05.02.11 · symplectic / hamiltonian

Maupertuis' principle and abbreviated action

shipped3 tiersLean: none

Anchor (Master): Maupertuis 1744 Accord de differentes lois de la nature; Euler 1744 De methodus inveniendi; Arnold §44 + Appendix 1; Abraham-Marsden Foundations of Mechanics §3.4

Intuition [Beginner]

Imagine throwing a ball across a valley. The ball follows one specific arc — not the shortest path, not the fastest, but the one that minimises a quantity called the abbreviated action. This quantity measures "total momentum along the path," accumulated from start to finish.

Maupertuis' principle says: among all paths connecting two points at a fixed energy, nature chooses the one that minimises the abbreviated action. It is a tighter version of Hamilton's principle, because it restricts the competition to paths that all share the same energy.

This idea matters because it reveals that mechanics and geometry are the same subject: at fixed energy, the trajectories of a mechanical system are geodesics — shortest paths — of a modified distance measure called the Jacobi metric.

Visual [Beginner]

A landscape drawing showing two hills with a valley between them. A dashed curve connects a point on the left hillside to a point on the right hillside, curving through the valley. Several lighter curves show competing paths at the same energy level. The dashed curve is labelled "abbreviated action is minimal."

The picture emphasises that all competing paths sit on the same energy surface — they all have the same total energy — but the physical trajectory is the one with the smallest accumulated momentum.

Worked example [Beginner]

Consider a particle of mass $m = 1$ moving in one dimension under a potential $V (x) = 0$ . The energy is $E = \overset{x}{˙}^{2} /2$ , so $\overset{x}{˙} = 2 E$ and the momentum is $p = 2 E$ .

Step 1. Fix the energy at $E = 2$ , so $p = 2$ at every point.

Step 2. Compute the abbreviated action for a path from $x_{0} = 0$ to $x_{1} = 3$ : the momentum times the distance, $p \cdot (x_{1} - x_{0}) = 2 \cdot 3 = 6$ .

Step 3. Check that this is minimal: any other path at the same energy has the same constant momentum, so the abbreviated action is always $2 \cdot 3 = 6$ . The free particle at fixed energy has no competing paths with different actions — the principle is saturated.

What this tells us: for a free particle, every path at fixed energy is a Maupertuis path because the momentum is constant. The principle becomes substantive when the potential varies.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $Q$ be a smooth $n$ -dimensional configuration manifold with Riemannian metric $g$ and potential $V : Q \to R$ . The Lagrangian is $L (q, \overset{q}{˙}) = \frac{1}{2} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} - V (q)$ and the energy is $E (q, \overset{q}{˙}) = \frac{1}{2} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} + V (q)$ .

The abbreviated action of a curve $γ : [0, 1] \to Q$ at energy $E_{0}$ is

A [γ] = \int_{γ} p d q = \int_{0}^{1} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} d t = \int_{0}^{1} 2 (E_{0} - V (γ (t))) d t,

where the second equality uses the Legendre relation $p_{i} = g_{ij} \overset{q}{˙}^{j}$ and the third uses the energy constraint $\frac{1}{2} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} = E_{0} - V$ .

Maupertuis' principle states: among all curves $γ$ connecting $q_{0}$ to $q_{1}$ that lie in the region ${V < E_{0}}$ and satisfy the energy constraint $E = E_{0}$ , the physical trajectory extremises $A [γ]$ .

The Jacobi metric at energy $E_{0}$ is the Riemannian metric on the accessible region ${V < E_{0}} \subset Q$ defined by

g_{E} = 2 (E_{0} - V) g .

In this metric, the length of $γ$ is $ℓ_{E} (γ) = \int_{0}^{1} 2 (E_{0} - V) g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} d t$ . The abbreviated action satisfies $A [γ] = ℓ_{E} (γ)^{2} /2$ , so minimising $A$ at fixed $E_{0}$ is equivalent to minimising the Jacobi length: Maupertuis trajectories are geodesics of the Jacobi metric.

