Carl Gustav Jacob Jacobi

How to use this course

Carl Gustav Jacob Jacobi worked for barely twenty-five years and left his name on an unusual range of objects: Jacobi elliptic functions, the Jacobi triple product, the Jacobian determinant, the Jacobi identity, the Hamilton–Jacobi equation, the Jacobi symbol, the Jacobi integral of the three-body problem, Jacobi’s eigenvalue method and the Jacobian variety. The common thread is a habit he recommended to his students: when a problem resists, turn it around and study the inverse question.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations or proof sketches, and exercises with folded solutions. Lessons 04–06 cover elliptic and abelian functions (1827–1834); Lessons 07–09 cover mechanics and analysis (1836–1843); Lesson 10 covers number theory.

Timeline

Potsdam and Berlin, 1804–1826

Jacobi was born in Potsdam on 10 December 1804, the second of four children of Simon Jacobi, a prosperous Jewish banker. He was named Jacques Simon, in the French style then fashionable in Prussia, and later took the name Carl Gustav Jacob. His elder brother Moritz Hermann Jacobi became a physicist and engineer in Russia, known for the electric motor and electrotyping.

A maternal uncle taught him classical languages and mathematics at home. He entered the Potsdam Gymnasium around his twelfth birthday and was moved to the final class within a few months. Since the university would not admit him before sixteen, he spent four years in that class, reading Euler and trying to solve the general quintic by radicals.

In 1821 he entered the University of Berlin. For two years he divided his time between philology, philosophy and mathematics, then committed himself to mathematics. The Berlin lectures were weak, so he read Euler, Lagrange and Laplace on his own. By 1824 he had passed the state examinations to teach mathematics, Greek and Latin.

In 1825 he received his doctorate with a dissertation on partial fractions, Disquisitiones analyticae de fractionibus simplicibus; Enno Dirksen chaired the examination. In the same year he converted to Christianity, without which a university career was closed to him, and completed his habilitation. In the winter of 1825–26 he lectured in Berlin on the theory of curves and surfaces with conspicuous success.

Königsberg and the race with Abel, 1826–1829

Prussian authorities, seeing his promise, sent him in May 1826 to the University of Königsberg in East Prussia, where Bessel directed the observatory and Franz Neumann was building mathematical physics. Jacobi’s first Königsberg work was on cubic residues and on the transformation of elliptic integrals: when can \(\int dy/\sqrt{(1-y^2)(1-\lambda^2y^2)}\) be turned into \(M\int dx/\sqrt{(1-x^2)(1-k^2x^2)}\) by a rational substitution \(y=U(x)/V(x)\)?

In 1827 he announced results on transformations of all orders in notes to Heinrich Schumacher’s Astronomische Nachrichten, and on 5 August he wrote to Legendre, then 74, who had spent forty years on the subject. Legendre replied with delight and presented the results in Paris. The proofs Jacobi had at first were incomplete. In September 1827 the young Norwegian Niels Henrik Abel published in Crelle’s Journal the first part of his Recherches sur les fonctions elliptiques, which inverted the elliptic integral and found the double periodicity of the inverse function. Jacobi adopted the inversion at once, and the two men advanced in parallel through 1828, each pushing the other.

The result was Jacobi’s Fundamenta nova theoriae functionum ellipticarum (Königsberg, 1829), the first systematic treatise on elliptic functions. It introduced the functions \(\mathrm{sn},\mathrm{cn},\mathrm{dn}\), expanded them in infinite products and series, built the theory on four theta functions, and applied it to the arithmetic of sums of squares. Abel died in April 1829. Jacobi, who had been promoted to extraordinary professor in December 1827, insisted on Abel’s priority in the decisive idea of inversion.

In the summer of 1829 he travelled to Göttingen to see Gauss, and to Paris, where he met Legendre, Fourier and Poisson. In a letter to Legendre the next year, commenting on Fourier’s view that mathematics should serve physics, he wrote that its goal was l’honneur de l’esprit humain, the honour of the human mind.

The Königsberg school, 1829–1843

Jacobi married Marie Schwinck on 11 September 1831; they had eight children. Sources date his promotion to ordinary professor to 1829 or 1832; the formal step, a four-hour disputation in Latin, took place in 1832. He was elected to the Royal Society of London in 1833 and became a corresponding member of the Paris Académie in 1830.

