02.12.12 · analysis / ode

First integrals / conserved quantities

shipped3 tiersLean: none

Anchor (Master): Arnold *Mathematical Methods of Classical Mechanics* (2nd ed., Springer GTM 60, 1989) Ch.4 §20 and Ch.10 §49 (Liouville-Arnold integrability theorem and the action-angle coordinates produced by $n$ independent first integrals in involution on a $2n$-dimensional symplectic manifold); Liouville 1855 *Note sur l'intégration des équations différentielles de la dynamique* (J. Math. Pures Appl. (2) 20, 137-138) — originator of the integrability theorem; Jacobi 1843 *Vorlesungen über Dynamik* (Berlin lectures, published posthumously 1866 by Clebsch) — originator of the canonical theory and the last-multiplier method; Noether 1918 *Invariante Variationsprobleme* (Nachr. Königl. Ges. Wiss. Göttingen Math.-Phys. Kl., 235-257) — symmetry-to-conservation correspondence; Poincaré 1892-1899 *Les méthodes nouvelles de la mécanique céleste* (Gauthier-Villars; volumes I-III) — non-existence of additional analytic first integrals for the perturbed three-body problem; Hörmander *The Analysis of Linear Partial Differential Operators I* (Springer Grundlehren 256, 1983) §8.1 (first integrals as solutions of the homogeneous first-order linear PDE $Xu = 0$)

Intuition [Beginner]

Think of a marble rolling without friction on a curved bowl. As the marble swings up one side, it slows; as it swings back down, it speeds up. Throughout the motion something stays the same — the sum of kinetic energy and gravitational potential energy. The marble's position and velocity change every moment, but this combined quantity does not. A function on phase space that stays put as the system evolves is called a first integral, or a conserved quantity. It labels the trajectory the marble is on: every marble starting with the same total energy traces the same curve in position-velocity space, just at different speeds and starting points.

The idea is older than calculus made it rigorous. Galileo noticed that a pendulum's amplitude on one side matches the amplitude on the other; Huygens turned this into an energy statement; Newton's laws made the conservation a derived consequence. A first integral is the modern way of saying "this is the thing the dynamics preserves". Once you have one, the motion is constrained: instead of wandering through the full phase space, the trajectory is confined to a level set of the conserved quantity, one dimension smaller. Find another independent conserved quantity and the trajectory is confined further. Find enough of them and the motion becomes geometrically simple.

Why does it matter? A vector field on a manifold of dimension $n$ defines an ODE; solving the ODE means describing the trajectories. First integrals are the geometric way to slice the phase space into invariant subsets, each one a possible trajectory or family of trajectories. Locally — near any point where the field does not vanish — there are exactly $n - 1$ functionally independent first integrals, just enough to label the trajectories by their starting position transverse to the flow. Globally the picture is harder and is the subject of integrability theory.

Visual [Beginner]

A schematic phase portrait on the plane shows a family of closed loops nested around the origin — the level curves of a conserved quantity, say the energy of an undamped pendulum. Arrows along each loop indicate the direction of motion. The arrows always run tangent to the loops: no trajectory crosses a level curve, and every trajectory stays on exactly one. A small inset labels three loops with their energy values $E_{1} < E_{2} < E_{3}$ , illustrating that the conserved quantity labels which loop a particular initial condition belongs to.

The picture conveys the structural content: a first integral is a function constant along trajectories, and its level sets are unions of trajectories. The dynamics moves points around inside a level set but never off it. Counting: a first integral cuts the phase space dimension by one. Several independent first integrals cut it by several, leaving the trajectories confined to a small invariant subset whose dimension and shape are read directly from the level-set geometry.

Worked example [Beginner]

Consider the simple harmonic oscillator on the plane: a unit mass on a spring with spring constant one, modelled by the planar ODE $\overset{x}{˙} = v$ , $\overset{v}{˙} = - x$ . The phase space is $R^{2}$ with coordinates $(x, v)$ .

Step 1. Guess the conserved quantity. The total energy is $E (x, v) = \frac{1}{2} v^{2} + \frac{1}{2} x^{2}$ , the sum of kinetic and potential. Energy is the natural candidate.

Step 2. Verify $E$ is conserved along trajectories. Compute the rate of change of $E$ along a solution. Differentiating $E (x (t), v (t)) = \frac{1}{2} v (t)^{2} + \frac{1}{2} x (t)^{2}$ in time gives $v \overset{v}{˙} + x \overset{x}{˙}$ , and substituting the ODE $\overset{x}{˙} = v$ , $\overset{v}{˙} = - x$ produces $v \cdot (- x) + x \cdot v = 0$ . The two terms cancel exactly. The function $E$ is conserved along every trajectory of the oscillator.

Step 3. Read off the trajectories. The level set ${E = c}$ is the circle $x^{2} + v^{2} = 2 c$ of radius $2 c$ for $c > 0$ , and the single point $(0, 0)$ for $c = 0$ . Every trajectory of the oscillator lies on exactly one such circle. The motion along a circle is uniform rotation with angular frequency one.

Step 4. Count the conserved quantities. The phase space has dimension $2$ . One independent conserved quantity $E$ confines a trajectory to a one-dimensional level set — a single circle. This is the maximal number of independent first integrals: a second independent first integral would confine the trajectory to a point, contradicting the actual circular motion.

What this tells us: a single first integral plus a non-vanishing vector field on a two-dimensional phase space gives a complete description of the trajectories. The energy is the conserved quantity, the level sets are the trajectories, and the dynamics is uniform motion along each level set. The pattern generalises: on a phase space of dimension $n$ , the maximal number of independent first integrals near a regular point is $n - 1$ , and finding them all reduces the dynamics to motion along a single labelled curve.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

On a phase space of dimension $n$ , what is the maximum number of functionally independent first integrals near a regular point of a vector field $X$ ?

A. $1$
B. $n - 1$
C. $n$
D. There is no upper bound

Hint

A trajectory through a regular point has dimension one. Each independent first integral confines the trajectory to a level set one dimension smaller than the ambient space.

Answer

B. $n - 1$ . A trajectory has dimension one, and $n - 1$ independent first integrals confine the trajectory to a one-dimensional intersection of level sets — exactly the dimension of a trajectory. An $n$ -th independent first integral would confine the trajectory to a point, contradicting the non-vanishing of $X$ at a regular point. The number $n - 1$ is attained: the rectification theorem produces $n - 1$ first integrals near every regular point.

Formal definition [Intermediate+]

Let $M$ be a smooth $n$ -dimensional manifold and $X$ a $C^{1}$ vector field on $M$ . A first integral (or conserved quantity) of $X$ on an open set $U \subseteq M$ is a $C^{1}$ function $ϕ : U \to R$ such that $$ X(\phi)(p) = d\phi_p(X(p)) = 0 \qquad \text{for every } p \in U. $$ Here $X (ϕ)$ denotes the Lie derivative of $ϕ$ along $X$ , which agrees with the directional derivative $d ϕ (X)$ since $ϕ$ is scalar-valued. Equivalently, $ϕ$ is constant along every integral curve of $X$ in $U$ : if $γ : I \to U$ satisfies $\overset{γ}{˙} (t) = X (γ (t))$ , then $$ \frac{d}{dt} \phi(\gamma(t)) = d\phi_{\gamma(t)}(\dot\gamma(t)) = d\phi_{\gamma(t)}(X(\gamma(t))) = 0, $$ so $ϕ \circ γ$ is constant on $I$ .