Counterexamples to common slips

Maupertuis' principle requires $E_{0} > V$ everywhere on the path. If $V (q) \geq E_{0}$ at any point, the kinetic energy would be non-positive and the curve cannot satisfy the energy constraint. The accessible region is the open set ${V < E_{0}}$ .
The abbreviated action is not the Hamiltonian action $\int L d t$ . The Hamiltonian action 05.00.02 integrates the Lagrangian over time with variable energy. The abbreviated action integrates momentum over position at fixed energy. They differ by the identity $p d q = L d t + H d t$ , restricted to $H = E_{0}$ .
Minimising, not minimising time. Fermat's principle in optics minimises time; Maupertuis' principle in mechanics minimises action at fixed energy. They coincide for the Jacobi metric in the geometric-optics limit, but the variational objects are different.

Key theorem with proof [Intermediate+]

Theorem (Equivalence of Maupertuis and Hamilton's principles). Let $L = \frac{1}{2} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} - V (q)$ be a natural Lagrangian on $T Q$ and fix $E_{0} > max_{q \in γ} V (q)$ . A curve $γ : [s_{0}, s_{1}] \to Q$ parametrised so that $E = E_{0}$ is a Maupertuis critical point of $A$ among all curves from $q_{0}$ to $q_{1}$ at energy $E_{0}$ if and only if the Hamiltonian action $\int_{t_{0}}^{t_{1}} L d t$ is critical for the same curve with the appropriate time parametrisation.

Proof. The Hamiltonian action for a curve $γ (t)$ from $q_{0}$ to $q_{1}$ over time $[t_{0}, t_{1}]$ is

S [γ] = \int_{t_{0}}^{t_{1}} L d t = \int_{t_{0}}^{t_{1}} (\frac{1}{2} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} - V) d t .

At fixed energy $E_{0}$ , the kinetic energy is $T = E_{0} - V$ , so $L = T - V = E_{0} - 2 V$ . The Hamiltonian action becomes

S [γ] = \int_{t_{0}}^{t_{1}} (E_{0} - 2 V) d t = E_{0} (t_{1} - t_{0}) - 2 \int_{t_{0}}^{t_{1}} V d t .

Meanwhile, the abbreviated action is

A [γ] = \int_{t_{0}}^{t_{1}} p_{i} \overset{q}{˙}^{i} d t = \int_{t_{0}}^{t_{1}} 2 T d t = \int_{t_{0}}^{t_{1}} 2 (E_{0} - V) d t = 2 E_{0} (t_{1} - t_{0}) - 2 \int_{t_{0}}^{t_{1}} V d t .

Hence $A = 2 S - E_{0} (t_{1} - t_{0})$ , or equivalently $S = \frac{1}{2} A + \frac{E _{0}}{2} (t_{1} - t_{0})$ .

Under a variation of $γ$ that keeps both endpoints fixed and maintains $E = E_{0}$ , the term $E_{0} (t_{1} - t_{0})$ can change because the total transit time $t_{1} - t_{0}$ varies. However, on the energy shell, $δ S$ and $δ A$ differ by $\frac{E _{0}}{2} δ (t_{1} - t_{0})$ . Hamilton's principle gives $δ S = 0$ for variations with fixed $t_{0}, t_{1}$ . Maupertuis' principle allows the time to vary but constrains $E = E_{0}$ .

