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Concept in celestial mechanics
In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular
Jacobi_integral
German mathematician (1804–1851)
example inverting elliptic integrals and focusing on the nature of elliptic and theta functions. In his 1835 paper, Jacobi proved the following basic
Carl_Gustav_Jacob_Jacobi
Physics problem related to laws of motion and gravity
a 4-dimensional phase space, but only one conserved quantity, the Jacobi integral. It was shown by Heinrich Bruns that there are no more algebraic conserved
Three-body_problem
English actor (born 1938)
Sir Derek George Jacobi (/ˈdʒækəbi/; born 22 October 1938) is an English actor. Known for his roles on stage and screen as well as for his work at the
Derek_Jacobi
Mathematical function
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as
Jacobi_elliptic_functions
Region in which an astronomical body dominates the attraction of satellites
zero-velocity surface in space which cannot be passed, the contour of the Jacobi integral.[not verified in body] When the object's energy is low, the zero-velocity
Hill_sphere
functions Jacobi field Jacobi's four-square theorem Jacobi form Jacobi's formula Jacobi group Jacobian ideal Jacobi identity Jacobi integral Jacobi's logarithm
List of things named after Carl Gustav Jacob Jacobi
List_of_things_named_after_Carl_Gustav_Jacob_Jacobi
mathematics, the Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials P n α
Jacobi_transform
Surface a body of energy cannot cross
momentum are not conserved separately in this coordinate system, but the Jacobi integral remains constant: C = ω 2 ( x 2 + y 2 ) + 2 ( μ 1 r 1 + μ 2 r 2 ) −
Zero-velocity_surface
Special function defined by an integral
Legendre's trigonometric form of the elliptic integral; substituting t = sin θ and x = sin φ, one obtains Jacobi's algebraic form: F ( x ; k ) = ∫ 0 x d t (
Elliptic_integral
Formulation of classical mechanics
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics
Hamilton–Jacobi_equation
Problem in physics and celestial mechanics
of n particles. Celestial mechanics Gravitational two-body problem Jacobi integral Lunar theory Natural units Numerical model of the Solar System Stability
N-body_problem
quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form ∫ − 1 1 f ( x ) ( 1 − x ) α ( 1 + x
Gauss–Jacobi_quadrature
Orbit parameter conserved in the three-body problem
quasi-conservation of Tisserand's invariant is derived as the limit of the Jacobi integral away from the main two bodies (usually the star and planet). Numerical
Tisserand's_parameter
Problem in physics and astronomy
elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. Carter constant Hydrogen molecular ion Jacobi integral Lagrangian
Euler's_three-body_problem
Construction in algebraic geometry
In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry
Abel–Jacobi_map
Physical quantity of dimension energy × time
the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides
Action_(physics)
Shape taken by a self-gravitating fluid body rotating at constant velocity
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating, fluid body of uniform density
Jacobi_ellipsoid
Special functions of several complex variables
as the most general 2 quasi-period function. The Jacobi theta functions have the following integral representations: ϑ 00 ( z ; τ ) = − i ∫ i − ∞ i +
Theta_function
Class of periodic mathematical functions
Carl Gustav Jacobi. Abel discovered elliptic functions by taking the inverse function φ {\displaystyle \varphi } of the elliptic integral function α (
Elliptic_function
Property of mathematical knots
polynomials. Jacobi diagrams were introduced as analogues of Feynman diagrams when Kontsevich defined knot invariants by iterated integrals in the first
Kontsevich_invariant
transform Hermite transform Hilbert transform Hilbert–Schmidt integral operator Jacobi transform Laguerre transform Laplace transform Inverse Laplace
List_of_transforms
Matrix of partial derivatives of a vector-valued function
referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi (1804-1851). The Jacobian matrix is the natural generalization of the derivative
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Mapping involving integration between function spaces
In mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via
Integral_transform
Concept in differential calculus
solving variational problems, such as the Legendre–Clebsch condition and the Jacobi necessary condition detailed below. Much of the calculus of variations relies
Second_variation
Polynomial sequence
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are
Jacobi_polynomials
polynomials q-Hahn polynomials q-Jacobi polynomials: Big q-Jacobi polynomials Continuous q-Jacobi polynomials Little q-Jacobi polynomials q-Krawtchouk polynomials