Jacobi changed how mathematics was taught in Germany. Instead of repeating standard material, he lectured on his own current research, and he set students problems that led to publishable work. In 1834, with Neumann, he founded the mathematical-physical seminar at Königsberg, one of the first research seminars in any German university. His students included Friedrich Richelot, Otto Hesse, Carl Wilhelm Borchardt, Johann Rosenhain, Philipp Ludwig von Seidel and Wilhelm Scheibner. Through them and through Neumann’s students, Königsberg became a centre of mathematics in the generation before Karl Weierstrass’s Berlin.

The range of these years is extraordinary. In 1832 he formulated the inversion problem for abelian integrals (Lesson 06). In 1834 he showed that a rotating mass of fluid can be in equilibrium as an ellipsoid with three unequal axes, correcting Lagrange, and that a single-valued function cannot have three independent periods. He read William Rowan Hamilton’s papers on dynamics in 1834 or early 1835 and transformed them (Lesson 07). He found the Jacobi integral of the three-body problem (1836), the conjugate-point condition (1837), the Jacobi symbol (1837), the Canon arithmeticus (1839), and the general theory of determinants and the Jacobian (1841). In the winter of 1842–43 he lectured on dynamics; the notes, edited by Alfred Clebsch in 1866, became a classic.

Elliptic functions by inversion

For the general reader. The arcsine is the inverse of the integral \(\int dx/\sqrt{1-x^2}\), and its inverse, the sine, is a periodic function with simple rules. Abel and Jacobi applied the same idea to Legendre’s harder elliptic integral. The inverse function turned out to be periodic in two independent complex directions, and it obeys addition rules as neat as those for sine and cosine.

Definitions

Fix a modulus \(0<k<1\) and \(k'=\sqrt{1-k^2}\). Let

Jacobi called \(\varphi=\mathrm{am}\,u\) the amplitude and defined \(\mathrm{sn}\,u=\sin\varphi\), \(\mathrm{cn}\,u=\cos\varphi\), \(\mathrm{dn}\,u=\sqrt{1-k^2\sin^2\varphi}\). Thus \(\mathrm{sn}^2+\mathrm{cn}^2=1\) and \(\mathrm{dn}^2+k^2\mathrm{sn}^2=1\). Since \(du/d\varphi=1/\mathrm{dn}\,u\),

As \(k\to0\), \(\mathrm{sn}\to\sin\) and \(\mathrm{cn}\to\cos\); as \(k\to1\), \(\mathrm{sn}\to\tanh\).

Idea. Along the real axis, \(u\) runs from 0 to \(K\) as \(x\) runs from 0 to 1, and \(\mathrm{sn}\) oscillates with period \(4K\), like a sine. Jacobi’s imaginary transformation \(\mathrm{sn}(iu,k)=i\,\mathrm{sn}(u,k')/\mathrm{cn}(u,k')\) shows that along the imaginary axis \(\mathrm{sn}\) behaves like a function with real period \(4K'\) and a pole at \(iK'\), which gives the second period \(2iK'\).

This is Euler’s addition theorem for elliptic integrals, read through the inverse function. Algebraically it expresses the group law on the elliptic curve \(y^2=(1-x^2)(1-k^2x^2)\), with \(u\mapsto(\mathrm{sn}\,u,\ \mathrm{cn}\,u\,\mathrm{dn}\,u)\) a parametrization of the curve by \(\mathbb C/(4K\mathbb Z+2iK'\mathbb Z)\).

The pendulum

For \(\ddot\theta=-\omega_0^2\sin\theta\) with amplitude \(\theta_0<\pi\), set \(k=\sin(\theta_0/2)\). Conservation of energy gives \(\dot\theta^2=4\omega_0^2\big(k^2-\sin^2(\theta/2)\big)\), and the substitution \(\sin(\theta/2)=k\sin\varphi\) reduces it to \(\dot\varphi=\omega_0\sqrt{1-k^2\sin^2\varphi}\). Hence

Theta functions and sums of squares

For the general reader. In how many ways can a number be written as a sum of four squares? Lagrange proved that it always can be, at least once. Jacobi found an exact formula for the number of ways, as a by-product of rewriting elliptic functions as quotients of rapidly converging series called theta functions. Theta functions are now everywhere, from the heat equation to string theory.