A collection of first integrals $ϕ^{1}, \dots, ϕ^{k}$ on $U$ is functionally independent at a point $p \in U$ when the differentials $d ϕ_{p}^{1}, \dots, d ϕ_{p}^{k}$ are linearly independent elements of the cotangent space $T_{p}^{*} M$ . The collection is functionally independent on $U$ when it is functionally independent at every point of $U$ . Equivalently, the map $Φ = (ϕ^{1}, \dots, ϕ^{k}) : U \to R^{k}$ is a submersion on $U$ .

A complete set of local first integrals at a regular point $p$ of $X$ is a collection $ϕ^{1}, \dots, ϕ^{n - 1}$ of functionally independent first integrals on a neighbourhood of $p$ . The maximal number $n - 1$ is dictated by the dimension count: a trajectory through a regular point is a one-dimensional integral curve, and $n - 1$ independent first integrals cut the ambient $n$ -dimensional space down to a one-dimensional intersection of level sets, exactly the dimension of a trajectory.

In the Hamiltonian setting (foreshadowed here, treated fully in chapter 05), a first integral of a Hamiltonian flow $X_{H}$ on a symplectic manifold $(M^{2 n}, ω)$ is a function $ϕ$ with $X_{H} (ϕ) = 0$ , equivalently ${ϕ, H} = 0$ where ${\cdot, \cdot}$ is the Poisson bracket. The Hamiltonian itself is always a first integral of its own flow (energy conservation), and two first integrals are said to be in involution when their Poisson bracket vanishes — the key hypothesis of the Liouville-Arnold integrability theorem.

Counterexamples to common slips

A function constant on a single trajectory need not be a first integral. The defining property requires constancy along every trajectory in the domain, not just one. A first integral is a global (on its domain) attribute of the function, not a per-trajectory observation.
Functional independence is a stronger condition than ordinary linear independence of values. Two first integrals can take linearly independent values at a point while having proportional differentials there, in which case they are functionally dependent: one is locally a function of the other.
The number $n - 1$ is the maximum near a regular point on a generic manifold. At an equilibrium, the rectification theorem fails and the count can be different — every $C^{1}$ function constant in a neighbourhood of the equilibrium is a first integral, so first integrals can be uncountably many but functionally dependent. Globally, the count can be smaller than $n - 1$ : ergodic flows on a manifold admit no non-constant first integral at all.
A Hamiltonian's status as a first integral of its own flow is automatic from the structure $X_{H} (ϕ) = {ϕ, H}$ and ${H, H} = 0$ . The substantive content is the search for other first integrals, which the Poisson-bracket criterion converts into the algebraic question of which functions Poisson-commute with $H$ .

Key theorem with proof [Intermediate+]

Theorem (local existence of first integrals near a regular point; Arnold ODE Ch.2 §10). Let $M$ be a smooth $n$ -manifold and $X$ a $C^{k}$ vector field on $M$ for some $k \geq 1$ . For every regular point $p$ of $X$ there is a neighbourhood $U$ of $p$ on which there exist $n - 1$ functionally independent $C^{k}$ first integrals $ϕ^{2}, \dots, ϕ^{n} : U \to R$ of $X$ . The intersection of level sets ${ϕ^{2} = c_{2}, \dots, ϕ^{n} = c_{n}}$ on $U$ is exactly the integral curve of $X$ through any point of $U$ with those values of $ϕ^{2}, \dots, ϕ^{n}$ .

Proof. The argument is the rectification theorem 02.12.05 followed by an immediate corollary on the rectifying coordinates.

Step 1: apply rectification. Since $X (p) \neq = 0$ , the rectification theorem produces a $C^{k}$ chart $Φ : V \to U$ on a neighbourhood $V$ of $0$ in $R^{n}$ with $Φ (0) = p$ , $U \subseteq M$ a neighbourhood of $p$ , and $Φ_{*} (\partial / \partial y^{1}) = X$ on $U$ . In the rectified coordinates $(y^{1}, \dots, y^{n}) = Φ^{- 1}$ , the vector field $X$ is the constant coordinate field $$ X = \frac{\partial}{\partial y^1}. $$

Step 2: identify candidate first integrals. Define $ϕ^{j} (q) = y^{j} (q)$ for $j \in {2, \dots, n}$ and $q \in U$ , where $y^{j}$ is the $j$ -th rectified coordinate of $q$ . Each $ϕ^{j}$ is a $C^{k}$ function on $U$ since $Φ^{- 1}$ is a $C^{k}$ diffeomorphism.

Step 3: verify the conservation law. For each $j \in {2, \dots, n}$ , the directional derivative of $ϕ^{j}$ along $X$ is $$ X(\phi^j) = \frac{\partial \phi^j}{\partial y^1} = \frac{\partial y^j}{\partial y^1} = 0 $$ because $y^{j}$ depends only on the $j$ -th rectified coordinate, not on $y^{1}$ . So $ϕ^{j}$ is a first integral of $X$ on $U$ .

Step 4: verify functional independence. The differentials $d ϕ^{j} = d y^{j}$ in rectified coordinates form $n - 1$ of the $n$ coordinate differentials $d y^{1}, \dots, d y^{n}$ , which are linearly independent at every point of $V$ . Pushing forward through $Φ$ , the differentials $d ϕ^{2}, \dots, d ϕ^{n}$ are linearly independent at every point of $U$ . So $ϕ^{2}, \dots, ϕ^{n}$ are functionally independent on $U$ .

Step 5: identify the level-set intersection with the integral curve. In rectified coordinates, the joint level set ${y^{j} = c_{j} : j = 2, \dots, n}$ on $V$ is the horizontal line ${(y^{1}, c_{2}, \dots, c_{n}) : (y^{1}, c_{2}, \dots, c_{n}) \in V}$ . The flow of $X = \partial / \partial y^{1}$ acts by $y^{1} \mapsto y^{1} + t$ , so the integral curve through $(0, c_{2}, \dots, c_{n})$ is exactly this horizontal line. Pushing forward through $Φ$ , the intersection ${ϕ^{2} = c_{2}, \dots, ϕ^{n} = c_{n}}$ on $U$ is exactly the integral curve of $X$ through any point of $U$ with those values of the $ϕ^{j}$ . $□$

Theorem (maximality of the count). Let $X$ be a $C^{1}$ vector field on $M$ and $p$ a regular point. On any neighbourhood $U$ of $p$ on which $X$ does not vanish, the maximal number of functionally independent first integrals of $X$ is $n - 1$ .

Proof. Suppose $ϕ^{1}, \dots, ϕ^{n}$ are $n$ first integrals on $U$ with $d ϕ_{q}^{1}, \dots, d ϕ_{q}^{n}$ linearly independent at every $q \in U$ . The vector $X (q) \in T_{q} M$ satisfies $d ϕ_{q}^{j} (X (q)) = X (ϕ^{j}) (q) = 0$ for every $j$ , so $X (q)$ lies in the intersection of the $n$ kernels $ker d ϕ_{q}^{j}$ . Each kernel is an $(n - 1)$ -dimensional subspace of $T_{q} M$ , and the intersection of $n$ such subspaces with linearly independent defining covectors is ${0}$ . So $X (q) = 0$ , contradicting regularity of $p$ . Hence no $n$ functionally independent first integrals can exist on a regular neighbourhood. The construction from the previous theorem attains $n - 1$ , so this is the maximum. $□$

Theorem (functional dependence criterion). Let $ϕ^{1}, \dots, ϕ^{k}$ be $C^{1}$ functions on an open set $U \subseteq M$ with $k \leq n$ . The collection is functionally independent on $U$ if and only if the differentials $d ϕ^{1}, \dots, d ϕ^{k}$ are linearly independent at every point of $U$ , equivalently the map $Φ = (ϕ^{1}, \dots, ϕ^{k}) : U \to R^{k}$ is a submersion on $U$ .