List_of_q-analogs
In mathematics, the term Pseudo Jacobi polynomials was introduced by Lesky for one of three finite sequences of orthogonal polynomials y. Since they form
Pseudo_Jacobi_polynomials
Formula for the Legendre polynomials
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde
Rodrigues'_formula
It is a generalization of the Lie bracket from an operation on the tangent bundle
bracket satisfies the Jacobi identity in the case p = 0 {\displaystyle p=0} . The curvature of a circle bundle always represents an integral cohomology class
Courant_bracket
Term used in the theories of Riemann surfaces and algebraic curves
to integrals that generalise the elliptic integrals to all curves over the complex numbers. They include for example the hyperelliptic integrals of type
Differential of the first kind
Differential_of_the_first_kind
P_{n}(x)} as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function f ( x ) {\displaystyle f(x)}
Legendre transform (integral transform)
Legendre_transform_(integral_transform)
Shortest paths on a bounded deformed sphere-like quadric surface
Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal. Königsberg, 28th Dec. '38. The solution given by Jacobi (Jacobi 1839)
Geodesics_on_an_ellipsoid
Type of symmetric polynomials in mathematics
0)}(x_{1},x_{2},\dots ,x_{n})}}.} This is known as the bialternant formula of Jacobi. It is a special case of the Weyl character formula. This is a symmetric
Schur_polynomial
2025 historical drama film by Chloé Zhao
Paul Mescal as Agnes and William, alongside Emily Watson, Joe Alwyn, and Jacobi Jupe in supporting roles. Hamnet had its world premiere at the 52nd Telluride
Hamnet_(film)
Branch of mathematics
mechanics. In a sense, q-calculus dates back to Leonhard Euler and Carl Gustav Jacobi, but has only recently begun to find usefulness in quantum mechanics, given
Quantum_calculus
Polynomial with all terms of degree two
classification of real quadratic forms under a linear change of variables. Jacobi proved that, for every real quadratic form, there is an orthogonal diagonalization;
Quadratic_form
Mathematical function
generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly
Ramanujan_theta_function
are generic Incomplete Elliptical Integrals of the first and second kind. Jacobi Zeta Functions being kinds of Jacobi theta functions have applications
Jacobi_zeta_function
Polynomial sequence
Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. Plot of the Gegenbauer
Gegenbauer_polynomials
polynomial F. H. Jackson: Jackson derivative Jackson integral Carl Gustav Jakob Jacobi: Jacobi polynomial, Jacobi theta function Joseph Marie Kampe de Feriet (1893–1982):
List of eponyms of special functions
List_of_eponyms_of_special_functions
German mathematician (1805–1859)
well as Jacobi and other liberal professors, as "the red contingent of the staff". In 1849 Dirichlet participated, together with his friend Jacobi, in the
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Polynomials used in approximation theory
{\displaystyle H(\varphi )} is the Jacobi eta function F ( φ | κ ) {\displaystyle F(\varphi |\kappa )} is the incomplete elliptic integral of the first kind K ( κ
Zolotarev_polynomials
Reformulation of general relativity
In general relativity, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the Hamiltonian formulation
Hamilton–Jacobi–Einstein equation
Hamilton–Jacobi–Einstein_equation
Physical quantity conserved throughout a motion
shown mathematically to be conserved throughout the motion. The Hamilton–Jacobi equations provide a commonly used and straightforward method for identifying
Constant_of_motion
Integral transform useful in probability theory, physics, and engineering
Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually t {\displaystyle
Laplace_transform
Measure of sustained displacement of an object from its initial position
constant as the object resides at the initial position. It is the first time-integral of the displacement (i.e. absement is the area under a displacement vs
Absement
Abel function Abel's integral equation Abel's identity Abel's inequality Abel's irreducibility theorem Abel–Jacobi map Abel–Jacobi theorem Abel polynomials
List of things named after Niels Henrik Abel
List_of_things_named_after_Niels_Henrik_Abel
distribution theory of holomorphic functions Line integral Cauchy's integral theorem Cauchy's integral formula Residue theorem Liouville's theorem (complex
List of complex analysis topics
List_of_complex_analysis_topics
Free swinging suspended body
large amplitudes. Equivalently, the angle can be given in terms of the Jacobi elliptic function cd {\displaystyle \operatorname {cd} } with modulus k
Pendulum_(mechanics)