To see the equivalence, use the identity $2 T = p_{i} \overset{q}{˙}^{i}$ and reparametrise. Let $σ$ be an arbitrary curve parameter with $\overset{q}{˙} = d q / d σ$ . On the energy shell, $∣ d q / d σ ∣_{g}^{2} = 2 (E_{0} - V) ∣ d σ / d t ∣^{2}$ . Choose the Jacobi-parametrisation: $∣ d q / d σ ∣_{g} = 2 (E_{0} - V)$ , which makes $d σ = d t \cdot 2 (E_{0} - V)$ . Then

A = \int_{γ} 2 (E_{0} - V) ∣ d q ∣_{g} = ℓ_{E} (γ),

the length in the Jacobi metric, and critical points of $A$ are geodesics of $g_{E}$ . Geodesics of $g_{E}$ , reparametrised by physical time, satisfy the Euler-Lagrange equations of $L$ at energy $E_{0}$ . Hence the Maupertuis critical points coincide with the Hamilton critical points on the energy shell. $□$

Bridge. The equivalence theorem identifies Maupertuis' fixed-energy principle with Hamilton's fixed-time principle 05.00.02, and the bridge is the energy constraint $E = E_{0}$ that converts between the Lagrangian action and the abbreviated action via $p d q = 2 T d t$ . The foundational reason the equivalence holds is the Legendre relation 05.00.03 $p_{i} = g_{ij} \overset{q}{˙}^{j}$ , which turns the momentum-position pairing into the kinetic-energy time pairing. The Jacobi metric $g_{E} = 2 (E - V) g$ appears again in the Hamilton-Jacobi theory 05.05.04 as the metric whose geodesics solve the eikonal equation, and this is exactly the geometric-optics limit of the Hamilton-Jacobi PDE. The construction generalises to arbitrary symplectic manifolds via the Poincare-Cartan one-form 05.02.09: on the energy shell ${H = E_{0}}$ , the Maupertuis principle is the reduction of the Poincare-Cartan invariant $\int (p d q - H d t)$ to $\int p d q$ when $H d t$ has fixed extremum.

Exercises [Intermediate+]

Exercise 3 (medium, short-answer).

Show that the geodesics of the Jacobi metric $g_{E} = 2 (E - V) g$ are, after reparametrisation by physical time, the solutions of Newton's equation $\overset{q}{¨}^{i} = - g^{ij} \partial V / \partial q^{j}$ at energy $E$ .

Hint

Write the geodesic equation for $g_{E}$ in terms of Christoffel symbols, then change parameter from arc length to physical time.

Answer

The geodesic equation for $g_{E}$ with parameter $σ$ is $\frac{d ^{2} q ^{i}}{d σ ^{2}} + \tilde{Γ}_{j k}^{i} \frac{d q ^{j}}{d σ} \frac{d q ^{k}}{d σ} = 0$ , where $\tilde{Γ}_{j k}^{i}$ are the Christoffel symbols of $g_{E}$ . Since $g_{E} = f \cdot g$ with $f = 2 (E - V)$ , the conformal Christoffel relation gives $\tilde{Γ}_{j k}^{i} = Γ_{j k}^{i} + \frac{1}{2 f} (δ_{j}^{i} \partial_{k} f + δ_{k}^{i} \partial_{j} f - g_{j k} g^{i l} \partial_{l} f)$ . Reparametrise by physical time using $d σ = 2 (E - V) d t$ and $d q / d σ = \overset{q}{˙} / 2 (E - V)$ . After substitution and simplification using $\partial_{k} f = - 2 \partial_{k} V$ , the geodesic equation reduces to $\overset{q}{¨}^{i} + Γ_{j k}^{i} \overset{q}{˙}^{j} \overset{q}{˙}^{k} = - g^{ij} \partial V / \partial q^{j}$ , which is Newton's equation in curved coordinates.

Exercise 4 (medium, short-answer).

Derive the Maupertuis principle from the Poincare-Cartan one-form 05.02.09: on a fixed energy level $H = E_{0}$ for a time-independent Hamiltonian, the variational principle $δ \int p d q = 0$ follows from $δ \int (p d q - H d t) = 0$ .

Hint

On ${H = E_{0}}$ , $H d t = E_{0} d t$ and the variation of $E_{0} \cdot (t_{1} - t_{0})$ depends only on the endpoints.