Proof. Forward: if the differentials are linearly independent at $p \in U$ , the Jacobian of $Φ$ at $p$ has rank $k$ , so $Φ$ is a submersion at $p$ . The implicit function theorem 02.05.04 applied to $Φ$ gives local coordinates $(ϕ^{1}, \dots, ϕ^{k}, ψ^{k + 1}, \dots, ψ^{n})$ on a neighbourhood of $p$ , in which $ϕ^{1}, \dots, ϕ^{k}$ are $k$ of the coordinate functions and no non-degenerate relation $F (ϕ^{1}, \dots, ϕ^{k}) = 0$ can hold identically.

Backward: if some $d ϕ_{p}^{j}$ is a linear combination of the others, $d ϕ_{p}^{j} = \sum_{i \neq = j} λ_{i} d ϕ_{p}^{i}$ , the implicit function theorem gives a local relation $ϕ^{j} = f (ϕ^{1}, \dots, \hat{ϕ}^{j}, \dots, ϕ^{k})$ on a neighbourhood of $p$ where $f$ is a $C^{1}$ function and the hat means "omit". So the collection is functionally dependent at $p$ . $□$

Theorem (Hamiltonian setting, statement only). Let $(M^{2 n}, ω)$ be a symplectic manifold and $H : M \to R$ a $C^{1}$ Hamiltonian function. The Hamiltonian flow $X_{H}$ defined by $ι_{X_{H}} ω = d H$ has $H$ itself as a first integral: $X_{H} (H) = 0$ on $M$ . More generally, a function $ϕ : M \to R$ is a first integral of $X_{H}$ if and only if its Poisson bracket with $H$ vanishes: ${ϕ, H} = 0$ on $M$ .

The proof appears in the symplectic chapter at 05.00.04 (Noether's theorem) and 05.01.* (Poisson bracket); the present unit records the statement as the foreshadowing of the Hamiltonian conservation theorem. Energy conservation is the special case $ϕ = H$ .

Bridge. The local-existence theorem for first integrals builds toward the integrability theory of Hamiltonian systems on a symplectic manifold: every Hamiltonian flow has at least one first integral — the Hamiltonian itself — and the question of whether enough additional first integrals exist to reduce the dynamics to a geometrically simple form is the central question of integrability. The foundational reason every regular point admits $n - 1$ local first integrals is exactly the rectification theorem 02.12.05: the rectifying coordinates that straighten the field to $\partial / \partial y^{1}$ produce, automatically, $n - 1$ coordinate functions that the field annihilates. This is exactly the geometric repackaging of the smooth-dependence-on-initial-conditions theorem: the rectifying chart identifies the local family of integral curves with the family of horizontal lines in $R^{n}$ , and the labels of those lines are the first integrals. The central insight is that the local theory of first integrals is identical to the local theory of vector-field rectification — they are two viewpoints on the same data.

This pattern appears again in 05.00.04 Noether's theorem, where a continuous symmetry of a variational system produces a first integral of the Euler-Lagrange equations, upgrading the count from rectification's local $n - 1$ to a globally meaningful list of conserved quantities indexed by the symmetry group. Putting these together, rectification produces local first integrals as labels on integral curves; Noether produces global first integrals as labels on symmetry orbits; the bridge is the recognition that both are instances of the same conservation principle stated at different levels of generality. The Hamiltonian setting refines the picture further by equipping the phase space with a symplectic structure, which generalises Noether's theorem to the Poisson-bracket criterion ${ϕ, H} = 0$ and identifies the conserved quantities with the Poisson-commuting functions.

Exercises [Intermediate+]

Exercise 4 (medium, short-answer).

Although the three components $L_{1}, L_{2}, L_{3}$ of $L = r \times p$ are all first integrals of central-force motion, they are only functionally dependent enough to yield $2$ independent integrals. Identify the dependence.

Hint

A central-force trajectory lies in a single plane orthogonal to $L$ . The plane is determined by two parameters; the magnitude of $L$ is a third. How are these related?

Answer

The three components are functionally independent as functions on $R^{6}$ : their differentials $d L_{1}, d L_{2}, d L_{3}$ are linearly independent on a generic open set (excluding $L = 0$ ). But they impose only two independent constraints on the trajectory: the direction $\hat{L}$ (two parameters on the sphere $S^{2}$ ) plus the magnitude $∣ L ∣$ (one parameter), and any two of these determine the third on a constraint surface. More precisely, the level sets of $L$ on phase space have codimension three, so the three components impose three constraints — but on a generic central-force trajectory, the level set of $L$ is a four-dimensional submanifold (six minus two angular plus radial parameters), reducing from six to three with two genuine reductions coming from the planar confinement and one from the magnitude. Counted properly: $L$ gives three components, but $∣ L ∣^{2}$ is one combination already accounted for in the radial-energy reduction; the energy $H$ provides a third independent constraint. The trajectory is confined to a three-dimensional invariant submanifold cut out by three independent first integrals: two components of $\hat{L}$ and the energy $H$ , or equivalently $L_{3}$ , $∣ L ∣^{2}$ , and $H$ . Rubric: full credit for explaining that the three components carry the same geometric information as the plane normal and the magnitude, reducing to two independent constraints in the angular variables plus one from energy.

Exercise 5 (medium, symbolic).

Let $X = y \partial_{x} - x \partial_{y}$ on $R^{2}$ — the rotation field generating uniform circular motion. Find a first integral and check functional independence with itself.

Hint

A trajectory is a circle around the origin. What function is constant on circles?

Answer

Take $ϕ (x, y) = x^{2} + y^{2}$ . Then $X (ϕ) = (y) (2 x) + (- x) (2 y) = 0$ , so $ϕ$ is a first integral. The level sets ${x^{2} + y^{2} = c}$ for $c > 0$ are concentric circles, exactly the trajectories of $X$ . With dimension $n = 2$ , the count $n - 1 = 1$ is attained: one independent first integral, namely $ϕ$ , suffices. Functional independence of a single function with itself is automatic provided $d ϕ \neq = 0$ , which holds away from the origin since $d ϕ = 2 x d x + 2 y d y$ vanishes only at $(0, 0)$ . Rubric: full credit for $ϕ = x^{2} + y^{2}$ with the verification $X (ϕ) = 0$ and the comment on $d ϕ \neq = 0$ away from the origin.

Exercise 6 (medium, short-answer).

Prove: if $ϕ$ is a first integral of $X$ and $f : R \to R$ is $C^{1}$ , then $f \circ ϕ$ is also a first integral of $X$ . Conclude that the algebra of first integrals on a domain is closed under composition with smooth functions.

Hint

Compute $X (f \circ ϕ)$ using the chain rule.

Answer

$X (f \circ ϕ) = f^{'} (ϕ) \cdot X (ϕ) = f^{'} (ϕ) \cdot 0 = 0$ by the chain rule and the hypothesis that $ϕ$ is a first integral. So $f \circ ϕ$ is also a first integral. More generally, if $ϕ^{1}, \dots, ϕ^{k}$ are first integrals and $F : R^{k} \to R$ is $C^{1}$ , then $X (F (ϕ^{1}, \dots, ϕ^{k})) = \sum_{j} \partial_{j} F \cdot X (ϕ^{j}) = 0$ by the multivariable chain rule. The algebra of first integrals is closed under arbitrary $C^{1}$ recombination. Functional independence detects exactly when two first integrals are not such a recombination of others. Rubric: full credit for the chain-rule computation plus the closure conclusion.