Approximation of the definite integral of a function
modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken
Gaussian_quadrature
Family of solutions to related differential equations
_{n=1}^{\infty }J_{n}(nz).} Another important relation for integer orders is the Jacobi–Anger expansion: e i z cos ϕ = ∑ n = − ∞ ∞ i n J n ( z ) e i n ϕ {\displaystyle
Bessel_function
Integral of a comparatively larger force over a short time interval
mass by a varying force acting from time t1 to t2 is defined to be the integral of the force F with respect to time: J = ∫ t 1 t 2 F d t . {\displaystyle
Impulse_(physics)
Mathematical functions
article; in references, notation for general Jacobi elliptic functions is used instead. The lemniscate integral and lemniscate functions satisfy an argument
Lemniscate_elliptic_functions
Fundamental mechanical principles
and in tandem Carl Gustav Jacob Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation. In 1915, David Hilbert
Action_principles
Formulation of classical mechanics using momenta
mechanics Dynamical systems theory Hamiltonian system Hamilton–Jacobi equation Hamilton–Jacobi–Einstein equation Lagrangian mechanics Maxwell's equations
Hamiltonian_mechanics
Ancient Greek cult practice
meaningful katabasis ... its object the restoration of the whole man". Jolande Jacobi added that "this 'great Nekyia' ... is interwoven with innumerable lesser
Nekyia
Equations describing classical electromagnetism
magnetic field corresponds to the negative curl of an electric field. In integral form, it states that the work per unit charge required to move a charge
Maxwell's_equations
Physical theory describing classical fields
have a continuous mass distribution ρ instead, the sum is replaced by an integral, g ( r ) = − G ∭ V ρ ( x ) d 3 x ( r − x ) | r − x | 3 , {\displaystyle
Classical_field_theory
Result used in the theory of propagation of waves
by a two-sided plane wave in the z {\displaystyle z} direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers
Sommerfeld_identity
functions: Un is the nth up/down number, Bn is the nth Bernoulli number in Jacobi elliptic functions, q = e − π K ( 1 − m ) K ( m ) {\displaystyle q=e^{-\pi
List_of_periodic_functions
Mathematical symbol used for partial derivatives and other concepts
Gustav Jacob Jacobi in 1841, whose usage became widely adopted. The symbol is variously referred to as "partial", "curly d" or "Jacobi's delta", or as
Partial_differential
Overview of mechanics based on the least action principle
topology. In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields. Routhian mechanics is
Analytical_mechanics
Hypercomplex number system
Media, ISBN 978-3-7643-9893-4 (Graves 1845) Cayley, Arthur (1845), "On Jacobi's Elliptic functions, in reply to the Rev. Brice Bronwin; and on Quaternions"
Octonion
methods such as Generalized Davidson and Jacobi-Davidson. Conjugate gradient methods such as LOBPCG. A contour integral solver (CISS). Interface to some external
SLEPc
same year he became aware of Carl Gustav Jacobi and his works on new transformations of elliptic integrals. Abel finishes then a second part of his article
Abel_elliptic_functions
Numerical analysis concept
quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval [−1, 1], the rule takes
Gauss–Legendre_quadrature
Number, approximately 3.14
) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} . An example is the Jacobi theta function θ ( z , τ ) = ∑ n = − ∞ ∞ e 2 π i n z + π i n 2 τ , {\displaystyle
Pi
Mathematics concept
polynomials was put forward by Raposo, with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme. It seems more consistent to
Romanovski_polynomials
Concept in Lie algebra representation theory
non‑associative algebra with a bilinear, antisymmetric bracket satisfying the Jacobi identity), then instead of requiring multiplicativity of a character, one
Weight (representation theory)
Weight_(representation_theory)
Function that is holomorphic on the whole complex plane
that of the sigma function. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential
Entire_function
Geometric model of the physical space
a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity. For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf
Three-dimensional_space
Formulation of classical mechanics
the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies r = r2 − r1 and the location of
Lagrangian_mechanics
Index of articles associated with the same name
a certain property. Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign
Comparison_theorem
Statement relating differentiable symmetries to conserved quantities
mathematician Emmy Noether in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can
Noether's_theorem
French mathematician (1809–1882)