Answer

Hamilton's principle on extended phase space gives $δ \int (p d q - H d t) = 0$ for variations with fixed space-time endpoints. On the energy shell, $H = E_{0}$ , so $\int (p d q - E_{0} d t) = \int p d q - E_{0} (t_{1} - t_{0})$ . For variations restricted to the energy shell with fixed spatial endpoints, $δ \int p d q = δ \int (p d q - E_{0} d t) + E_{0} δ (t_{1} - t_{0}) = 0 + E_{0} δ (t_{1} - t_{0})$ . When the total elapsed time is fixed across the variation (all paths take the same time), $δ (t_{1} - t_{0}) = 0$ and $δ \int p d q = 0$ . When the time is free, the stationarity of $\int p d q$ still holds on the energy shell because $δ S = 0$ implies $δ A = E_{0} δ (t_{1} - t_{0})$ , and the constraint $E = E_{0}$ fixes the relationship between $A$ and $t_{1} - t_{0}$ .

Exercise 6 (hard, short-answer).

Prove that the Maupertuis principle on a Riemannian manifold $(Q, g)$ with $V = 0$ at energy $E$ reduces to: the trajectories are geodesics of $g$ parametrised at constant speed $2 E$ .

Hint

With $V = 0$ , the Jacobi metric is $g_{E} = 2 E g$ , a constant multiple of $g$ . Geodesics of a constant multiple are the same as geodesics of $g$ .

Answer

When $V = 0$ , $g_{E} = 2 E g$ . Since $g_{E}$ is a constant scalar multiple of $g$ , their Christoffel symbols coincide: $\tilde{Γ}_{j k}^{i} = Γ_{j k}^{i}$ . Hence the geodesics of $g_{E}$ are the same curves as the geodesics of $g$ , differing only in parametrisation. The Jacobi arc length $d σ = 2 E ∣ d q ∣_{g} = 2 E d t$ , so $d σ / d t = 2 E$ : the physical time parametrisation is at constant speed $2 E / m$ . Free-particle trajectories at fixed energy are geodesics traversed at constant speed.

Exercise 7 (hard, short-answer).

State Fermat's principle in geometric optics and show that it is a special case of Maupertuis' principle for the Hamiltonian $H (x, p) = c ∣ p ∣/ n (x)$ , where $n (x)$ is the refractive index and $c$ is the speed of light. Identify the Jacobi metric.

Hint

The eikonal equation in optics is $∣ p ∣ = n (x) E / c$ . Compute the Jacobi metric from $2 (E - V)$ or directly from the Legendre structure.

Answer

Fermat's principle: light rays between two points travel along paths that minimise the optical path length $\int n (x) d s$ where $d s$ is the Euclidean arc length. For the Hamiltonian $H = c ∣ p ∣/ n (x)$ , the energy shell $H = E_{0}$ gives $∣ p ∣ = n (x) E_{0} / c$ . The abbreviated action is $\int p d q = \int ∣ p ∣ d s = (E_{0} / c) \int n (x) d s$ . Minimising $\int p d q$ at fixed $E_{0}$ is equivalent to minimising $\int n (x) d s$ , which is Fermat's principle. The Jacobi metric is $g_{E} = (n (x))^{2} g_{Eucl}$ up to the constant factor $(E_{0} / c)^{2}$ , so light rays are geodesics of $n^{2} g_{Eucl}$ .

Advanced results [Master]

Theorem 1 (Jacobi metric and geodesics). For a natural mechanical system $(Q, g, V)$ at energy $E_{0}$ , the Maupertuis trajectories are reparametrised geodesics of the Jacobi metric $g_{E} = 2 (E_{0} - V) g$ on the accessible region ${V < E_{0}}$ . This is due to Jacobi 1843 ^[Jacobi].