Let \(q=e^{i\pi\tau}\) with \(\operatorname{Im}\tau>0\), so \(|q|<1\). The theta constants are

In the Fundamenta nova Jacobi proved that with the nome \(q=e^{-\pi K'/K}\),

and \(\mathrm{sn}\,u\) is a ratio of two theta functions of the variable \(z=\pi u/(2K)\). Because \(q\) is usually small, theta series converge extremely fast, which made them the practical tool of computation. In his Königsberg lectures of the 1830s Jacobi rebuilt the whole theory starting from theta functions.

Proof sketch. Call the product \(P(z)\). It is holomorphic on \(\mathbb C^\times\), and replacing \(z\) by \(zq^2\) shifts the factors so that \(P(zq^2)=(zq)^{-1}P(z)\). The series \(S(z)\) satisfies the same functional equation. Writing \(P(z)=\sum c_n(q)z^n\), the equation forces \(c_n=q^{n^2}c_0\), so \(P=c_0(q)S\). Evaluating at special values (for example \(z=i\), comparing with \(z=-1\) at \(q^4\)) shows \(c_0=1\).

Since \(\sum_n r_s(n)q^n=\vartheta_3(q)^s\), these are identities between power series:

Jacobi obtained them by comparing the product and series expansions of his elliptic functions. Today they are seen as the statement that \(\vartheta_3^4\) is a modular form of weight 2 for \(\Gamma_0(4)\), equal to an Eisenstein series.

Periods and the inversion problem

For the general reader. After elliptic integrals come integrals of square roots of polynomials of higher degree. Inverting them naively fails: the inverse would need more periods than a function of one variable can have. Jacobi proved that impossibility and found the way out: add up several integrals at once and invert them together, as functions of several variables. This is the birth of the Jacobian variety.

Proof. \(\Lambda\) is a subgroup. If it had an accumulation point, it would contain arbitrarily small non-zero periods \(\omega_n\to0\). Pick \(z_0\) not a pole; then \(f(z_0+\omega_n)=f(z_0)\), so \(f-f(z_0)\) has zeros accumulating at \(z_0\) and is identically zero. A discrete subgroup of \(\mathbb R^2\) is a lattice of rank at most 2.

Abelian integrals and the inversion problem

Let \(P(x)\) be a polynomial of degree 5 or 6 without repeated roots. The curve \(y^2=P(x)\) has genus 2, and its two holomorphic differentials are \(dx/y\) and \(x\,dx/y\). Each integral \(\int dx/y\) has four independent periods, so its inverse cannot be a function in the sense above. Abel’s theorem (1826–29) says that sums of such integrals behave well. In 1832 Jacobi proposed to invert the pair

and conjectured that the symmetric functions \(x_1+x_2\) and \(x_1x_2\) are single-valued meromorphic functions of \((u_1,u_2)\in\mathbb C^2\) with four independent periods. Adolph Göpel and Johann Rosenhain solved the genus-2 case in the 1840s using theta functions of two variables. Weierstrass solved the hyperelliptic case in 1854, and Bernhard Riemann the general case in 1857.

Hamilton–Jacobi theory

For the general reader. Newton’s laws describe a system one step at a time. Hamilton showed in 1834 that all the trajectories of a system can be encoded in a single function, much as light rays are encoded in wavefronts. Jacobi simplified Hamilton’s construction: one partial differential equation, solved in any way at all, gives every orbit by differentiation. A century later the same equation became the classical limit of Schrödinger’s quantum mechanics.

From Hamilton to Jacobi

In On a general method in dynamics (1834) and its sequel (1835), Hamilton expressed the motion through a principal function \(S\), the action along the true path as a function of its initial and final points and of time. \(S\) had to satisfy two partial differential equations, one in each set of endpoints. Jacobi, in papers of 1837 and in his lectures of 1842–43, showed that one equation suffices, that any complete integral works (not only Hamilton’s particular one), that the Hamiltonian may depend on time, and that the method is a general technique for first-order partial differential equations.

The equation

Let \(H(q,p,t)\) be a Hamiltonian with \(q,p\in\mathbb R^n\). A function \(S(q,\alpha,t)\) of \(q\), of \(n\) constants \(\alpha\), and of \(t\), generates a canonical transformation \((q,p)\to(\beta,\alpha)\) through

If the new Hamiltonian \(K\) is zero, the new variables are constants of motion. This gives the Hamilton–Jacobi equation

Proof. Differentiate \(\partial S/\partial\alpha_i=\beta_i\) along \(q(t)\): \(\sum_j S_{\alpha_iq_j}\dot q_j+S_{\alpha_it}=0\). Differentiate the HJ equation in \(\alpha_i\): \(S_{t\alpha_i}+\sum_j H_{p_j}S_{q_j\alpha_i}=0\). Subtracting and using the invertibility of \(S_{q\alpha}\) gives \(\dot q_j=\partial H/\partial p_j\). Then differentiate \(p_i=S_{q_i}\) along the path and the HJ equation in \(q_i\): \(\dot p_i=S_{q_it}+\sum_jS_{q_iq_j}\dot q_j\) and \(S_{tq_i}+H_{q_i}+\sum_jH_{p_j}S_{q_jq_i}=0\), so \(\dot p_i=-\partial H/\partial q_i\).

