For a smooth function , the Hamiltonian vector field is defined by , with the sign depending on convention.
Compactly supported Hamiltonian vector fields can attain any prescribed tangent vector at a point of a symplectic manifold: prescribe the derivative of a local Hamiltonian and multiply by a bump function. Composing flows for a tangent basis gives an open orbit by the inverse function theorem. Every orbit is open, so on a connected manifold the compactly supported Hamiltonian transformations act transitively.
On a symplectic manifold, use and for the Poisson bracket. Cartan's magic formula gives . Taking the interior product of a differential form with then yields , proving the homomorphism. Its kernel consists of locally constant functions, since precisely when . On a connected manifold these are the constants. The quotient of smooth functions by this kernel is therefore the Lie algebra of Hamiltonian vector fields. A different contraction sign requires a compatible bracket sign.
For independent integrals , the Hamiltonian agrees with to first order on the common zero level. On a nonempty level , the vector-field difference is and is nowhere zero by independence of the differentials. Nonemptiness is necessary if one expresses the conclusion as inequality between restricted vector fields.
A Hamiltonian function is a smooth function whose differential determines a Hamiltonian vector field.
A Hamiltonian isotopy is a smooth path of symplectomorphisms , with , generated by a smooth time-dependent Hamiltonian function. In the convention , it satisfiesThe flow must exist throughout the chosen time interval. This is automatic on a closed symplectic manifold; on an open one, uniformly compactly supported Hamiltonians also suffice. The Hamiltonian flow preserves the symplectic form by Cartan's magic formula.
Its time-one map is a Hamiltonian diffeomorphism, an element of the group of Hamiltonian diffeomorphisms. A path and its endpoint are different data: a full rotation of the symplectic plane gives a nonconstant Hamiltonian path whose endpoint is the identity. Distinct paths can therefore represent the same Hamiltonian diffeomorphism.
A Hamiltonian diffeomorphism is the time-one map of a Hamiltonian isotopy beginning at the identity on a symplectic manifold. The isotopy is generated by a smooth time-dependent Hamiltonian function . Its flow must exist for the whole interval; on an open manifold a common group convention requires compact support throughout that interval.
In the convention , the infinitesimal preservation law isThus every Hamiltonian diffeomorphism is a symplectomorphism. The conclusion follows from closedness of the symplectic form, not from a special choice of coordinates.
The identity is generated by the zero Hamiltonian. Inverses and compositions are again Hamiltonian endpoints, forming the group of Hamiltonian diffeomorphisms. A generating function path and its endpoint are different data: different isotopies can have the same time-one map. The Hamiltonian condition imposes more structure than only preserving the symplectic form.
If and is its flow, then Cartan's magic formula givesThus every defined time slice of a Hamiltonian isotopy is a symplectomorphism.
Given a bounded set in the symplectic plane and a translation vector , the linear Hamiltonian generates translation by . Multiplying it by a cutoff that equals one on the entire swept region produces a compactly supported Hamiltonian diffeomorphism that agrees with the translation on the set.
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In the context of Hamiltonian mechanics, a Hamiltonian vector field is a vector field that is derived from a Hamiltonian function, which typically represents the total energy of a physical system. The Hamiltonian formulation of classical mechanics describes the evolution of a system in phase space using this vector field. Suppose we have a Hamiltonian function \( H(q, p) \), where \( q \) represents generalized coordinates (position variables) and \( p \) represents generalized momenta.