesimal condition is exactly Lie's theorem that invariance of a set under a connected group is equivalent to tangency of the generator ^{[Olver §2.3]}. $\square$

Proposition (prolongation is a Lie-algebra homomorphism). For vector fields $v,w$ on $E$ and each $k$, ${\mathrm{pr}}^{(k)}[v,w]=[{\mathrm{pr}}^{(k)}v,{\mathrm{pr}}^{(k)}w]$. Consequently the infinitesimal symmetries of a system form a Lie algebra.

Proof. Both sides are vector fields on $J^k(E)$. Prolongation is defined by the requirement that the flow of ${\mathrm{pr}}^{(k)}v$ be the prolongation ${\mathrm{pr}}^{(k)}(\exp \epsilon v)$ of the flow of $v$, and prolongation of diffeomorphisms is functorial: ${\mathrm{pr}}^{(k)}(\Phi \circ \Psi )={\mathrm{pr}}^{(k)}\Phi \circ {\mathrm{pr}}^{(k)}\Psi$, since the $k$-jet of a composite section is determined by the $k$-jets of the factors. The Lie bracket of two fields is the second-order term of the commutator of their flows, $$ [v, w] = \frac{d}{ds}\bigg|{0} (\Psi{-\sqrt s})*!(\text{flow}), $$ more precisely $[v,w] = \tfrac{1}{2}\tfrac{d^2}{d\varepsilon^2}\big|0 (\Phi^w{-\varepsilon}\Phi^v{-\varepsilon}\Phi^w_{\varepsilon}\Phi^v_{\varepsilon})$.Applying$\mathrm{pr}^{(k)}$totheflowsandusingfunctoriality,theprolongedcommutatorofflowsisthecommutatoroftheprolongedflows,sothesecond-ordergeneratoroneachsideagrees:$\mathrm{pr}^{(k)}[v,w] = [\mathrm{pr}^{(k)} v, \mathrm{pr}^{(k)} w]$.Nowif$v, w$areinfinitesimalsymmetries,thenon$\mathcal{S}$ $$ \mathrm{pr}^{(k)}[v,w][\Delta_\nu] = \mathrm{pr}^{(k)} v\bigl[\mathrm{pr}^{(k)} w[\Delta_\nu]\bigr] - \mathrm{pr}^{(k)} w\bigl[\mathrm{pr}^{(k)} v[\Delta_\nu]\bigr]. $$ Each inner bracket equals ${\sum }_{\mu }{\lambda }^{\mu }{\Delta }_{\mu }$ for smooth multipliers (criterion plus maximal rank), and applying the other prolonged field and restricting to $\mathcal{S}$ leaves a combination of the ${\Delta }_{\mu }$, which vanishes on $\mathcal{S}$. Hence $[v,w]$ satisfies the criterion and the symmetries are closed under bracket. $\square$

Connections Master

The jet bundle and total derivative of 05.05.05 are the substrate of this unit: the prolongation formula ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^i{u}_{J,i}^{\alpha }$ is the same total derivative $D_J$ acting on the characteristic, and the equation submanifold $\mathcal{S}\subset J^k(E)$ and its prolongations ${\mathcal{S}}^{(l)}$ are exactly the total-derivative-generated varieties defined there.

The characteristics $Q^{\alpha }$ produced here are the inputs to 05.00.04 (Noether's theorem): a variational symmetry of an action, expressed in evolutionary form, pairs with the Euler-Lagrange expression to yield a conserved current whose divergence vanishes as a total-derivative identity, so the symmetry-to-conservation-law correspondence runs through the characteristic computed in this unit.

The infinitesimal generators integrate to one-parameter groups by the phase-flow construction of 02.12.02, and the resulting finite-dimensional symmetry algebra is a Lie algebra whose exponential is a Lie group in the sense of 03.03.01; the determining equations are thus the linear shadow of an underlying Lie-group action on the equation manifold.

Historical & philosophical context Master

Sophus Lie created the theory of continuous transformation groups in the 1870s and 1880s expressly to do for differential equations what Galois had done for polynomial equations: to organise their solvability through symmetry. His 1881 paper Über die Integration durch bestimmte Integrale einer Klasse linearer partieller Differentialgleichungen in the Archiv for Mathematik og Naturvidenskab ^{[Lie 1881]} worked out the invariance of differential equations under one-parameter groups and introduced the prolongation of the group action to the derivative variables, written out by hand in coordinates. The infinitesimal criterion — that invariance of the equation is equivalent to the tangency of the prolonged generator — is Lie's, and his determining equations were the first systematic algorithm for finding all continuous symmetries of a given equation.

The twentieth century recast Lie's coordinate computations in the geometric language of jet bundles, following Ehresmann's 1951 abstraction of jets. Lev Ovsiannikov's 1982 Group Analysis of Differential Equations ^{[Ovsiannikov Ch. 2]} systematised the determining-equation method for applications in continuum mechanics and gas dynamics, and George Bluman and Sukeyuki Kumei's 1989 Symmetries and Differential Equations ^{[Bluman-Kumei Ch. 2]} developed the algorithm and its extensions (nonclassical and potential symmetries) for a broad applied audience. Peter Olver's Applications of Lie Groups to Differential Equations (1986; second edition 1993) ^{[Olver §2.3]} gave the modern presentation in terms of the characteristic, the evolutionary representative, and the prolongation formula proved through the total derivative, which is the form used in this unit. Lie's classification that every linear second-order scalar ODE admits the eight-dimensional algebra $\mathfrak{sl}(3,\mathbb{R})$, instantiated above by the harmonic oscillator, remains the prototype of using a symmetry algebra to detect structural features such as linearisability.

Bibliography Master

@article{Lie1881Integration,
  author  = {Lie, Sophus},
  title   = {{\"U}ber die Integration durch bestimmte Integrale einer Klasse linearer partieller Differentialgleichungen},
  journal = {Archiv for Mathematik og Naturvidenskab},
  volume  = {6},
  year    = {1881},
  pages   = {328--368}
}

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

@book{Hydon2000,
  author    = {Hydon, Peter E.},
  title     = {Symmetry Methods for Differential Equations: A Beginner's Guide},
  series    = {Cambridge Texts in Applied Mathematics},
  publisher = {Cambridge University Press},
  year      = {2000}
}

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