05.05.03 · symplectic / lagrangian

Generating functions for symplectomorphisms

shipped3 tiersLean: none

Anchor (Master): Hörmander 1971 (originator GFQI); Viterbo 1992 *Symplectic topology as the geometry of generating functions*; Cannas §22; McDuff-Salamon Ch. 9 + 11

Intuition [Beginner]

A symplectomorphism is a map of phase space that preserves the symplectic pairing — the geometric quantity that makes Hamilton's equations of motion sensible. Generating functions are a way to encode such a map by a single function, instead of by writing out the map's components. One scalar quantity, two-variable input, captures the entire dynamical transformation.

The picture that makes this work: feed the map's graph into a Lagrangian submanifold of a doubled phase space. That graph turns out to be the level set of an exact one-form $d F$ . The function $F$ is the generating function, and its derivatives reproduce the map.

The pay-off is enormous. Fixed points of a symplectomorphism become critical points of $F$ . Counting fixed points becomes Morse theory of $F$ . Hamilton-Jacobi becomes a nonlinear PDE for $F$ . The whole vocabulary of classical mechanics translates into a single function on a base manifold.

Visual [Beginner]

A schematic: phase space drawn twice, side-by-side, with the graph of a symplectomorphism stretched between them. Above the graph sits a function $F$ on the base; arrows from $F$ 's derivatives down to the graph indicate that the map is recovered from $F$ alone.

The picture records two facts: the graph is half-dimensional (Lagrangian) inside the doubled phase space, and a Lagrangian close to the zero section of a cotangent bundle is the graph of an exact one-form, hence comes from a function.

Worked example [Beginner]

Take phase space $T^{*} R = R^{2}$ with coordinates $(q, p)$ . Consider the time- $1$ flow of the simple harmonic oscillator: a rotation by $90$ degrees in the $(q, p)$ -plane, sending $(q_{1}, p_{1})$ to $(q_{2}, p_{2}) = (p_{1}, - q_{1})$ .

Try the candidate generating function $F (q_{1}, q_{2}) = q_{1} q_{2}$ on the doubled base $R \times R$ . Compute its derivatives: $F_{q_{1}} = q_{2}$ and $F_{q_{2}} = q_{1}$ . The recipe says the map is recovered by $p_{1} = - F_{q_{1}} = - q_{2}$ and $p_{2} = F_{q_{2}} = q_{1}$ .

Solve: $p_{1} = - q_{2}$ rearranges to $q_{2} = - p_{1}$ , but the rotation gives $q_{2} = p_{1}$ . The sign is off because rotations are not "type 1" everywhere; the rotation by $90$ degrees turns the graph perpendicular to the base. Switching sign — using $F = - q_{1} q_{2}$ — corrects this and recovers the rotation cleanly.

What this tells us: the generating function works locally, captures the entire map in one scalar, and is sensitive to which projection from the doubled phase space onto a base is used.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

Why does the graph of a symplectomorphism become Lagrangian in the doubled phase space?

A. Because every map has a Lagrangian graph
B. Because pulling the symplectic form back along the diagonal cancels with the form on the second factor when a symplectomorphism is used
C. Because the dimension of the graph is twice that of phase space
D. Because Hamiltonians vanish on it

Hint

Consider the form $ω \oplus - ω$ on the product, and what the graph of a map with $ϕ^{*} ω = ω$ does to it.

Answer

B. Because pulling the symplectic form back along the diagonal cancels with the form on the second factor when a symplectomorphism is used.

Feedback-correct: yes — the graph has dimension equal to half the doubled phase space, and the form $ω \oplus - ω$ vanishes on it because $ϕ^{*} ω = ω$ . Feedback-wrong: option C inverts the dimension count; A is false; D conflates with Hamiltonian vector fields.

Formal definition [Intermediate+]

Let $M$ be a smooth manifold and $T^{*} M$ its cotangent bundle with the canonical Liouville one-form $λ$ and symplectic form $ω = - d λ$ . The product $T^{*} M \times T^{*} M$ carries the form $ω \oplus - ω$ — the first factor's form minus the second's. A diffeomorphism $ϕ : T^{*} M \to T^{*} M$ is a symplectomorphism when $ϕ^{*} ω = ω$ , equivalent to its graph $Γ_{ϕ} := {(x, ϕ (x)) : x \in T^{*} M}$ being Lagrangian in $(T^{*} M \times T^{*} M, ω \oplus - ω)$ .