Separation of variables

If \(H\) does not depend on \(t\), try \(S=W(q,\alpha)-Et\), giving \(H(q,\partial W/\partial q)=E\). If \(W\) splits as \(\sum_iW_i(q_i)\), each piece obeys an ordinary differential equation. Jacobi used elliptic coordinates to separate the geodesic problem on a triaxial ellipsoid and the attraction by two fixed centres, problems that had resisted direct methods. Separability is coordinate-dependent, and finding the right coordinates is the art.

Optics and quantum mechanics

For one particle, the surfaces \(S=\text{const}\) are wavefronts and trajectories are their normals, since \(p=\nabla S\). In 1926 Erwin Schrödinger used this analogy: writing \(\psi=e^{iS/\hbar}\) in his equation and letting \(\hbar\to0\) returns the Hamilton–Jacobi equation, which is the starting point of the WKB approximation.

Dynamics: brackets, multipliers, integrals

For the general reader. Jacobi looked for structure in the equations of motion: what makes conserved quantities combine into new ones, how much must be known before the rest follows, and which quantity survives in the three-body problem when energy alone is not conserved. His answers became the algebra of modern mechanics and of quantum theory.

Poisson brackets and the Jacobi identity

For functions on phase space, the Poisson bracket is \(\{f,g\}=\sum_i\big(\partial_{q_i}f\,\partial_{p_i}g-\partial_{p_i}f\,\partial_{q_i}g\big)\), and along a Hamiltonian flow \(\dot f=\{f,H\}+\partial_tf\).

Proof. Every term of the sum contains exactly one second derivative. Write \(D_g=\{g,\cdot\}\), a first-order differential operator. The terms with second derivatives of \(f\) are \(\{g,\{h,f\}\}-\{h,\{g,f\}\}=[D_g,D_h]f\). A commutator of two first-order operators is again first order, so these terms contain no second derivatives of \(f\) and must cancel. The same holds for \(g\) and \(h\), so the sum vanishes. For Poisson’s theorem, put \(h=H\): \(\{\{f,g\},H\}=\{f,\{g,H\}\}-\{g,\{f,H\}\}=0\). Poisson had found the theorem in 1809; Jacobi gave the identity that explains it, in work published after his death. Sophus Lie later took the identity as the defining axiom of a Lie algebra.

The last multiplier

For a system \(\dot x=X(x)\) in \(\mathbb R^n\), a function \(M>0\) with \(\operatorname{div}(MX)=0\) is a Jacobi multiplier: the measure \(M\,dx\) is invariant under the flow.

Hamiltonian vector fields have \(\operatorname{div}X_H=\sum_i(\partial_{q_i}\partial_{p_i}H-\partial_{p_i}\partial_{q_i}H)=0\), so \(M=1\): this is Liouville’s theorem on the conservation of phase volume.

The Jacobi integral (1836)

In the circular restricted three-body problem, a small body moves in the plane of two masses \(1-\mu\) and \(\mu\) circling their centre of mass. In units where the separation, total mass and angular velocity are 1, and in the rotating frame,

Energy is not conserved, but \(C=2\Omega-\dot x^2-\dot y^2\) is. Since \(\dot x^2+\dot y^2\ge0\), the motion is confined to the region \(2\Omega\ge C\), bounded by the zero-velocity curves of George William Hill.

Jacobi’s form of least action

For a natural system with kinetic energy \(T=\tfrac12g_{ij}(q)\dot q^i\dot q^j\) and potential \(V\), trajectories of energy \(E\) are, after reparametrization, the geodesics of the metric \(2(E-V)\,g_{ij}\,dq^idq^j\) on the region \(V<E\). This turned dynamics into Riemannian geometry, a link later exploited by Henri Poincaré and in the study of chaotic geodesic flows. In the same spirit Jacobi showed in 1834 that a uniformly rotating fluid mass can be in equilibrium as an ellipsoid with three unequal axes; these Jacobi ellipsoids branch off the Maclaurin spheroids, and Poincaré’s study of the branching founded bifurcation theory.