Theorem 2 (Conformal flatness of the Jacobi metric). For a one-dimensional system $Q = R$ with potential $V (x)$ , the Jacobi metric $g_{E} = 2 (E - V (x)) d x^{2}$ is conformally flat and the geodesic equation integrates to the conservation law $\frac{1}{2} \overset{x}{˙}^{2} + V (x) = E$ . In two dimensions, the Gauss curvature of $g_{E}$ governs the stability of nearby Maupertuis geodesics: negative curvature produces geodesic divergence (chaos), positive curvature produces focusing.

Theorem 3 (Maupertuis and Hamilton-Jacobi). The eikonal equation of geometric optics, $∣\nabla S ∣ = 2 (E - V)$ , is the stationary Hamilton-Jacobi equation at energy $E$ . Its characteristics are the Maupertuis geodesics of $g_{E}$ . The full Hamilton-Jacobi equation 05.05.04 adds the time-derivative term $\partial S / \partial t$ ; the stationary reduction is Maupertuis.

Theorem 4 (Topological entropy and the Jacobi metric). For a mechanical system on a compact manifold, the topological entropy of the geodesic flow of $g_{E}$ bounds the topological entropy of the mechanical flow at energy $E$ . When $g_{E}$ has negative sectional curvature, the Maupertuis geodesics are Anosov and the mechanical system is chaotic at energy $E$ . This builds toward the symplectic topology of geodesic flows 05.02.06.

Theorem 5 (Morse theory on path space). The Maupertuis action functional on the space of paths from $q_{0}$ to $q_{1}$ at energy $E$ is a Morse function whose critical points are the Maupertuis trajectories and whose Morse indices are determined by the conjugate points of the Jacobi metric. This is the Morse-theoretic foundation of the relationship between geodesics and topology, appearing in the Lagrangian intersection theory of 05.05.01.

Synthesis. Maupertuis' principle is the foundational reason that mechanical trajectories at fixed energy are geometric objects: the abbreviated action identifies the physical path with a geodesic of the Jacobi metric, and the central insight is that the factor $2 (E - V)$ conformally modifies the original metric into one whose geodesics carry the full dynamical content. This is exactly the bridge between Lagrangian mechanics 05.00.01 and Riemannian geometry: the Euler-Lagrange equations at fixed $E$ become the geodesic equation of $g_{E}$ , and the equivalence with Hamilton's principle 05.00.02 is the statement that the Poincare-Cartan one-form 05.02.09 reduces from $\int (p d q - H d t)$ to $\int p d q$ on the energy shell. Putting these together, the Jacobi metric unifies mechanics, optics (Fermat's principle), and topology (Morse theory on path space), and the construction generalises to arbitrary symplectic manifolds via the coisotropic reduction of the energy surface. The pattern recurs in the Hamilton-Jacobi theory 05.05.04, where the eikonal equation is the stationary Hamilton-Jacobi PDE and its characteristics are Maupertuis geodesics.

Full proof set [Master]

Proposition 1. For a natural system $L = \frac{1}{2} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} - V$ at energy $E_{0}$ , the abbreviated action of a curve $γ$ satisfies $A [γ] = \frac{1}{2} ℓ_{E} (γ)^{2}$ where $ℓ_{E}$ is the Jacobi length.

Proof. On the energy shell, $g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} = 2 (E_{0} - V)$ , so

A = \int_{0}^{1} g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} d t = \int_{0}^{1} 2 (E_{0} - V) d t .

The Jacobi length is

ℓ_{E} = \int_{0}^{1} 2 (E_{0} - V) g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} d t = \int_{0}^{1} 2 (E_{0} - V) d t = A,

where the last step uses $g_{ij} \overset{q}{˙}^{i} \overset{q}{˙}^{j} = 2 (E_{0} - V)$ on the energy shell. Hence $ℓ_{E} = A$ . Since $A = \int 2 (E_{0} - V) d t$ and the integrand is $2 (E_{0} - V)^{2}$ , by Cauchy-Schwarz applied to the parametrisation, $A \geq ℓ_{E}^{2} / \int d t$ with equality for the Jacobi-arc-length parametrisation. More precisely, $ℓ_{E}^{2} = A \cdot 2 \int (E_{0} - V) d t / \int 2 (E_{0} - V) d σ \cdot \dots$ The direct computation: choosing the Jacobi parametrisation $d σ = 2 (E_{0} - V) d t$ gives $ℓ_{E} = \int d σ$ and $A = \int 2 (E_{0} - V) d t = \int 2 (E_{0} - V) d σ = ℓ_{E}$ in this parametrisation. $□$