The bundle isomorphism

T^{*} M \times T^{*} M ≅ T^{*} (M \times M), (q_{1}, p_{1}, q_{2}, p_{2}) \leftrightarrow ((q_{1}, q_{2}), (p_{1}, - p_{2})),

intertwines $ω \oplus - ω$ with the canonical symplectic form on $T^{*} (M \times M)$ . Under this identification, $Γ_{ϕ}$ becomes a Lagrangian submanifold of $T^{*} (M \times M)$ .

Definition (type-1 generating function). A symplectomorphism $ϕ : T^{*} M \to T^{*} M$ is of type 1 when the projection $Γ_{ϕ} \to M \times M$ (forgetting the cotangent fibres) is a diffeomorphism onto an open subset. In that case $Γ_{ϕ}$ is the image of an exact one-form $d F$ for some smooth $F : M \times M \to R$ , called the generating function of $ϕ$ . The map $ϕ$ is recovered from $F$ via

ϕ (q_{1}, p_{1}) = (q_{2}, p_{2}) ⟺ p_{1} = - \frac{\partial F}{\partial q _{1}} (q_{1}, q_{2}), p_{2} = \frac{\partial F}{\partial q _{2}} (q_{1}, q_{2}) .

Definition (generating function for a Lagrangian). Let $L \subset T^{*} M$ be a Lagrangian submanifold near the zero section. By 05.05.02, $L$ is the graph of a closed one-form $β \in Ω^{1} (M)$ . When $β$ is exact, $β = d S$ for some $S : M \to R$ called the generating function of $L$ . In Darboux coordinates,

L = {(q, p) \in T^{*} M : p = d S_{q}} .

Counterexamples to common slips.

The identity symplectomorphism $ϕ = id$ is not type 1: its graph is the diagonal, which projects to $M \times M$ as the diagonal, not as a graph. Capturing the identity requires a different type (typically type 2, $S (q_{1}, p_{2})$ ).
Rotations of $T^{*} R^{n}$ by $90$ degrees rotate the graph perpendicular to the base $M \times M$ , so the type-1 projection degenerates. The four classical types (Goldstein §9.1) handle the cases distinguished by which Legendre transform makes the graph projectable.
A closed one-form $β$ on a non-simply-connected $M$ gives a Lagrangian without a global generating function — only a multivalued one. Generating functions globalise only on the universal cover.

Key theorem with proof [Intermediate+]

Theorem (existence of type-1 generating functions). *Let $ϕ : T^{*} M \to T^{*} M$ be a symplectomorphism and suppose its graph $Γ_{ϕ}$ , viewed as a Lagrangian in $T^{*} (M \times M)$ via the identification above, projects diffeomorphically onto an open subset $U \subset M \times M$ . Then there exists $F : U \to R$ , unique up to additive constant on each connected component, such that $Γ_{ϕ}$ is the graph of $d F$ . Equivalently, $ϕ$ satisfies $p_{1} d q_{1} - p_{2} d q_{2} = d F$ on $Γ_{ϕ}$ .*

Proof. Three steps: identify $Γ_{ϕ}$ with a Lagrangian section of $T^{*} (M \times M) \to M \times M$ , apply the closed-form criterion for sections, then integrate.

Step 1: graph as Lagrangian section. The projection assumption $Γ_{ϕ} \to M \times M$ is a diffeomorphism onto $U$ — call its inverse $σ : U \to T^{*} (M \times M)$ . The image $σ (U) = Γ_{ϕ}$ is Lagrangian by hypothesis, and $σ$ realises it as the image of a section of the cotangent bundle over $U$ . By 05.05.02, a section of $T^{*} (M \times M)$ is Lagrangian if and only if the section is a closed one-form on $U$ .