colleagues, including William Thomson (Lord Kelvin), Carl Gustav Jacob Jacobi, and Peter Gustav Lejeune Dirichlet. As a lecturer, he offered support and
Joseph_Liouville
Function defined by a hypergeometric series
{c-1}{2}}P_{-a}^{1-c}(1-2z)} Several orthogonal polynomials, including Jacobi polynomials P(α,β) n and their special cases Legendre polynomials, Chebyshev
Hypergeometric_function
Mathematical function
the series expansion has integral coefficients. The Jacobi triple product implies that the eta is (up to a factor) a Jacobi theta function for special
Dedekind_eta_function
Mathematical model for sequential decision making under uncertainty
continuous, the optimal criterion could be found by solving the Hamilton–Jacobi–Bellman (HJB) partial differential equation. In order to discuss the HJB
Markov_decision_process
a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece
History_of_calculus
Set of polynomials where any two are orthogonal to each other
the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev
Orthogonal_polynomials
Principle of least length in physics
V(\mathbf {q} )} . In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length s = ∫ d s {\textstyle s=\int
Maupertuis's_principle
Special function in mathematics
Adolf Hurwitz, who introduced it in 1882. The Hurwitz zeta function has an integral representation ζ ( s , a ) = 1 Γ ( s ) ∫ 0 ∞ x s − 1 e − a x 1 − e − x
Hurwitz_zeta_function
Mathematical measure of a function's variability
just shows that the Lagrange equations (or, equivalently, the Hamilton–Jacobi equations) provide the basic tools for obtaining extremal solutions. Dirichlet's
Dirichlet_energy
French mathematician (born 1956)
Hamilton-Jacobi equations, by regularizing sub- or super-solutions. Using such techniques, Crandall and Lions extended their analysis of Hamilton-Jacobi equations
Pierre-Louis_Lions
Dutch mathematician (1856–1894)
attending lectures, he spent his student years reading the works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were two
Thomas_Joannes_Stieltjes
Lattice in 8-dimensional space with special properties
q^{12}+O(q^{14}).} The E8 theta function may be written in terms of the Jacobi theta functions as follows: Θ Γ 8 ( τ ) = 1 2 ( θ 2 ( q ) 8 + θ 3 ( q )
E8_lattice
German mathematician (1826–1866)
he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex
Bernhard_Riemann
Matrix decomposition
as a Jacobi rotation, M ← M J ( p , q , θ ) , {\displaystyle M\leftarrow MJ(p,q,\theta ),} where the angle θ {\displaystyle \theta } of the Jacobi rotation
Singular_value_decomposition
Mathematical function of two positive real arguments
compute elliptic integrals, which are used, for example, in elliptic filter design. The arithmetic–geometric mean is connected to the Jacobi theta function
Arithmetic–geometric_mean
2001 mystery film directed by Robert Altman
Bates, Charles Dance, Stephen Fry, Michael Gambon, Richard E. Grant, Derek Jacobi, Kelly Macdonald, Helen Mirren, Jeremy Northam, Clive Owen, Ryan Phillippe
Gosford_Park
There are a number of notational systems for the Jacobi theta functions. The notations given in the Wikipedia article define the original function ϑ 00
Jacobi theta functions (notational variations)
Jacobi_theta_functions_(notational_variations)
indexed by prime numbers of a Dirichlet series Euler pseudoprime Euler–Jacobi pseudoprime Euler's totient function (or Euler phi (φ) function) in number
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Differential calculus on function spaces
functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or
Calculus_of_variations
Quantum field theory with four-point interactions
{sn}}(p\cdot x+\theta ,i),} where s n {\displaystyle \,{\rm {sn\!}}} is the Jacobi elliptic sine function and μ , θ {\displaystyle \,\mu ,\theta } are two
Quartic_interaction
Function in q-analog theory
1216/RMJ-1984-14-2-403 Mező, István (2012), "A q-Raabe formula and an integral of the fourth Jacobi theta function", Journal of Number Theory, 133 (2): 692–704
Q-gamma_function
Mathematical function
distributions related to the beta function Jacobi sum, the analogue of the beta function over finite fields. Nørlund–Rice integral Yule–Simon distribution Davis,
Beta_function
Analytic function in mathematics
Philippe; Pitman, Jim; Yor, Marc (2001). "Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions". Bulletin of
Riemann_zeta_function
American adult animated comedy series
The support of comedian Daniel Tosh was integral in getting the series picked up.
Brickleberry
Complex-differentiable part of a Maass wave function
indefinite lattices of dimension 2, and to Appell–Lerch sums, and to meromorphic Jacobi forms. Zwegers's fundamental result shows that mock theta functions are
Mock_modular_form
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