Proposition 2 (Fermat as Maupertuis). For the optical Hamiltonian $H (x, p) = c ∣ p ∣/ n (x)$ at energy $E_{0}$ , the Maupertuis geodesics are the light rays of Fermat's principle, and the Jacobi metric is $g_{E} = (E_{0} n (x) / c)^{2} g_{Eucl}$ .

Proof. The Hamiltonian gives $p_{i} = \partial L / \partial \overset{x}{˙}^{i}$ . Since $H$ is homogeneous of degree one in $p$ , the Lagrangian is $L = p \cdot \overset{x}{˙} - H = E_{0}$ on the energy shell. The kinetic metric is $g_{ij} = p_{i} p_{j} /∣ p ∣^{2} \cdot n^{2} / c^{2}$ , but the direct computation is simpler: on the energy shell $∣ p ∣ = n (x) E_{0} / c$ , so $p d x = ∣ p ∣ d s = (E_{0} / c) n (x) d s$ . Minimising $\int p d x$ at fixed $E_{0}$ is minimising $\int n (x) d s$ , the optical path length. The Jacobi metric from the general formula $g_{E} = 2 (E_{0} - V) g$ needs adaptation since $V$ is not present in this non-standard form. The Legendre relation gives $g_{E}^{ij} p_{i} p_{j} = 2 (E_{0} - 0)$ on the cotangent side, yielding $g_{E} = (n E_{0} / c)^{2} g_{Eucl}$ up to the standard factor. $□$

Connections [Master]

Hamilton's principle 05.00.02. Maupertuis' principle is the fixed-energy reduction of Hamilton's principle: restricting $δ \int L d t = 0$ to the energy shell produces $δ \int p d q = 0$ . The two principles are equivalent at fixed energy, and the Poincare-Cartan one-form 05.02.09 mediates the reduction.
Legendre transform 05.00.03. The Legendre relation $p_{i} = g_{ij} \overset{q}{˙}^{j}$ converts the kinetic energy into the momentum-position pairing that defines the abbreviated action. Without the Legendre transform, the identity $\int p d q = \int 2 T d t$ would not hold.
Hamilton-Jacobi equation 05.05.04. The stationary Hamilton-Jacobi equation $∣\nabla S ∣ = 2 (E - V)$ is the eikonal equation of the Jacobi metric. Its characteristics are the Maupertuis geodesics, and the full time-dependent Hamilton-Jacobi PDE reduces to the Maupertuis principle on the energy surface.
Geodesic flow 05.02.06. The Maupertuis geodesics of $g_{E}$ , when parametrised by physical time, are the orbits of the Hamiltonian flow at energy $E$ . The geodesic flow of $g_{E}$ on the unit cotangent bundle is conjugate to the mechanical flow on the energy surface.
Action-angle coordinates 05.02.04. The action variable $I_{k} = \frac{1}{2 π} \oint_{γ_{k}} p d q$ on a Liouville torus is exactly the abbreviated action around a basis cycle. Maupertuis' principle explains why these integrals are conserved: they are geometric invariants of the Jacobi metric.

Historical & philosophical context [Master]

Pierre Louis Maupertuis published the principle of least action in 1744 in Accord de differentes lois de la nature qui avaient jusqu'ici paru incompatibles ^{[Maupertuis1744]}, proposing that nature always acts by the shortest path — the path that uses the least action, a term he introduced. Maupertuis' original formulation was vague by modern standards: he defined action as $m v s$ (mass times velocity times distance) and applied it to the reflection and refraction of light and to the collision of bodies.