Step 2: identify the closed one-form. Under the bundle isomorphism $T^{*} M \times T^{*} M ≅ T^{*} (M \times M)$ given by $(q_{1}, p_{1}, q_{2}, p_{2}) \leftrightarrow ((q_{1}, q_{2}), (p_{1}, - p_{2}))$ , the section $σ$ corresponds to the one-form

σ_{*} : (q_{1}, q_{2}) \mapsto p_{1} d q_{1} - p_{2} d q_{2},

where $(p_{1}, p_{2})$ are determined by $ϕ (q_{1}, p_{1}) = (q_{2}, p_{2})$ . The Lagrangian condition on $Γ_{ϕ}$ translates to $d (p_{1} d q_{1} - p_{2} d q_{2}) = 0$ along the graph — equivalently, $σ_{*}$ is a closed one-form on $U$ .

Step 3: Poincaré primitive. On any contractible component of $U$ , the closed one-form $σ_{*}$ admits a primitive $F$ with $d F = σ_{*}$ , unique up to additive constant. The relations $\partial F / \partial q_{1} = p_{1}$ and $\partial F / \partial q_{2} = - p_{2}$ (rearranged) recover the standard sign convention $p_{1} = - \partial F / \partial q_{1}$ , $p_{2} = \partial F / \partial q_{2}$ when one writes the form as $- p_{1} d q_{1} + p_{2} d q_{2} = - d F$ . The two sign conventions appear in the literature interchangeably; we adopt the convention $p_{1} d q_{1} - p_{2} d q_{2} = d F$ throughout this unit. $□$

Bridge. The type-1 theorem builds toward 05.08.01 (Arnold conjecture): fixed points of $ϕ$ are exactly the points where $ϕ (q_{1}, p_{1}) = (q_{1}, p_{1})$ , equivalently $q_{2} = q_{1}$ and $p_{2} = p_{1}$ , which in terms of $F$ on the diagonal becomes $\partial F / \partial q_{1} + \partial F / \partial q_{2} = 0$ — the critical-point equation for the restriction of $F$ to the diagonal $Δ_{M} \subset M \times M$ . The same identification appears again in 05.07.01 (Gromov non-squeezing) where Hamiltonian capacities are compared by tracking the critical-value spectrum of associated generating functions across symplectic embeddings. Putting these together, the central insight is that a generating function turns the dynamical question "what are the fixed points of $ϕ$ ?" into the variational question "what are the critical points of $F$ ?", and this is the foundational reason Morse theory becomes a tool for symplectic dynamics. The bridge is between two equivalent formulations of the same data — graph of a symplectomorphism, graph of $d F$ — connected by the cotangent-bundle identification.

Exercises [Intermediate+]

Exercise 3 (medium, short-answer).

Show that fixed points of a type-1 symplectomorphism $ϕ$ correspond to critical points of $G (q) := F (q, q)$ , where $F$ is the generating function and $G$ is its restriction to the diagonal.

Hint

Compute $d G / d q$ using $F$ 's partial derivatives and the chain rule, then impose $ϕ (q, p) = (q, p)$ .

Answer

A fixed point satisfies $ϕ (q, p) = (q, p)$ , equivalently $q_{2} = q_{1} = q$ and $p_{2} = p_{1} = p$ . The generating-function relations give $p = - \partial F / \partial q_{1} (q, q)$ and $p = \partial F / \partial q_{2} (q, q)$ . Subtracting: $\partial F / \partial q_{2} (q, q) + \partial F / \partial q_{1} (q, q) = 0$ . Computing $d G / d q$ via the chain rule: $G^{'} (q) = \partial F / \partial q_{1} (q, q) + \partial F / \partial q_{2} (q, q)$ . Hence $G^{'} (q) = 0$ at fixed points. Conversely, a critical point of $G$ on the diagonal supplies a $p$ via either partial-derivative equation, recovering a fixed point.

Rubric: full credit for the diagonal restriction calculation and the equivalence in both directions.

Exercise 5 (medium, short-answer).

Let $H : T^{*} M \to R$ be a smooth Hamiltonian and $ϕ_{t}$ its time- $t$ flow. Derive (formally, ignoring questions of small-time existence) the Hamilton-Jacobi equation that the type-1 generating function $F_{t}$ of $ϕ_{t}$ satisfies.

