| Version 3 |
Version 2 |
| Let $(M,\omega)$ be a symplectic manifold, and $\tom:TM\to T^*M$ be the isomorphism from the tangent |
Let $(M,\omega)$ be a symplectic manifold, and $\tom:TM\to T^*M$ be the isomorphism from the tangent |
| bundle to the cotangent bundle $$X\mapsto\om(\cdot,X)$$ and let $f:M\to\R$ is a smooth function. |
bundle to the cotangent bundle $$X\mapsto\om(\cdot,X)$$ and let $f:M\to\R$ is a smooth function. |
| Then $H_f=\tom^{-1}(df)$ |
Then $H_f=\tom^{-1}(df)$ |
| is the {\em Hamiltonian vector field} of $f$. The vector field $H_f$ is \PMlinkname{symplectic}{SymplecticVectorField}, |
is the {\em Hamiltonian vector field} of $f$. The vector field $H_f$ is \PMlinkname{symplectic}{SymplecticVectorField}, |
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and a symplectic vector field $X$ is \PMlinkid{6410}{Hamiltonian} if and only if the 1-form $\tom(X)=
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and a symplectic vector field $X$ is Hamiltonian if and only if the 1-form $\tom(X)=
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| \om(\cdot,X)$ is exact. |
\om(\cdot,X)$ is exact. |
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| If $T^*Q$ is the cotangent bundle of a manifold $Q$, which is naturally identified with the phase |
If $T^*Q$ is the cotangent bundle of a manifold $Q$, which is naturally identified with the phase |
| space of one particle on $Q$, and $f$ is the Hamiltonian, then the flow of the Hamiltonian |
space of one particle on $Q$, and $f$ is the Hamiltonian, then the flow of the Hamiltonian |
| vector field $H_f$ is the time flow of the physical system. |
vector field $H_f$ is the time flow of the physical system. |