Exercise 7 (hard, short-answer).

Show that the first-integral equation $X (ϕ) = 0$ is the homogeneous first-order linear PDE $\sum_{i} a^{i} (x) \partial ϕ / \partial x^{i} = 0$ in local coordinates $(x^{1}, \dots, x^{n})$ where $X = \sum_{i} a^{i} \partial / \partial x^{i}$ . Use the rectification theorem to write down the general $C^{1}$ solution near a regular point.

Hint

Apply the rectification theorem to $X$ . In rectified coordinates the PDE becomes the elementary equation $\partial ϕ / \partial y^{1} = 0$ , whose solutions are arbitrary $C^{1}$ functions of $(y^{2}, \dots, y^{n})$ .

Answer

The PDE $X (ϕ) = 0$ in local coordinates reads $\sum_{i} a^{i} \partial ϕ / \partial x^{i} = 0$ . At a regular point $p$ ( $X (p) \neq = 0$ ), the rectification theorem produces local coordinates $(y^{1}, \dots, y^{n})$ in which $X = \partial / \partial y^{1}$ . The PDE becomes $$ \frac{\partial \phi}{\partial y^1} = 0, $$ whose general $C^{1}$ solution is $ϕ (y^{1}, \dots, y^{n}) = F (y^{2}, \dots, y^{n})$ for an arbitrary $C^{1}$ function $F$ of $n - 1$ variables. The first integrals are exactly the functions of the $n - 1$ transverse rectifying coordinates. The method of characteristics for first-order linear PDEs is the global form of this local construction: the integral curves of $X$ are the characteristics, and a solution is constant along each. Rubric: full credit for the rectification step, the elementary PDE $\partial ϕ / \partial y^{1} = 0$ , and the form $ϕ = F (y^{2}, \dots, y^{n})$ .

Exercise 8 (hard, short-answer).

The central-force problem in $R^{3}$ has phase space of dimension $6$ and admits $3$ functionally independent first integrals: $H$ , $L_{3}$ , and $∣ L ∣^{2}$ (or equivalently $H, L_{3}, L_{3}^{2} + L_{1}^{2} + L_{2}^{2}$ ). Show that the trajectories lie on a $3$ -dimensional invariant submanifold and explain the resulting reduction to a one-dimensional radial problem.

Hint

Three independent first integrals on a $6$ -dimensional phase space confine trajectories to a $6 - 3 = 3$ -dimensional invariant submanifold. Use the angular-momentum vector to fix the orbital plane and the radial-energy decomposition.

Answer

Fix values $E, ℓ_{3}, ℓ^{2}$ of the three independent first integrals. The level set ${H = E, L_{3} = ℓ_{3}, ∣ L ∣^{2} = ℓ^{2}}$ is a $3$ -dimensional submanifold of $R^{6}$ (codimension $3$ ). The condition $L_{3} = ℓ_{3}$ and $∣ L ∣^{2} = ℓ^{2}$ fixes the direction of $L$ up to a one-parameter rotation about the $z$ -axis and the magnitude of $L$ ; combined with the perpendicularity $L \cdot r = 0$ , the motion is confined to a fixed plane through the origin.

In this plane, introduce polar coordinates $(r, θ)$ and conjugate momenta $(p_{r}, p_{θ})$ . The conservation $p_{θ} = ℓ = ℓ^{2}$ is a fourth conservation law (the planar angular momentum), but only one of $L_{3}, ∣ L ∣^{2}$ is independent of the planar reduction; the count holds because the planar reduction was already implicit. The reduced Hamiltonian is $$ H_\text{eff}(r, p_r) = \frac{p_r^2}{2m} + \frac{\ell^2}{2 m r^2} + V(r), $$ a one-dimensional Hamiltonian in $(r, p_{r})$ with the effective potential $V_{eff} (r) = ℓ^{2} / (2 m r^{2}) + V (r)$ . The radial motion is governed by $\overset{r}{˙} = p_{r} / m$ , $\overset{p}{˙}_{r} = - V_{eff}^{'} (r)$ , which is a one-dimensional conservative system solvable by quadrature: $t - t_{0} = \int d r / (2/ m) (E - V_{eff} (r))$ . The angular motion is recovered by $\dot{θ} = ℓ / (m r^{2})$ and a single quadrature. The three independent first integrals have reduced the six-dimensional Kepler-style problem to a one-dimensional integrable problem plus two quadratures. This is the foundational reduction of central-force mechanics; in the Liouville-Arnold framework it is the canonical example of a completely integrable system on a $6$ -dimensional symplectic manifold. Rubric: full credit for the codimension count, the effective potential, and the quadrature reduction.

Lean formalization [Intermediate+]

Mathlib has the infrastructure for vector fields on smooth manifolds (Mathlib.Geometry.Manifold.VectorField, Mathlib.Geometry.Manifold.IntegralCurve) but does not package first integrals as a named concept. The intended formalisation reads schematically:

import Mathlib.Geometry.Manifold.IntegralCurve
import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv

/-- A first integral of a vector field X on an open set U is a C^1
    function whose Lie derivative along X vanishes on U. -/
def IsFirstIntegral
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (X : E → E) (U : Set E) (φ : E → ℝ) : Prop :=
  ContDiffOn ℝ 1 φ U ∧ ∀ p ∈ U, fderiv ℝ φ p (X p) = 0

/-- Functional independence: the differentials of the φʲ are linearly
    independent at every point of U. -/
def AreFunctionallyIndependent
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (U : Set E) (φ : Fin k → (E → ℝ)) : Prop :=
  ∀ p ∈ U, LinearIndependent ℝ (fun j => fderiv ℝ (φ j) p)

/-- Local existence of first integrals at a regular point:
    n - 1 functionally independent C^k first integrals exist on a
    neighbourhood of every regular point of X. -/
theorem first_integrals_exist_near_regular_point
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E]
    (X : E → E) (hX : ContDiff ℝ 1 X) (p : E) (hp : X p ≠ 0) :
    ∃ (U : Set E) (hU : IsOpen U) (hpU : p ∈ U)
      (φ : Fin (Module.finrank ℝ E - 1) → (E → ℝ)),
      (∀ j, IsFirstIntegral X U (φ j)) ∧
      AreFunctionallyIndependent U φ :=
  sorry  -- apply rectificationTheorem and take φʲ = yʲ for j ≥ 2

The proof gap is the named rectification theorem (also unformalised in Mathlib, see 02.12.05) followed by the immediate identification of the rectifying coordinates $y^{2}, \dots, y^{n}$ as functionally independent first integrals. Each piece is formalisable from existing Mathlib infrastructure but has not been packaged. The manifold version of the theorem (with $E$ replaced by a smooth manifold and the field a section of the tangent bundle) is a further packaging target, predicated on Mathlib's existing SmoothManifoldWithCorners framework.

Advanced results [Master]

Theorem (Hamiltonian first integrals via Poisson brackets; Arnold MMCM Ch.4 §20). Let $(M^{2 n}, ω)$ be a symplectic manifold and $H, F \in C^{1} (M)$ . The function $F$ is a first integral of the Hamiltonian flow $X_{H}$ if and only if the Poisson bracket ${F, H}$ vanishes identically on $M$ . The Hamiltonian itself is a first integral of its own flow since ${H, H} = 0$ from the antisymmetry of the Poisson bracket.

