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A symplectic manifold equipped with a Hamiltonian function whose Hamiltonian vector field generates the evolution. An autonomous Hamiltonian is conserved along its own flow, subject to the chosen sign convention.
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Hamiltonian systems specialize continuous dynamics by adding a symplectic state space and a Hamiltonian whose vector field generates the flow.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
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Supplying a smooth function on a symplectic manifold determines the Hamiltonian vector field by up to the declared sign convention, and hence a local Hamiltonian evolution; global evolution additionally requires completeness.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
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Hamiltonian flows supply the microscopic phase-space evolution used in classical statistical mechanics, including invariant Liouville measure; ensemble reasoning adds probabilistic and macroscopic structure.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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Hamiltonian dynamical systems provide the mathematical phase-flow structure used by Hamiltonian mechanics; the physical interpretation and Hamiltonian are additional physical input.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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