Hamilton–Jacobi equation
E182751
The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Hamilton–Jacobi theory | 3 |
| Hamilton–Jacobi equation canonical | 2 |
| Hamilton–Jacobi–Bellman equations | 1 |
| classical Hamilton–Jacobi equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1615219 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hamilton–Jacobi equation Context triple: [Carl Gustav Jacob Jacobi, notableWork, Hamilton–Jacobi equation]
-
A.
Euler–Lagrange equation
The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.
-
B.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
C.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
D.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
-
E.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hamilton–Jacobi equation Target entity description: The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
-
A.
Euler–Lagrange equation
The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.
-
B.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
-
C.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
D.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
-
E.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hamiltonian formulation of mechanics
ⓘ
equation in classical mechanics ⓘ partial differential equation ⓘ |
| appliesTo |
conservative mechanical systems
ⓘ
time-dependent Hamiltonian systems ⓘ |
| basedOn | principle of least action ⓘ |
| connectsTo |
classical limit of quantum mechanics
ⓘ
ray optics ⓘ |
| describes | time evolution of a mechanical system ⓘ |
| domain | configuration space ⓘ |
| expresses | Hamiltonian as function of coordinates and derivatives of action ⓘ |
| field |
analytical mechanics
ⓘ
classical mechanics ⓘ mathematical physics ⓘ |
| framework | phase space ⓘ |
| goal |
find canonical transformation to constant momenta
ⓘ
reduce dynamics to quadratures ⓘ |
| hasForm | H(q,∂S/∂q,t)+∂S/∂t=0 ⓘ |
| historicalDevelopment | 19th century ⓘ |
| influenced |
Feynman path integral
ⓘ
surface form:
Feynman path integral formulation
WKB approximation ⓘ |
| mathematicalType | first-order nonlinear partial differential equation ⓘ |
| namedAfter |
Carl Gustav Jacob Jacobi
ⓘ
William Rowan Hamilton ⓘ |
| providesBridgeTo | quantum mechanics ⓘ |
| relatedTo |
Hamiltonian mechanics
ⓘ
Lagrangian mechanics ⓘ Schrödinger equation ⓘ action-angle variables ⓘ canonical quantization ⓘ eikonal equation ⓘ geometrical optics ⓘ integrable systems ⓘ symplectic geometry ⓘ variational principles ⓘ |
| requires | differentiability of the action function ⓘ |
| solutionCalled |
Hamilton’s characteristic function
ⓘ
Hamilton’s principal function ⓘ |
| usedFor |
analysis of integrable Hamiltonian systems
ⓘ
construction of action-angle coordinates ⓘ semi-classical approximations in quantum mechanics ⓘ separation of variables in mechanics ⓘ |
| usesConcept |
Hamiltonian (time translation generator)
ⓘ
surface form:
Hamiltonian function
action functional ⓘ canonical transformation ⓘ characteristics method ⓘ generating function ⓘ principal function ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hamilton–Jacobi equation Description of subject: The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hamilton–Jacobi theory
this entity surface form:
classical Hamilton–Jacobi equation
subject surface form:
Carl Gustav Jacob Jacobi
this entity surface form:
Hamilton–Jacobi theory
subject surface form:
Carl Gustav Jacob Jacobi
this entity surface form:
Hamilton–Jacobi theory
this entity surface form:
Hamilton–Jacobi–Bellman equations