The proof is a direct computation using the identity $X_{H} (F) = {F, H}$ that defines the Poisson bracket on a symplectic manifold. The result is the algebraic criterion that converts the analytic problem "is $F$ conserved?" into the algebraic problem "does $F$ Poisson-commute with $H$ ?". It is the foundational reason why first integrals of Hamiltonian flows form a Poisson subalgebra of $C^{1} (M)$ and why the search for additional conserved quantities becomes a search for elements of the centraliser of $H$ in the Poisson algebra.

Theorem (Noether's theorem in the variational setting; Noether 1918). Let $L : T Q \to R$ be a $C^{2}$ Lagrangian on the tangent bundle of a smooth manifold $Q$ , and suppose a one-parameter group of diffeomorphisms $φ_{s} : Q \to Q$ preserves $L$ in the sense that $L \circ T φ_{s} = L$ for every $s$ . The infinitesimal generator $ξ = \partial φ_{s} / \partial s ∣_{s = 0}$ produces a first integral of the Euler-Lagrange equations: $$ J_\xi(q, \dot q) = \frac{\partial L}{\partial \dot q^i} \xi^i(q), $$ satisfying $\dot{J}_{ξ} = 0$ along every solution of the Euler-Lagrange equations.

The proof appears in 05.00.04. The statement here records the Noether mapping from continuous symmetries to first integrals — the source of energy conservation (time-translation symmetry), momentum conservation (space-translation symmetry), and angular-momentum conservation (rotational symmetry) in classical mechanics. The Hamiltonian form of the theorem is the Poisson-bracket criterion: a symmetry generated by the Hamiltonian vector field $X_{F}$ produces a conserved quantity $F$ , with ${F, H} = 0$ encoding the invariance of $H$ under the flow of $X_{F}$ .

Theorem (Liouville-Arnold integrability theorem; Liouville 1855, Arnold 1963). Let $(M^{2 n}, ω)$ be a symplectic manifold and $F_{1} = H, F_{2}, \dots, F_{n}$ be $n$ functionally independent $C^{k}$ first integrals of $X_{H}$ in involution, meaning ${F_{i}, F_{j}} = 0$ for every $i, j$ . If a level set $M_{c} = {F_{1} = c_{1}, \dots, F_{n} = c_{n}}$ is compact and connected, $M_{c}$ is diffeomorphic to an $n$ -torus $T^{n}$ , and on a neighbourhood of $M_{c}$ there exist action-angle coordinates $(I_{1}, \dots, I_{n}, θ_{1}, \dots, θ_{n})$ in which the Hamiltonian flow is uniform rotation $\dot{θ} = ω (I)$ , $\dot{I} = 0$ .

The compactness hypothesis produces the torus structure; the non-compact case (e.g. one-dimensional unbounded motion) produces cylindrical phase manifolds. The theorem is the gold-standard reduction in integrable-systems theory: $n$ Poisson-commuting first integrals on a $2 n$ -dimensional symplectic manifold reduce the dynamics to motion on an $n$ -torus, with the dynamics in action-angle coordinates a simple linear flow. Proof in 05.09.*.

Theorem (reduction of order by first integrals; Arnold ODE Ch.2 §12). Let $X$ be a $C^{k}$ vector field on $M^{n}$ and $ϕ^{1}, \dots, ϕ^{k}$ functionally independent first integrals of $X$ on an open set $U$ . The intersection $N_{c} = {ϕ^{1} = c_{1}, \dots, ϕ^{k} = c_{k}}$ is a $C^{k}$ submanifold of $U$ of dimension $n - k$ , invariant under the flow of $X$ , and $X ∣_{N_{c}}$ is a $C^{k}$ vector field on $N_{c}$ . The dynamics of $X$ on $U$ is, by restriction to level sets, an $(n - k)$ -dimensional family of $(n - k)$ -dimensional dynamics.

The proof is the regular-value theorem applied to the submersion $Φ = (ϕ^{1}, \dots, ϕ^{k}) : U \to R^{k}$ , yielding $N_{c}$ as a smooth submanifold, plus the conservation laws to confirm invariance. The theorem is the reduction principle: each independent first integral reduces the effective dimension by one.

Theorem (non-integrability of the three-body problem; Poincaré 1892, Bruns 1887). The restricted three-body problem on the planar phase space does not admit any analytic first integral beyond the classical ones (energy, the three components of angular momentum, the three components of linear momentum). Bruns' 1887 result rules out algebraic first integrals; Poincaré's 1892 result extends to single-valued analytic first integrals.

The proof technique is the Poincaré non-existence theorem: small-divisor obstructions in the perturbation series for a putative additional integral force the formal series to diverge, and a careful analytic argument shows that no convergent additional integral can exist. The result is the foundational negative theorem in the theory of integrability: not every Hamiltonian system is integrable, and the three-body problem — historically the testbed of celestial mechanics — exhibits the precise pattern of obstruction. The theorem motivated the development of KAM theory (05.09.04) and the modern theory of chaotic dynamics. Proof technique in 05.09.*.

Theorem (Whittaker's reduction by ignorable coordinates). Let $L (q^{1}, \dots, q^{n}, \overset{q}{˙}^{1}, \dots, \overset{q}{˙}^{n})$ be a Lagrangian independent of the coordinate $q^{1}$ (an ignorable coordinate). The conjugate momentum $p_{1} = \partial L / \partial \overset{q}{˙}^{1}$ is a first integral, and the dynamics on the level set $p_{1} = c$ is governed by the Routhian $R = p_{1} \overset{q}{˙}^{1} - L$ regarded as a function of $(q^{2}, \dots, q^{n}, \overset{q}{˙}^{2}, \dots, \overset{q}{˙}^{n})$ with $\overset{q}{˙}^{1}$ eliminated via $p_{1} = c$ . This reduces the dynamics from $2 n$ -dimensional to $(2 n - 2)$ -dimensional.

The procedure is the variational form of the reduction-by-first-integral theorem. Each ignorable coordinate contributes one conserved conjugate momentum, and the Routhian reduction removes both the coordinate and its conjugate momentum from the dynamics. The procedure was the workhorse of 19th-century Hamiltonian mechanics and the foundation of the modern symplectic-reduction theory of Marsden-Weinstein.

Synthesis. The local theory of first integrals is the foundational reason every non-equilibrium point of a smooth vector field has a $(n - 1)$ -parameter family of conserved quantities, and the central insight is that these conserved quantities are exactly the rectifying coordinates produced by the flow-box theorem 02.12.05. The Hamiltonian setting adds a layer of structure: the Poisson bracket converts the analytic equation $X_{H} (F) = 0$ into the algebraic equation ${F, H} = 0$ , and the search for additional first integrals becomes the search for Poisson-commuting functions. Putting these together, the geometric theory identifies first integrals with rectifying coordinates near regular points; the Hamiltonian theory identifies them with Poisson-commuting functions globally; and Noether's theorem identifies them with infinitesimal symmetries of the underlying variational principle. The bridge is the recognition that these three descriptions — geometric (rectification), algebraic (Poisson commutation), and variational (Noether) — agree wherever they overlap and that the global theory of integrable systems is the consistent extension of the local rectification picture to symplectic phase spaces.