The Jacobian and conjugate points

For the general reader. When you change coordinates, areas and volumes stretch by a factor given by a determinant of derivatives. Jacobi made this determinant a central tool in 1841, and it now bears his name. A few years earlier he had solved a problem Legendre left open: when does a curve satisfying the equations of the calculus of variations actually minimize? The answer depends on whether nearby solutions cross it again.

Functional determinants (1841)

For \(f=(f_1,\dots,f_n):U\subseteq\mathbb R^n\to\mathbb R^n\), the Jacobian is \(\dfrac{\partial(f_1,\dots,f_n)}{\partial(x_1,\dots,x_n)}=\det\big(\partial f_i/\partial x_j\big)\). Augustin-Louis Cauchy had used such determinants in 1815, but Jacobi’s memoirs De formatione et proprietatibus determinantium and De determinantibus functionalibus (1841) made them a systematic tool. He showed that the Jacobian of a composition is the product of Jacobians, that it governs the change of variables in multiple integrals, and that it detects functional dependence.

Proof. \(\det\) is linear in each entry, and \(\partial\det A/\partial a_{ij}\) is the cofactor \(C_{ij}=\operatorname{adj}(A)_{ji}\). The chain rule gives \(\sum_{i,j}C_{ij}\dot a_{ij}=\operatorname{tr}(\operatorname{adj}(A)\dot A)\).

Conjugate points (1837)

Return to Legendre’s second variation (see the Legendre course): for \(J[y]=\int_a^bF(x,y,y')dx\) along an extremal, \(\delta^2J=\int_a^b(Ph'^2+Qh^2)dx\) with \(P=F_{y'y'}\), \(Q=F_{yy}-\frac{d}{dx}F_{yy'}\). Jacobi linearized Legendre’s Riccati equation by the substitution \(w=-Pu'/u\), obtaining the Jacobi equation

A point \(c\in(a,b]\) is conjugate to \(a\) if the solution with \(u(a)=0\), \(u'(a)=1\) vanishes at \(c\). Geometrically, it is where neighbouring extremals from the same starting point cross the given one again.

Jacobi stated these results with brief indications of proof in 1837; complete proofs were given by later authors, including Otto Hesse and Weierstrass. In Riemannian geometry, conjugate points along geodesics are zeros of Jacobi fields, and the same theorem governs when a geodesic stops minimizing length.

The Jacobi symbol and Jacobi sums

For the general reader. Legendre’s symbol answers “is this a square modulo a prime?”. Jacobi extended it to odd moduli that need not be prime. The extended symbol no longer answers that question exactly, but it obeys the same reciprocity law, which makes it far faster to compute. Today it sits inside primality tests used in cryptography. Jacobi also introduced character sums that later became central in number theory.

Let \(n\) be an odd positive integer with factorization \(n=p_1p_2\cdots p_r\) (primes repeated). Jacobi defined, in 1837,

a product of Legendre symbols.

Proof. Each formula is multiplicative in \(n\) (and in \(m\)), and so is the right side, because \((m-1)/2\) and \((n^2-1)/8\) are additive modulo 2 under multiplication of odd numbers: \(\frac{ab-1}2\equiv\frac{a-1}2+\frac{b-1}2\pmod 2\). The prime case is quadratic reciprocity. As a result, \(\big(\frac an\big)\) can be computed by a Euclid-like algorithm, flipping and reducing, without factoring anything. If \(\big(\frac an\big)=-1\), then \(a\) is not a square mod \(n\); the converse fails.

Jacobi sums

For multiplicative characters \(\chi,\psi\) of \(\mathbb F_p^\times\), extended by \(\chi(0)=0\), the Jacobi sum is \(J(\chi,\psi)=\sum_{a\in\mathbb F_p}\chi(a)\psi(1-a)\). Jacobi used such sums in his work on cyclotomy and higher reciprocity.

If \(p\equiv1\pmod4\) and \(\chi\) has order 4, then \(J(\chi,\chi)=a+bi\in\mathbb Z[i]\) with \(a^2+b^2=p\): an explicit proof of Fermat’s two-square theorem. Jacobi sums count points on curves such as \(x^m+y^m=1\) over finite fields, and in this role they led to André Weil’s conjectures.

