Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: Darboux coordinates | Statement: [Darboux theorem, guaranteesExistenceOf, Darboux coordinates]

Target entity: Darboux coordinates
Context triple: [Darboux theorem, guaranteesExistenceOf, Darboux coordinates]

• A. Darboux theorem
  The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
• B. 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.
• C. Kovalevskaya integral
  The Kovalevskaya integral is an additional conserved quantity that makes the motion of the Kovalevskaya top exactly integrable in classical rigid body dynamics.
• D. Jacobi manifold
  A Jacobi manifold is a smooth manifold equipped with a Lie bracket on its space of smooth functions that satisfies a generalized Leibniz rule, extending the notion of Poisson manifolds.
• E. Liouville–Arnold theorem
  The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Darboux coordinates
Target entity description: Darboux coordinates are local canonical coordinate systems on a symplectic manifold in which the symplectic form takes a standard constant form, simplifying the study of Hamiltonian mechanics and symplectic geometry.

• A. Darboux theorem chosen
  The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
• B. 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.
• C. Kovalevskaya integral
  The Kovalevskaya integral is an additional conserved quantity that makes the motion of the Kovalevskaya top exactly integrable in classical rigid body dynamics.
• D. Jacobi manifold
  A Jacobi manifold is a smooth manifold equipped with a Lie bracket on its space of smooth functions that satisfies a generalized Leibniz rule, extending the notion of Poisson manifolds.
• E. Liouville–Arnold theorem
  The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.
• F. None of above.