The theorem identifies several pictures that look distinct at first inspection. The local-existence theorem for first integrals is identified with the rectification theorem applied to the field $X$ ; this is exactly the geometric repackaging of smooth dependence on initial conditions. Functional independence of first integrals is identified with linear independence of differentials, which by the implicit function theorem 02.05.04 is the same as the submersion property of the map $Φ = (ϕ^{1}, \dots, ϕ^{k})$ . The maximality count $n - 1$ generalises in the Hamiltonian setting to the count $n$ in involution on a $2 n$ -dimensional symplectic manifold — the Liouville-Arnold integrability hypothesis — and the bridge is the recognition that the symplectic structure converts the rectification count of $n - 1$ (which would be $2 n - 1$ on a $2 n$ -dimensional phase space) down to the integrability count of $n$ via the involution constraint. The Noether correspondence between symmetries and conservation laws is identified with the recognition that an infinitesimal symmetry $ξ$ corresponds to a flow that preserves $H$ , hence to a Hamiltonian vector field $X_{F}$ with ${F, H} = 0$ , hence to a conserved quantity $F$ . The pattern recurs in modern guise as the symplectic-reduction theorem of Marsden-Weinstein, which identifies the quotient of a level set of a momentum map by the symmetry group with a new symplectic manifold on which the reduced dynamics live.

Full proof set [Master]

Proposition (local existence of first integrals), proof. Stated and proved in the Intermediate-tier section: apply the rectification theorem 02.12.05 at the regular point $p$ to obtain coordinates $(y^{1}, \dots, y^{n})$ in which $X = \partial / \partial y^{1}$ , define $ϕ^{j} = y^{j}$ for $j \geq 2$ , verify $X (ϕ^{j}) = \partial y^{j} / \partial y^{1} = 0$ , and confirm linear independence of $d y^{2}, \dots, d y^{n}$ by their coordinate-differential structure. $□$

Proposition (maximality of the count), proof. Stated and proved in the Intermediate-tier section: $n$ functionally independent first integrals would force the regular vector $X (q)$ into the intersection of $n$ kernels of linearly independent covectors, which is ${0}$ , contradicting regularity. $□$

Proposition (functional dependence criterion), proof. Stated and proved in the Intermediate-tier section: the implicit function theorem 02.05.04 applied to the map $Φ = (ϕ^{1}, \dots, ϕ^{k})$ either produces local coordinates extending $ϕ^{1}, \dots, ϕ^{k}$ (when the differentials are linearly independent) or produces a functional relation $ϕ^{j} = f (ϕ^{1}, \dots, \hat{ϕ}^{j}, \dots, ϕ^{k})$ (when they are not). $□$

Proposition (Hamiltonian flow conserves the Hamiltonian). Let $(M^{2 n}, ω)$ be a symplectic manifold, $H \in C^{1} (M)$ , and $X_{H}$ the Hamiltonian vector field defined by $ι_{X_{H}} ω = d H$ . Then $X_{H} (H) = 0$ on $M$ .

Proof. Compute $X_{H} (H) = d H (X_{H}) = ι_{X_{H}} ω (X_{H}) = ω (X_{H}, X_{H}) = 0$ by the antisymmetry of $ω$ . Equivalently, $X_{H} (H) = {H, H} = 0$ by the antisymmetry of the Poisson bracket. The conservation of $H$ along its own flow is the statement of energy conservation in Hamiltonian mechanics. $□$

Proposition (composition closure of first integrals), proof. Stated and proved in Exercise 6. If $ϕ^{1}, \dots, ϕ^{k}$ are first integrals and $F : R^{k} \to R$ is $C^{1}$ , the chain rule gives $X (F (ϕ^{1}, \dots, ϕ^{k})) = \sum_{j} \partial_{j} F \cdot X (ϕ^{j}) = 0$ . The algebra of first integrals is closed under $C^{1}$ composition. $□$

Proposition (level-set submanifold). Let $ϕ^{1}, \dots, ϕ^{k}$ be $C^{k}$ functions on $U \subseteq M^{n}$ with $d ϕ^{1}, \dots, d ϕ^{k}$ linearly independent at every point of $U$ . The intersection $N_{c} = {ϕ^{1} = c_{1}, \dots, ϕ^{k} = c_{k}}$ for $c \in R^{k}$ in the image is a $C^{k}$ submanifold of $U$ of dimension $n - k$ .

Proof. The map $Φ = (ϕ^{1}, \dots, ϕ^{k}) : U \to R^{k}$ has Jacobian of rank $k$ at every point of $U$ since the rows are the differentials $d ϕ^{j}$ and are linearly independent by hypothesis. So $Φ$ is a $C^{k}$ submersion on $U$ . By the regular-value theorem, $N_{c} = Φ^{- 1} (c)$ is a $C^{k}$ submanifold of $U$ of dimension $n - k$ . The tangent space at $q \in N_{c}$ is $T_{q} N_{c} = ⋂_{j} ker d ϕ_{q}^{j}$ , the joint kernel of the differentials. $□$

Proposition (invariance of level sets under the flow). Let $X$ be a $C^{1}$ vector field on $M$ and $ϕ^{1}, \dots, ϕ^{k}$ first integrals of $X$ . The level set $N_{c} = {ϕ^{1} = c_{1}, \dots, ϕ^{k} = c_{k}}$ is forward-invariant and backward-invariant under the flow of $X$ wherever defined.

Proof. Let $γ : I \to M$ be an integral curve of $X$ with $γ (0) \in N_{c}$ . Each $ϕ^{j} \circ γ$ satisfies $d (ϕ^{j} \circ γ) / d t = d ϕ^{j} (\overset{γ}{˙}) = d ϕ^{j} (X) = X (ϕ^{j}) = 0$ , so $ϕ^{j} \circ γ$ is constant on $I$ . Since $ϕ^{j} (γ (0)) = c_{j}$ , $ϕ^{j} (γ (t)) = c_{j}$ for every $t \in I$ . So $γ (I) \subseteq N_{c}$ , proving invariance. $□$

Proposition (Lie-derivative formulation). Let $ϕ$ be a $C^{1}$ function on $M$ and $X$ a $C^{1}$ vector field. The following are equivalent: (a) $ϕ$ is a first integral of $X$ ; (b) $L_{X} ϕ = 0$ , where $L_{X}$ is the Lie derivative; (c) $ϕ$ is preserved by the flow $g^{t}$ of $X$ , i.e. $ϕ \circ g^{t} = ϕ$ wherever $g^{t}$ is defined.

Proof. Equivalences. (a) $\Leftrightarrow$ (b): the Lie derivative of a scalar function along $X$ is $L_{X} ϕ = X (ϕ) = d ϕ (X)$ , by definition; both sides vanish identically iff $ϕ$ is a first integral. (b) $\Leftrightarrow$ (c): differentiate the flow-preservation condition $ϕ \circ g^{t} = ϕ$ at $t = 0$ to get $L_{X} ϕ = (d / d t) ∣_{t = 0} (ϕ \circ g^{t}) = 0$ . Conversely, $L_{X} ϕ = 0$ gives $(d / d t) (ϕ \circ g^{t}) = (L_{X} ϕ) \circ g^{t} = 0$ , so $ϕ \circ g^{t}$ is constant in $t$ and equals its value at $t = 0$ , which is $ϕ$ . $□$

Connections [Master]

