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| {{expand Italian|date=December 2011|Campo vettoriale hamiltoniano}}
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| In [[mathematics]] and [[physics]], a '''Hamiltonian vector field''' on a [[symplectic manifold]] is a [[vector field]], defined for any '''energy function''' or '''Hamiltonian'''. Named after the physicist and mathematician [[William Rowan Hamilton|Sir William Rowan Hamilton]], a Hamiltonian vector field is a geometric manifestation of [[Hamilton's equations]] in [[classical mechanics]]. The [[integral curve]]s of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The [[diffeomorphism]]s of a symplectic manifold arising from the [[flow (mathematics)|flow]] of a Hamiltonian vector field are known as [[canonical transformation]]s in physics and (Hamiltonian) [[symplectomorphism]]s in mathematics.
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| Hamiltonian vector fields can be defined more generally on an arbitrary [[Poisson manifold]]. The [[Lie_bracket_of_vector_fields|Lie bracket]] of two Hamiltonian vector fields corresponding to functions ''f'' and ''g'' on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the
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| [[Poisson bracket]] of ''f'' and ''g''.
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| == Definition ==
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| Suppose that (''M'',''ω'') is a [[symplectic manifold]]. Since the [[symplectic form]] ''ω'' is nondegenerate, it sets up a ''fiberwise-linear'' [[isomorphism]]
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| : <math>\omega:TM\to T^*M, </math>
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| between the [[tangent bundle]] ''TM'' and the [[cotangent bundle]] ''T*M'', with the inverse
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| : <math>\Omega:T^*M\to TM, \quad \Omega=\omega^{-1}.</math>
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| Therefore, [[one-form]]s on a symplectic manifold ''M'' may be identified with [[vector field]]s and every [[differentiable function]] ''H'': ''M'' → '''R''' determines a unique [[vector field]] ''X<sub>H</sub>'', called the '''Hamiltonian vector field''' with the '''Hamiltonian''' ''H'', by requiring that for every vector field ''Y'' on ''M'', the identity
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| :<math>\mathrm{d}H(Y) = \omega(X_H,Y)\,</math>
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| must hold.
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| '''Note''': Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.
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| == Examples ==
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| Suppose that ''M'' is a 2''n''-dimensional symplectic manifold. Then locally, one may choose [[canonical coordinates]] (''q''<sup>1</sup>, ..., ''q<sup>n</sup>'', ''p''<sub>1</sub>, ..., ''p<sub>n</sub>'') on ''M'', in which the symplectic form is expressed as
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| :<math>\omega=\sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,</math>
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| where d denotes the [[exterior derivative]] and ∧ denotes the [[exterior product]]. Then the Hamiltonian vector field with Hamiltonian ''H'' takes the form
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| :<math>\Chi_H=\left( \frac{\partial H}{\partial p_i},
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| - \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,</math>
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| where ''Ω'' is a 2''n'' by 2''n'' square matrix
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| :<math>\Omega =
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| \begin{bmatrix}
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| 0 & I_n \\
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| -I_n & 0 \\
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| \end{bmatrix},</math>
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| and
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| :<math> \mathrm{d}H=\begin{bmatrix} \frac{\partial H}{\partial q^i} \\
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| \frac{\partial H}{\partial p_i} \end{bmatrix}.</math>
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| Suppose that ''M'' = '''R'''<sup>2''n''</sup> is the 2''n''-dimensional [[symplectic vector space]] with (global) canonical coordinates.