Illness, Berlin and 1848

By 1843 overwork had worn Jacobi down, and he was diagnosed with diabetes. His family’s bank had failed some years earlier, so he had no private means. Dirichlet, his close friend, asked Alexander von Humboldt to obtain help from King Frederick William IV, who granted funds for a stay in Italy. Jacobi travelled with Dirichlet and Borchardt, attended a scientific congress in Lucca, and reached Rome in November 1843, where he studied Vatican manuscripts of Diophantus.

The doctors judged the Königsberg climate too harsh, and in 1844 he moved to Berlin with a royal stipend. As a member of the Prussian Academy of Sciences he had the right to lecture at the university. He continued research on mechanics, number theory and numerical methods; his 1846 paper on computing the eigenvalues of a symmetric matrix by successive plane rotations is the origin of the Jacobi eigenvalue algorithm. In 1847–48 he lectured in Berlin on analytical mechanics, criticizing what he saw as gaps in Lagrange’s foundations.

In the revolution of 1848 he spoke at the Constitutional Club in Berlin and stood without success as a liberal candidate for parliament. His speech pleased neither monarchists nor republicans. In 1849 the government refused him a university post and withdrew the supplement to his stipend; he sent his family to Gotha, where living was cheaper. Humboldt intervened on his behalf. When the University of Vienna offered him a chair in 1850, Prussia restored his position in Berlin.

Death and legacy

In January 1851 Jacobi caught influenza. He had hardly recovered when he contracted smallpox, and he died in Berlin on 18 February 1851, aged 46. He is buried in the Dreifaltigkeitsfriedhof I in Berlin-Kreuzberg; the grave is an honorary grave of the city of Berlin. Dirichlet delivered his memorial address before the Berlin Academy in 1852. His collected works, the Gesammelte Werke, were edited by Borchardt and Weierstrass in seven volumes (1881–1891), and the lectures on dynamics appeared in 1866.

He was elected to many academies, including the Royal Society (1833), the Paris Académie (corresponding member 1830, foreign associate 1846), the Göttingen Academy (1840) and the American Academy of Arts and Sciences (1850). He received the Pour le Mérite for sciences and arts in 1842.

Legacy

• Elliptic and abelian functions: \(\mathrm{sn},\mathrm{cn},\mathrm{dn}\), theta functions, the triple product, and the inversion problem, which led through Riemann and Weierstrass to Jacobian varieties and modern algebraic geometry.
• Mechanics: the Hamilton–Jacobi equation, the Jacobi identity, the last multiplier, the Jacobi integral and the Jacobi metric are standard tools in classical mechanics, celestial mechanics and the passage to quantum theory.
• Analysis and computation: the Jacobian, Jacobi’s formula, Jacobi polynomials, the Jacobi iterative method and the Jacobi eigenvalue algorithm.
• Number theory: the Jacobi symbol, Jacobi sums, and the use of theta series to count representations by quadratic forms, the root of the theory of modular forms.
• Teaching: the research seminar, now universal in graduate education, and a lineage of students through Hesse, Borchardt and Richelot.
• Honours: the lunar crater Jacobi is named after him.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Carl Gustav Jacob Jacobi, MacTutor History of Mathematics.
2. Carl Gustav Jacob Jacobi, Wikipedia; German edition.
3. Carl Gustav Jacob Jacobi, Mathematics Genealogy Project.
4. L. Koenigsberger, Carl Gustav Jacob Jacobi, Teubner, Leipzig, 1904.
5. P. G. Lejeune Dirichlet, Gedächtnisrede auf Carl Gustav Jacob Jacobi, Abh. Berliner Akad. (1852).

Mathematics

1. C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Königsberg, 1829.
2. C. G. J. Jacobi, Vorlesungen über Dynamik, ed. A. Clebsch, Berlin, 1866; English translation by K. Balagangadharan, Hindustan Book Agency, 2009.
3. C. G. J. Jacobi, Gesammelte Werke, 7 vols., ed. C. W. Borchardt and K. Weierstrass, Berlin, 1881–1891.
4. M. Nakane and C. G. Fraser, The early history of Hamilton–Jacobi dynamics 1834–1837, Centaurus 44 (2002).
5. Hamilton–Jacobi equation and Jacobi ellipsoid, Wikipedia.
6. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978, chapters 8–9.
7. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge, 1927, chapters 21–22.
8. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1990, chapter 8.
9. D. Mumford, Tata Lectures on Theta I, Birkhäuser, 1983.