Phase space, vector field, integral curve 02.12.01. A first integral is defined relative to a vector field $X$ on phase space; the integral curves of $X$ are the trajectories along which a first integral is constant. The unit 02.12.01 provides the framework — vector fields as derivations of $C^{1}$ functions, integral curves as solutions of $\overset{γ}{˙} = X (γ)$ — within which first integrals are formulated as the functions $ϕ$ satisfying $X (ϕ) = 0$ .
Phase flow / one-parameter group 02.12.02. First integrals are equivalently the functions preserved by the flow $g^{t}$ of the field: $ϕ \circ g^{t} = ϕ$ . The unit 02.12.02 provides the flow as a one-parameter group of diffeomorphisms, and the conservation law $ϕ \circ g^{t} = ϕ$ expresses invariance of $ϕ$ under this group. The Lie-derivative formulation $L_{X} ϕ = 0$ is the infinitesimal form; the flow-invariance form is the integrated statement.
Rectification theorem 02.12.05. The local existence of $n - 1$ functionally independent first integrals near a regular point is a direct corollary of the rectification theorem: the rectifying coordinates $y^{2}, \dots, y^{n}$ are themselves first integrals. The proof of the local-existence theorem in the present unit is the rectification theorem followed by a one-line identification of the rectifying coordinates as the desired conserved quantities. The two units are dual viewpoints on the same content.
Implicit and inverse function theorems 02.05.04. Functional independence of first integrals is equivalent to linear independence of their differentials, which by the implicit function theorem corresponds to the submersion property of the map $Φ = (ϕ^{1}, \dots, ϕ^{k})$ to $R^{k}$ . The level-set submanifold structure is the regular-value theorem; the functional dependence criterion is the implicit function theorem applied to a relation among the $ϕ^{j}$ .
Noether's theorem 05.00.04. Noether's theorem upgrades the local existence of first integrals to a globally meaningful list indexed by continuous symmetries of a variational system. Where the present unit's rectification-based construction produces $n - 1$ local labels on integral curves, Noether's theorem produces conserved quantities tied to specific symmetry groups: energy from time-translation, momentum from space-translation, angular momentum from rotation. The bridge is the recognition that the variational structure converts a symmetry into a Hamiltonian vector field whose Hamiltonian Poisson-commutes with $H$ .
Smooth manifold 03.02.01. First integrals are most cleanly defined on a smooth manifold, where the Lie derivative $L_{X} ϕ$ and the differential $d ϕ$ are coordinate-free objects. The reduction theorem produces invariant submanifolds via the regular-value theorem, which is the standard manifold-theoretic submersion construction. The unit 03.02.01 provides the framework in which the theorem is naturally stated.
Lyapunov stability 02.12.08. A Lyapunov function $V$ with $\dot{V} \leq 0$ is the inequality version of a first integral $ϕ$ with $\dot{ϕ} = 0$ ; the boundary case $\dot{V} = 0$ recovers the conservation law. Conservative systems (no dissipation) are exactly the systems for which the natural energy candidate is a first integral; dissipative systems are exactly the systems for which it becomes a strict Lyapunov function. The two units describe the same geometric construction in two regimes — energy-method conservation versus energy-method stability.
Inhomogeneous linear ODE / variation of constants 02.12.13. The Wronskian determinant $W (t) = det Φ (t)$ of a fundamental matrix $Φ$ of the homogeneous linear system $\overset{x}{˙} = A (t) x$ is a first integral of the matrix flow on $Mat_{n} (R)$ exactly when $tr A \equiv 0$ ; for general $A$ , Liouville's formula $W (t) = W (t_{0}) exp (\int_{t_{0}}^{t} tr A (s) d s)$ expresses the logarithmic derivative of $W$ as the trace, making the Wronskian the natural conserved-quantity object on the homogeneous linear flow. The geometric Liouville formula $g^{t, s *} Ω = Ω \cdot exp (\int_{s}^{t} div_{Ω} X d r)$ is the manifold-level avatar, with the divergence of a Hamiltonian vector field vanishing identically (the trace-free condition becomes symplectic volume preservation). The unit 02.12.13 is the dedicated linear-ODE setting where this first-integral / volume-distortion picture is given its explicit determinant formula. Connection type: specialisation / instance — the Wronskian is a named first integral of the matrix flow associated to a linear vector field.

Historical & philosophical context [Master]

The systematic theory of first integrals begins with Lagrange's 1788 Mécanique Analytique (Paris, Chez la veuve Desaint; second edition Vve Courcier 1811-1815) ^{[Lagrange 1788]}, which derived energy and momentum conservation for variational systems via the principle of least action and the now-standard Euler-Lagrange equations. Lagrange's framing was already abstract — conservation followed from the variational structure rather than from coordinate-specific manipulations — and laid the groundwork for the symplectic theory that emerged a half-century later. Jacobi's 1842-1843 Berlin lectures, published posthumously in 1866 by Clebsch as Vorlesungen über Dynamik (Reimer, Berlin) ^{[Jacobi 1843]}, introduced the canonical theory of Hamiltonian mechanics with the systematic exploitation of the Hamilton-Jacobi equation, and the last-multiplier method for completing a partial set of first integrals to a full integrability statement. Jacobi's treatment made the search for first integrals a central problem of analytical dynamics. Liouville's 1855 Note sur l'intégration des équations différentielles de la dynamique (J. Math. Pures Appl. (2) 20, 137-138) ^{[Liouville 1855]} proved the integrability theorem in the form that bears his name: $n$ functionally independent first integrals in involution on a $2 n$ -dimensional symplectic phase space suffice to integrate the equations of motion by quadrature.

The geometric viewpoint emerged with Poincaré's 1881-1886 mémoires on phase portraits and his 1892-1899 Les méthodes nouvelles de la mécanique céleste (Gauthier-Villars, Paris; three volumes) ^{[Poincaré 1892-1899]}, which together established the framework for studying first integrals as geometric objects (level-set submanifolds, transverse sections, return maps) rather than as analytic expressions. Poincaré's most influential negative result — the non-existence of additional analytic first integrals for the perturbed three-body problem — exposed the obstruction to general integrability and motivated the development of KAM theory by Kolmogorov (1954), Arnold (1963), and Moser (1962). Emmy Noether's 1918 Invariante Variationsprobleme (Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse, 235-257) ^{[Noether 1918]} proved the symmetry-to-conservation correspondence in full generality, completing the conceptual loop opened by Lagrange: every continuous symmetry of a variational system produces a first integral of the Euler-Lagrange equations, and conversely. Noether's theorem is now the textbook starting point for conservation laws in classical and quantum field theory.

The 20th-century synthesis is Arnold's Ordinary Differential Equations (Russian 1971, English 1973) ^{[Arnold 1973]} and Mathematical Methods of Classical Mechanics (Russian 1974, English 1978, 2nd English ed. Springer GTM 60, 1989) ^{[Arnold MMCM]}. Arnold's exposition framed first integrals as the geometric content of conservation, separated the local existence theorem (a corollary of rectification) from the global integrability theorem (Liouville-Arnold with the action-angle construction), and integrated the Noether correspondence into the symplectic-geometric picture. Whittaker's A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge, 4th ed. 1937) ^{[Whittaker 1937]} is the canonical 19th-century reference, encoding the lore of ignorable coordinates, the Routhian reduction, and the systematic exploitation of conserved quantities in problems of celestial and rigid-body mechanics. The modern viewpoint, developed by Marsden, Weinstein, and the symplectic-geometry school from the 1970s onward, identifies first integrals with momentum maps for symplectic group actions, and the resulting symplectic-reduction theorem subsumes the classical ignorable-coordinate reduction as the abelian special case.

Through this lineage, the concept of a first integral has remained one of the most stable and useful ideas in dynamics: from Lagrange's variational derivation, through Jacobi's canonical theory and Liouville's integrability theorem, to Poincaré's geometric non-integrability, Noether's symmetry correspondence, and Arnold's modern synthesis, the same object — a function constant along trajectories — encodes the conservation laws of physics, the reduction principles of mechanics, and the geometric obstructions to integrability. The story is one of progressive abstraction: the same conservation law that Galileo observed in pendulum amplitudes becomes Lagrange's energy integral, Jacobi's separation constant, Liouville's involutive function, Noether's symmetry generator, and Arnold's action variable.