Hint

Differentiate $ϕ_{t}^{*} λ - λ$ in $t$ and use that the flow is generated by $X_{H}$ with $ι_{X_{H}} ω = d H$ .

Answer

The graph $Γ_{ϕ_{t}}$ is generated by $F_{t} (q_{1}, q_{2})$ with $p_{2} = \partial F_{t} / \partial q_{2}$ . Since $ϕ_{t}$ is generated by $H$ , the action $F_{t}$ satisfies $\partial F_{t} / \partial t = - H (q_{2}, p_{2}) = - H (q_{2}, \partial F_{t} / \partial q_{2})$ . This is the Hamilton-Jacobi equation: $F_{t}$ obeys the first-order PDE $\partial_{t} F_{t} + H (q_{2}, \nabla_{q_{2}} F_{t}) = 0$ with $F_{0} (q_{1}, q_{2})$ encoding the identity (or its type-2 substitute, since the identity itself is not type 1).

Rubric: full credit for the PDE in correct form; partial credit for the structural argument without the explicit form.

Exercise 6 (hard, short-answer).

State the Sikorav-Viterbo uniqueness theorem for generating functions quadratic at infinity, and explain why uniqueness up to stabilisation, fibre-preserving change, and additive constant is the correct equivalence relation.

Hint

Two GFQI presentations of the same Lagrangian must give the same critical-value invariants; the equivalence relation is the smallest one preserving those invariants.

Answer

Theorem (Sikorav 1986; Viterbo 1992). Let $L_{0}, L_{1} \subset T^{*} M$ be closed Lagrangians Hamiltonian-isotopic to the zero section, and let $S_{0}, S_{1}$ be GFQI presentations. Then $S_{1}$ is obtained from $S_{0}$ by a finite sequence of three moves: (a) stabilisation $S \mapsto S \oplus Q$ for $Q$ a non-degenerate quadratic form on a new fibre, (b) fibre-preserving diffeomorphism $S (q, ξ) \mapsto S (q, ψ_{q} (ξ))$ for $ψ_{q}$ smooth and fibrewise-diffeomorphic, (c) additive constant $S \mapsto S + c$ . Conversely, any sequence of these moves preserves the Lagrangian.

The equivalence is correct because: (a) stabilisation does not change the homotopy type of sublevel sets at infinity, so spectral invariants (defined via relative Morse / minimax over sublevels) are preserved; (b) fibre-preserving changes are automorphisms of the GFQI structure, manifestly preserving critical values up to reordering; (c) additive constants shift the entire spectrum but preserve differences. Any smaller relation fails to identify GFQIs that the construction itself produces from the same Lagrangian; any larger relation collapses Lagrangians the spectrum should distinguish.

Rubric: full credit for stating the three moves and explaining the spectral-invariant rationale; partial credit for the moves alone.

Exercise 7 (hard, short-answer).

Conley-Zehnder used generating functions to prove the Arnold conjecture for the standard $2 n$ -torus $T^{2 n}$ in 1983. Sketch the strategy: how does the discrete-action principle reduce a fixed-point count to a Morse-theoretic computation?

Hint

A Hamiltonian symplectomorphism $ϕ$ on $T^{2 n}$ lifts to a $Z^{2 n}$ -equivariant map of $R^{2 n}$ ; build a global generating function on the universal cover and read off fixed points as critical points modulo the deck transformation.

Answer

A Hamiltonian symplectomorphism $ϕ$ of $T^{2 n} = R^{2 n} / Z^{2 n}$ lifts to a $Z^{2 n}$ -equivariant Hamiltonian symplectomorphism $\tilde{ϕ}$ of $R^{2 n}$ . For $ϕ$ small enough (or, by a discrete-time argument, factored as a composition of small steps), each step admits a type-1 generating function $F_{k} : R^{2 n} \times R^{2 n} \to R$ . The composite generating function — sum of the discrete actions — is a function $Φ : (R^{2 n})^{N} \to R$ on a sequence space, $Z^{2 n}$ -equivariant in the natural diagonal action. Critical points of $Φ$ (mod the deck action) are exactly periodic discrete trajectories closing up to fixed points of $ϕ$ . Morse theory on the quotient $T^{2 n} \times \dots \times T^{2 n}$ then bounds the number of critical points below by the sum of Betti numbers, which for the torus is $\sum (k 2 n) = 4^{n}$ . This gives the Arnold lower bound $# Fix (ϕ) \geq 4^{n}$ (or $\geq 2 n + 1$ if only Morse-counting non-degenerate critical points).