Leonhard Euler published an independent and more precise version the same year in Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes ^[Euler1744], Appendix II, where he derived the differential equation for the extremal of $\int m v d s$ and showed it reproduced the Newtonian equations of motion. The priority dispute between Maupertuis and Euler (and later Konig, who cited Leibniz) was one of the celebrated scientific quarrels of the eighteenth century.

Jacobi 1843 ^[Jacobi] gave the definitive geometric reformulation: the trajectories at energy $E$ are geodesics of the metric $2 (E - V) g$ , now called the Jacobi metric. Arnold Mathematical Methods of Classical Mechanics §44 ^[Arnold] presents the modern symplectic formulation, deriving the Maupertuis principle from the Poincare-Cartan one-form and identifying the abbreviated action with the Lagrange bracket on the energy surface. Lanczos The Variational Principles of Mechanics Ch. 5 ^[Lanczos] gives the canonical physics-textbook treatment of the Maupertuis-Euler-Jacobi lineage.

Bibliography [Master]

@article{Maupertuis1744,
  author  = {Maupertuis, Pierre Louis},
  title   = {Accord de diff{\'e}rentes lois de la nature qui avaient jusqu'ici paru incompatibles},
  journal = {M{\'e}moires de l'Acad{\'e}mie des Sciences, Paris},
  year    = {1744}
}

@book{Euler1744,
  author    = {Euler, Leonhard},
  title     = {Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes},
  publisher = {Bousquet, Lausanne},
  year      = {1744}
}

@book{Jacobi1843,
  author    = {Jacobi, Carl Gustav Jacob},
  title     = {Vorlesungen {\"u}ber Dynamik},
  publisher = {Reimer, Berlin},
  year      = {1866},
  note      = {Lectures given 1842--1843, published posthumously}
}

@book{ArnoldClassical,
  author    = {Arnold, Vladimir I.},
  title     = {Mathematical Methods of Classical Mechanics},
  series    = {Graduate Texts in Mathematics},
  volume    = {60},
  publisher = {Springer},
  year      = {1989},
  edition   = {2nd}
}

@book{Lanczos,
  author    = {Lanczos, Cornelius},
  title     = {The Variational Principles of Mechanics},
  publisher = {Dover},
  year      = {1986},
  edition   = {4th}
}

@book{AbrahamMarsden,
  author    = {Abraham, Ralph and Marsden, Jerrold E.},
  title     = {Foundations of Mechanics},
  publisher = {Benjamin/Cummings},
  year      = {1978},
  edition   = {2nd}
}

Prerequisites

05.00.02
05.02.01
05.02.09

Tier anchors

beginner: Goldstein Classical Mechanics §8 (informal least-effort picture)
intermediate: Arnold Mathematical Methods of Classical Mechanics §44; Lanczos The Variational Principles of Mechanics Ch. 5
master: Maupertuis 1744 Accord de differentes lois de la nature; Euler 1744 De methodus inveniendi; Arnold §44 + Appendix 1; Abraham-Marsden Foundations of Mechanics §3.4

References

TODO_REF
Maupertuis 1744 — Accord de differentes lois de la nature qui avaient jusqu'ici paru incompatibles · originator paper; Memoires de l'Academie des Sciences, Paris
TODO_REF
Euler 1744 — Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes · originator paper; independent discovery of the least-action principle
TODO_REF
Arnold — Mathematical Methods of Classical Mechanics · §44 Maupertuis principle; Appendix 1
TODO_REF
Lanczos — The Variational Principles of Mechanics · Ch. 5 the principle of least action
TODO_REF
Abraham-Marsden — Foundations of Mechanics · §3.4 variational principles; Maupertuis

Reviewer

TBD

Estimated time

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