Bibliography [Master]

@book{Arnold1973ODE,
  author    = {Arnold, Vladimir I.},
  title     = {Ordinary Differential Equations},
  publisher = {MIT Press},
  year      = {1973},
  note      = {Translated from the 1971 Russian edition by Richard A. Silverman}
}

@book{Arnold1989MMCM,
  author    = {Arnold, Vladimir I.},
  title     = {Mathematical Methods of Classical Mechanics},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {60},
  edition   = {2},
  year      = {1989},
  note      = {Translated from the 1974 Russian edition by K. Vogtmann and A. Weinstein}
}

@article{Liouville1855,
  author  = {Liouville, Joseph},
  title   = {Note sur l'int{\'e}gration des {\'e}quations diff{\'e}rentielles de la dynamique},
  journal = {J. Math. Pures Appl. (2)},
  volume  = {20},
  year    = {1855},
  pages   = {137--138}
}

@book{Jacobi1866Dynamik,
  author    = {Jacobi, Carl Gustav Jacob},
  title     = {Vorlesungen {\"u}ber Dynamik},
  publisher = {Reimer, Berlin},
  year      = {1866},
  editor    = {Clebsch, Alfred},
  note      = {Berlin lectures of 1842-1843; published posthumously by A. Clebsch}
}

@article{Noether1918,
  author  = {Noether, Emmy},
  title   = {Invariante Variationsprobleme},
  journal = {Nachrichten von der K{\"o}niglichen Gesellschaft der Wissenschaften zu G{\"o}ttingen, Mathematisch-Physikalische Klasse},
  year    = {1918},
  pages   = {235--257}
}

@book{Poincare1892MNMC,
  author    = {Poincar{\'e}, Henri},
  title     = {Les m{\'e}thodes nouvelles de la m{\'e}canique c{\'e}leste},
  publisher = {Gauthier-Villars, Paris},
  year      = {1892},
  note      = {Volume I (1892), Volume II (1893), Volume III (1899)}
}

@book{Lagrange1788Mecanique,
  author    = {Lagrange, Joseph-Louis},
  title     = {M{\'e}canique Analytique},
  publisher = {Chez la veuve Desaint, Paris},
  year      = {1788},
  note      = {Second edition: V{\^v}e Courcier, Paris, 1811-1815}
}

@book{Whittaker1937,
  author    = {Whittaker, Edmund Taylor},
  title     = {A Treatise on the Analytical Dynamics of Particles and Rigid Bodies},
  publisher = {Cambridge University Press},
  edition   = {4},
  year      = {1937},
  note      = {Reprinted with foreword by W. McCrea, 1989}
}

@book{Hormander1983LPDO,
  author    = {H{\"o}rmander, Lars},
  title     = {The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {256},
  year      = {1983}
}

@book{Strogatz2015NonlinearDynamics,
  author    = {Strogatz, Steven H.},
  title     = {Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering},
  publisher = {Westview Press},
  edition   = {2},
  year      = {2015}
}

@book{MarsdenRatiu1999,
  author    = {Marsden, Jerrold E. and Ratiu, Tudor S.},
  title     = {Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems},
  publisher = {Springer-Verlag},
  series    = {Texts in Applied Mathematics},
  volume    = {17},
  edition   = {2},
  year      = {1999}
}

Prerequisites

02.05.04
02.12.01
02.12.02
02.12.05
03.02.01

Tier anchors

beginner: Arnold *Ordinary Differential Equations* Ch.2 §10 (first integrals as functions constant along trajectories, with the central-force angular momentum and energy as the canonical illustrations); Strogatz *Nonlinear Dynamics and Chaos* §6.5 informal energy-function reading for conservative planar systems
intermediate: Arnold *Ordinary Differential Equations* Ch.2 §10-§12 (first integrals near a regular point, functional independence, and the reduction to a lower-dimensional invariant manifold); Arnold *Mathematical Methods of Classical Mechanics* Ch.4 §20 (first integrals of a Hamiltonian flow and the Poisson-bracket criterion)
master: Arnold *Mathematical Methods of Classical Mechanics* (2nd ed., Springer GTM 60, 1989) Ch.4 §20 and Ch.10 §49 (Liouville-Arnold integrability theorem and the action-angle coordinates produced by $n$ independent first integrals in involution on a $2n$-dimensional symplectic manifold); Liouville 1855 *Note sur l'intégration des équations différentielles de la dynamique* (J. Math. Pures Appl. (2) 20, 137-138) — originator of the integrability theorem; Jacobi 1843 *Vorlesungen über Dynamik* (Berlin lectures, published posthumously 1866 by Clebsch) — originator of the canonical theory and the last-multiplier method; Noether 1918 *Invariante Variationsprobleme* (Nachr. Königl. Ges. Wiss. Göttingen Math.-Phys. Kl., 235-257) — symmetry-to-conservation correspondence; Poincaré 1892-1899 *Les méthodes nouvelles de la mécanique céleste* (Gauthier-Villars; volumes I-III) — non-existence of additional analytic first integrals for the perturbed three-body problem; Hörmander *The Analysis of Linear Partial Differential Operators I* (Springer Grundlehren 256, 1983) §8.1 (first integrals as solutions of the homogeneous first-order linear PDE $Xu = 0$)

References

TODO_REF
Arnold — Ordinary Differential Equations · Ch.2 §10-§12 first integrals, functional independence, and reduction to a lower-dimensional invariant manifold
TODO_REF
Arnold — Mathematical Methods of Classical Mechanics · Ch.4 §20 first integrals of a Hamiltonian flow and the Poisson-bracket criterion; Ch.10 §49 Liouville-Arnold integrability theorem and action-angle coordinates
TODO_REF
Liouville — Note sur l'intégration des équations différentielles de la dynamique · J. Math. Pures Appl. (2) 20 (1855) 137-138; statement of the integrability theorem for $n$ first integrals in involution on a $2n$-dimensional symplectic manifold
TODO_REF
Jacobi — Vorlesungen über Dynamik · Berlin lectures 1842-1843, posthumously edited by A. Clebsch (Reimer, Berlin 1866); the canonical theory, the last-multiplier method, and the systematic exploitation of the Hamilton-Jacobi equation
TODO_REF
Noether — Invariante Variationsprobleme · Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse, 1918, 235-257; symmetry-to-conservation correspondence for variational systems
TODO_REF
Poincaré — Les méthodes nouvelles de la mécanique céleste · Gauthier-Villars, Paris; Vol. I (1892), Vol. II (1893), Vol. III (1899); non-existence of additional analytic first integrals for the perturbed three-body problem, foundation of non-integrability theory
TODO_REF
Hörmander — The Analysis of Linear Partial Differential Operators I · Springer Grundlehren 256 (1983) §8.1; first integrals as solutions of the homogeneous first-order linear PDE $Xu = 0$, method of characteristics
TODO_REF
Lagrange — Mécanique Analytique · Première édition, Paris, Chez la veuve Desaint, 1788; second edition Vve Courcier 1811-1815; conservation of energy and momentum for variational systems via the principle of least action
TODO_REF
Whittaker — A Treatise on the Analytical Dynamics of Particles and Rigid Bodies · Cambridge University Press, 4th ed. 1937, reprinted 1989; canonical synthesis of the 19th-century theory of first integrals, ignorable coordinates, and reduction

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 35m
master: 70m