Rubric: full credit for the lift, the discrete-action sum, and the Morse-theoretic count on the torus; partial credit for any two of the three.

Lean formalization [Intermediate+]

A statement-level skeleton (Mathlib does not yet have the apparatus; the gap is detailed in lean_mathlib_gap):

[object Promise]

The sorry blocks the existence theorem on three Mathlib gaps: (1) the canonical-form identification on $T^{*} M \times T^{*} M ≅ T^{*} (M \times M)$ , (2) the Lagrangian-section criterion (a section of $T^{*} N$ is Lagrangian iff it is a closed one-form), and (3) the relative Poincaré lemma on contractible domains. All three are foundational and Mathlib-contribution-sized.

Advanced results [Master]

The type-1 theorem is the surface of a much deeper apparatus. The genuine working object in modern symplectic topology is the generating function quadratic at infinity (GFQI), due to Hörmander 1971 in the Fourier-integral-operator setting and developed for symplectic-topological purposes by Sikorav, Viterbo, Théret, and Eliashberg-Polterovich.

Definition (GFQI). A smooth function $S : M \times R^{N} \to R$ is quadratic at infinity if there exists a non-degenerate quadratic form $Q$ on $R^{N}$ and a compact $K \subset M \times R^{N}$ such that $S (q, ξ) = Q (ξ)$ for all $(q, ξ) \in / K$ . The Lagrangian generated by $S$ is

L_{S} := {(q, p) \in T^{*} M : \exists ξ with \frac{\partial S}{\partial ξ} (q, ξ) = 0 and p = \frac{\partial S}{\partial q} (q, ξ)} .

The condition $\partial S / \partial ξ = 0$ defines the fibre critical set $Σ_{S} \subset M \times R^{N}$ ; the projection $Σ_{S} \to T^{*} M$ via $(q, ξ) \mapsto (q, \partial S / \partial q)$ has Lagrangian image whenever $S$ is generic (Morse-Bott on each fibre).

Theorem (Hörmander 1971; Sikorav 1986; Viterbo 1992). *Every closed Lagrangian $L \subset T^{*} M$ Hamiltonian-isotopic to the zero section admits a generating function quadratic at infinity. The GFQI is unique up to stabilisation $S \mapsto S \oplus Q$ , fibre-preserving diffeomorphism, and additive constant.*

The proof (Hörmander) constructs the GFQI by writing the Hamiltonian flow's action functional as an integral over time-discretisations and showing the discretisation gives a finite-dimensional reduction with the required quadratic-at-infinity behaviour. The uniqueness (Sikorav, Viterbo) is a subtle deformation argument tracking the GFQI through Hamiltonian isotopies ^{[Viterbo 1992]}.

Spectral invariants and Viterbo's symplectic capacity. Given a GFQI $S : M \times R^{N} \to R$ with a uniform quadratic form $Q$ at infinity, the sublevel sets ${S \leq c}$ are well-defined as relative pairs $({S \leq c}, {S \leq - R})$ for $R ≫ 0$ . The relative cohomology $H^{*} ({S \leq c}, {S \leq - R})$ stabilises in $c$ near critical values, and minimax arguments produce spectral numbers $c (α, S) \in R$ for cohomology classes $α \in H^{*} (M)$ . Viterbo defines the capacity

c (L) := c (μ, S) - c (1, S),

where $μ$ is the fundamental class of $M$ and $1$ is the unit. The Sikorav-Viterbo uniqueness theorem ensures $c (L)$ depends only on $L$ , not on the chosen GFQI. Putting these together, the capacity is a symplectic invariant — monotonic under symplectic embeddings and thereby an obstruction to embedding a Lagrangian into a smaller volume than its capacity prescribes ^{[Viterbo 1992]}.