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| * If ''H'' = ''p<sub>i</sub>'' then <math>X_H=\partial/\partial q^i; </math>
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| * if ''H'' = ''q<sup>i</sup>'' then <math>X_H=-\partial/\partial p^i; </math>
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| * if <math>H=1/2\sum (p_i)^2</math> then <math>X_H=\sum p_i\partial/\partial q^i; </math>
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| * if <math>H=1/2\sum a_{ij} q^i q^j, a_{ij}=a_{ji} </math> then <math>X_H=-\sum a_{ij} q_i\partial/\partial p^j. </math>
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| == Properties ==
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| * The assignment ''f'' ↦ ''X<sub>f</sub>'' is [[linear map|linear]], so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
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| * Suppose that (''q''<sup>1</sup>, ..., ''q<sup>n</sup>'', ''p''<sub>1</sub>, ..., ''p<sub>n</sub>'') are canonical coordinates on ''M'' (see above). Then a curve γ(''t'')=''(q(t),p(t))'' is an [[integral curve]] of the Hamiltonian vector field ''X<sub>H</sub>'' if and only if it is a solution of the [[Hamilton's equations]]:
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| :<math>\dot{q}^i = \frac {\partial H}{\partial p_i}</math>
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| :<math>\dot{p}_i = - \frac {\partial H}{\partial q^i}.</math>
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| * The Hamiltonian ''H'' is constant along the integral curves, because <math><dH, \dot{\gamma}> = \omega(X_H(\gamma),X_H(\gamma)) = 0</math>. That is, ''H''(γ(''t'')) is actually independent of ''t''. This property corresponds to the [[conservation of energy]] in [[Hamiltonian mechanics]].
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| * More generally, if two functions ''F'' and ''H'' have a zero [[Poisson bracket]] (cf. below), then ''F'' is constant along the integral curves of ''H'', and similarly, ''H'' is constant along the integral curves of ''F''. This fact is the abstract mathematical principle behind [[Noether's theorem]].
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| *The [[symplectic form]] ω is preserved by the Hamiltonian flow. Equivalently, the [[Lie derivative]] <math>\mathcal{L}_{X_H} \omega= 0</math>
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| ==Poisson bracket==
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| The notion of a Hamiltonian vector field leads to a [[skew-symmetric]], bilinear operation on the differentiable functions on a symplectic manifold ''M'', the '''[[Poisson bracket]]''', defined by the formula
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|
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| :<math>\{f,g\} = \omega(X_g, X_f)= dg(X_f) = \mathcal{L}_{X_f} g</math> | |
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| where <math>\mathcal{L}_X</math> denotes the [[Lie derivative]] along a vector field ''X''. Moreover, one can check that the following identity holds:
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| : <math> X_{\{f,g\}}= [X_f,X_g], </math>
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| where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians ''f'' and ''g''. As a consequence (a proof at [[Poisson bracket]]), the Poisson bracket satisfies the [[Jacobi identity]]
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| : <math> \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0, </math> | |
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| which means that the vector space of differentiable functions on ''M'', endowed with the Poisson bracket, has the structure of a [[Lie algebra]] over '''R''', and the assignment ''f'' ↦ ''X<sub>f</sub>'' is a [[Lie algebra homomorphism]], whose [[kernel (linear algebra)|kernel]] consists of the locally constant functions (constant functions if ''M'' is connected).
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| == References ==
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| * {{cite book|last=Abraham|first=Ralph|authorlink=Ralph Abraham|coauthors=[[Jerrold E. Marsden|Marsden, Jerrold E.]]|title=Foundations of Mechanics|publisher=Benjamin-Cummings|location=London|year=1978|isbn=0-8053-1012-X {{Please check ISBN|reason=Check digit (X) does not correspond to calculated figure.}}}}''See section 3.2''.
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| * {{cite book|last=Arnol'd|first=V.I.|authorlink=Vladimir Arnold|title=Mathematical Methods of Classical Mechanics|publisher=Springer |location=Berlin etc|year=1997|isbn=0-387-96890-3}}
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| * {{cite book|last=Frankel|first=Theodore|title=The Geometry of Physics|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-38753-1}}
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| * {{cite book|last=McDuff|first=Dusa|coauthors=Salamon, D.|authorlink=Dusa McDuff|title=Introduction to Symplectic Topology|series=Oxford Mathematical Monographs|year=1998|isbn=0-19-850451-9}}
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| [[Category:Symplectic geometry]]
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| [[Category:Hamiltonian mechanics]]
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