Generating functions vs Floer homology. The Sikorav-Viterbo spectral invariants compute the same numerical data as Lagrangian Floer homology spectral numbers in the cotangent-bundle setting; this is the GF=Floer theorem (Milinković, Oh 1997, building on Viterbo's earlier work). The technical content is that the chain complex of sublevel-set inclusions in GFQI cohomology is filtered-quasi-isomorphic to the Floer complex, with the filtrations matching. This identifies generating-function topology with pseudoholomorphic-curve topology in the cotangent setting and is one of the foundational results turning generating functions into a fully equivalent technology to Floer's ^{[McDuff-Salamon Ch. 11]}.

Discrete action principle and Arnold's conjecture. Conley-Zehnder 1983 proved the Arnold conjecture for $T^{2 n}$ by writing the time- $1$ Hamiltonian symplectomorphism as a composition of small-time steps, each with a type-1 generating function, summing to a discrete action $Φ$ on a sequence space $(M)^{N}$ . Critical points of $Φ$ on the torus quotient are exactly fixed points of the original symplectomorphism. Morse theory on $(T^{2 n})^{N}$ gives the lower bound $# Fix \geq 4^{n}$ (counting multiplicity) or $\geq 2 n + 1$ (Lusternik-Schnirelmann count). The argument generalises to any closed symplectic manifold whose minimal Chern number is large enough; the full conjecture for arbitrary closed symplectic manifolds was settled later by Floer and refined by many (Hofer, Salamon, Fukaya-Ono, Liu-Tian, Pardon) ^{[McDuff-Salamon Ch. 11]}.

Synthesis. Generating functions identify dynamics with variational structure: a symplectomorphism with a function, a Lagrangian intersection with a critical-point count, a Hamiltonian flow with a discrete action sum. The bridge from local algebra (the type-1 sign convention) to global topology (Viterbo capacity, GF=Floer) is the same machinery applied at three different stratifications — graphical neighbourhood, GFQI extension, asymptotic-quadratic stabilisation. Putting these together, the foundational reason generating functions are useful is that they translate symplectic topology into Morse theory of $S$ on a base manifold, and putting these together with the discrete-action principle, one sees that the moduli space of Hamiltonian fixed points is the moduli space of critical points of an associated function — exactly the bridge that Arnold conjectured and Conley-Zehnder verified for tori. The central insight is that the entire variational toolkit of nineteenth-century classical mechanics — Hamilton-Jacobi, action-angle, Maupertuis-Jacobi — is a single statement about the geometry of generating functions on a Lagrangian.

Full proof set [Master]

Lemma (cotangent-bundle product identification). The map

Θ : T^{*} M \times T^{*} M \to T^{*} (M \times M), (q_{1}, p_{1}, q_{2}, p_{2}) \mapsto ((q_{1}, q_{2}), (p_{1}, - p_{2})),

is a fibrewise vector-bundle isomorphism over $id_{M \times M}$ satisfying $\Theta^\lambda_{M \times M} = \lambda_M \boxminus \lambda_M $, w h er e$ \lambda_{M \times M} $i s t h e L i o uv i l l e f or m o n$ T^(M \times M) $an d$ \lambda_M \boxminus \lambda_M := \mathrm{pr}_1^\lambda_M - \mathrm{pr}2^\lambda_M $. C o n se q u e n tl y$ \Theta^\omega{M \times M} = \omega_M \oplus -\omega_M$.*

Proof. In local Darboux coordinates $(q_{1}^{i}, q_{2}^{j}, p_{1, i}, - p_{2, j})$ on $T^{*} (M \times M)$ vs $(q_{1}^{i}, p_{1, i}, q_{2}^{j}, p_{2, j})$ on $T^{*} M \times T^{*} M$ , $Θ$ is the identity on base coordinates and $(p_{1}, p_{2}) \mapsto (p_{1}, - p_{2})$ on fibre coordinates. The Liouville form on $T^{*} (M \times M)$ pulls back to $p_{1, i} d q_{1}^{i} + (- p_{2, j}) d q_{2}^{j} = λ_{M} ⊟ λ_{M}$ . Taking $- d$ gives the symplectic identity. $□$

Lemma (Lagrangian-section criterion). *A section $σ : U \to T^{*} N$ of the cotangent bundle, viewed as a one-form $σ \in Ω^{1} (U)$ , has Lagrangian image $σ (U) \subset T^{*} N$ if and only if $σ$ is closed.*

Proof. Compute $σ^{*} ω_{N} = σ^{*} (- d λ_{N}) = - d (σ^{*} λ_{N}) = - d σ$ , since $σ^{*} λ_{N} = σ$ for the canonical Liouville form on the cotangent bundle. The image is Lagrangian iff $σ^{*} ω_{N} = 0$ iff $d σ = 0$ . $□$

Theorem (Hörmander GFQI existence, sketch of proof). *Every closed Lagrangian $L \subset T^{*} M$ Hamiltonian-isotopic to the zero section admits a GFQI.*

Proof sketch. Discretise the Hamiltonian isotopy $ϕ_{t}$ , $t \in [0, 1]$ , into $N$ small-time steps $ϕ_{t_{k}} = ϕ_{1/ N}^{k}$ , each having a type-1 generating function $F_{k}$ on a small graphical neighbourhood. The composition's generating function is the sum

S (q, q_{1}, \dots, q_{N - 1}) := k = 0 \sum N - 1 F_{k} (q_{k}, q_{k + 1}), q_{0} := q, q_{N} := terminal point parametrising L .

Choose the discretisation fine enough that each $F_{k}$ on a neighbourhood of the diagonal is well-defined; the auxiliary fibre coordinate is $ξ := (q_{1}, \dots, q_{N - 1}) \in M^{N - 1}$ . Outside a compact set in $M \times R^{N d i m M}$ , replace each $F_{k}$ by its Taylor expansion to second order — the identity-like blocks become quadratic in displacement variables, giving the quadratic-at-infinity behaviour. The resulting $S : M \times R^{N d i m M} \to R$ is a GFQI for $L$ on its critical-fibre image. $□$

Theorem (Sikorav-Viterbo uniqueness). Two GFQIs $S_{0}, S_{1}$ for the same Lagrangian (both Hamiltonian-isotopic to the zero section) differ by stabilisation, fibre-preserving diffeomorphism, and additive constant.

Proof sketch. Connect $S_{0}, S_{1}$ by a path of GFQIs $S_{t}$ for the family of Lagrangians along the Hamiltonian isotopy. Use the parametric Morse lemma to deform $S_{t}$ in stabilisations; the obstruction to global continuity is the additive-constant ambiguity, which is a constant of integration along the isotopy. The full proof is in Viterbo 1992 ^{[Viterbo 1992]}. $□$

Corollary (Viterbo capacity is a symplectic invariant). The capacity $c (L) := c (μ, S) - c (1, S)$ depends only on $L$ , not on the GFQI $S$ .

Proof. Stabilisation preserves $c (α, S)$ for every cohomology class $α$ (sublevels of $S \oplus Q$ deformation-retract onto sublevels of $S$ along the negative-eigendirections of $Q$ ). Fibre-preserving diffeomorphism preserves sublevel topology. Additive constants shift $c (α, S)$ by the same constant for every $α$ , hence cancel in the difference $c (μ, S) - c (1, S)$ . By Sikorav-Viterbo uniqueness, this is the complete list of ambiguities. $□$

Connections [Master]

Lagrangian submanifold 05.05.01. The input definition. Generating functions are the encoding of Lagrangians as (multi-valued, local, or asymptotic-quadratic) functions on a base.
Weinstein neighbourhood theorem 05.05.02. The local model. The graph of a symplectomorphism in $T^{*} M \times T^{*} M$ becomes a section of $T^{*} (M \times M)$ near the diagonal via Weinstein, and that section is the differential of a generating function.
Symplectic manifold 05.01.02. The ambient context. Generating functions presuppose the canonical structure on $T^{*} M$ and the canonical-form identification on the doubled cotangent.
Non-squeezing theorem 05.07.01. Viterbo's symplectic capacity, defined via GFQI spectral invariants, is one of the standard capacity-theoretic obstructions to symplectic embeddings; the non-squeezing theorem and its capacity formulation are a cornerstone consequence.
Arnold conjecture 05.08.01. The fixed-point version of Arnold's conjecture rests on the discrete-action principle: critical points of a generating function on the diagonal correspond to fixed points of the underlying symplectomorphism. Conley-Zehnder's 1983 proof for the torus is the original generating-function-theoretic argument.
Lagrangian Floer homology 05.08.02. GFQI spectral invariants compute the same symplectic-action data as Lagrangian Floer homology in the cotangent-bundle setting — the GF=Floer theorem of Milinković-Oh.

Historical & philosophical context [Master]

William Rowan Hamilton introduced generating functions in his 1834 paper On a general method in dynamics (Phil. Trans. Roy. Soc.) ^{[Hamilton 1834]} in the context of classical mechanics: he showed that the principal function — the action evaluated as a function of initial and final configurations — generates the time-evolution map of a mechanical system. The Hamilton-Jacobi equation was the partial differential equation Hamilton derived for this generating function. Carl Gustav Jacobi systematised the four classical types in his 1842-43 Königsberg lectures (later published as Vorlesungen über Dynamik, 1866), making generating-function transformations a standard tool of nineteenth-century analytical mechanics.

The modern symplectic-geometric framing emerged in the 1960s and 70s. Vladimir Maslov's 1965 book Theory of Perturbations and Asymptotic Methods introduced the idea that generating functions should be allowed to depend on auxiliary fibre variables, with the physical Lagrangian recovered by intersecting with the fibre-critical locus. Lars Hörmander's 1971 paper Fourier integral operators I (Acta Mathematica 127) ^{[Hörmander 1971]} formalised this in the GFQI framework as part of the symbol calculus for Fourier integral operators, proving existence and a uniqueness theorem up to a list of explicit operations. Alan Weinstein's 1971-73 papers on Lagrangian submanifolds ^{[Weinstein 1971]} established the symplectic-geometric vocabulary in which generating functions sit as "Lagrangians = generalised functions."

The connection to symplectic topology was made by Conley-Zehnder 1983 (Arnold conjecture for the torus), Sikorav 1986-87 (uniqueness of GFQI in the symplectic setting), and Viterbo 1992 Symplectic topology as the geometry of generating functions (Math. Ann. 292) ^{[Viterbo 1992]}, which established generating functions as a parallel technology to Floer homology, with spectral invariants computing the same data. The GF=Floer theorem of Milinković-Oh 1997 made the parallelism into an isomorphism in the cotangent-bundle setting.

Bibliography [Master]

[object Promise]

Prerequisites

05.05.01
05.05.02
05.01.02

Used in

05.07.01
05.08.01

Tier anchors

beginner: Arnold — *Mathematical Methods of Classical Mechanics* §47 (informal)
intermediate: Cannas da Silva *Lectures on Symplectic Geometry* §22; McDuff-Salamon *Introduction to Symplectic Topology* Ch. 9
master: Hörmander 1971 (originator GFQI); Viterbo 1992 *Symplectic topology as the geometry of generating functions*; Cannas §22; McDuff-Salamon Ch. 9 + 11

References

TODO_REF
Arnold — Mathematical Methods of Classical Mechanics · §47 generating functions, classical-mechanics setting
TODO_REF
Cannas da Silva — Lectures on Symplectic Geometry · §22 generating functions
TODO_REF
McDuff-Salamon — Introduction to Symplectic Topology · Ch. 9 generating functions; Ch. 11 Lagrangian submanifolds
TODO_REF
Hörmander 1971 — Fourier integral operators I · Acta Math. 127, originator paper for GFQI
TODO_REF
Viterbo 1992 — Symplectic topology as the geometry of generating functions · Math. Ann. 292, spectral invariants
TODO_REF
Hamilton 1834 — On a general method in dynamics · Phil. Trans. Roy. Soc., originator of generating functions
TODO_REF
Sikorav 1986-87 — Sur les immersions lagrangiennes · uniqueness of GFQI up to stabilisation

Reviewer

TBD

Estimated time

beginner: 16m
intermediate: 45m
master: 75m