Is it possible for hamiltonian systems to have asymptotically stable rest points?
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Is it possible for hamiltonian (dynamical) systems to have asymptotically stable rest points? By Liouville's theorem I got that for symplectic hamiltonian systems only Lyapunov stability is allowed.
My knowledge about symplectic manifolds is however limited and my question would be if it is possible for "non symplectic" hamiltonian (dynamical) systems to exist and to have asymptotically stable rest points.
ordinary-differential-equations dynamical-systems stability-theory
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
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add a comment |
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Is it possible for hamiltonian (dynamical) systems to have asymptotically stable rest points? By Liouville's theorem I got that for symplectic hamiltonian systems only Lyapunov stability is allowed.
My knowledge about symplectic manifolds is however limited and my question would be if it is possible for "non symplectic" hamiltonian (dynamical) systems to exist and to have asymptotically stable rest points.
ordinary-differential-equations dynamical-systems stability-theory
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
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1
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Please tell more about these "'non-symplectic' Hamiltonian systems". $dot q=H_p,,dot p=-H_q$ is per construction symplectic for $omega=sum_i dp_iwedge dq_i$.
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– LutzL
Dec 20 '18 at 15:27
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@LutzL: It's not really clear, but maybe it's supposed to refer to Hamiltonian systems on Poisson manifolds (as opposed to symplectic manifolds)?
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– Hans Lundmark
Dec 20 '18 at 20:21
add a comment |
$begingroup$
Is it possible for hamiltonian (dynamical) systems to have asymptotically stable rest points? By Liouville's theorem I got that for symplectic hamiltonian systems only Lyapunov stability is allowed.
My knowledge about symplectic manifolds is however limited and my question would be if it is possible for "non symplectic" hamiltonian (dynamical) systems to exist and to have asymptotically stable rest points.
ordinary-differential-equations dynamical-systems stability-theory
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
$endgroup$
Is it possible for hamiltonian (dynamical) systems to have asymptotically stable rest points? By Liouville's theorem I got that for symplectic hamiltonian systems only Lyapunov stability is allowed.
My knowledge about symplectic manifolds is however limited and my question would be if it is possible for "non symplectic" hamiltonian (dynamical) systems to exist and to have asymptotically stable rest points.
ordinary-differential-equations dynamical-systems stability-theory
ordinary-differential-equations dynamical-systems stability-theory
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
asked Dec 20 '18 at 13:54
OzymandiasOzymandias
62
62
1
$begingroup$
Please tell more about these "'non-symplectic' Hamiltonian systems". $dot q=H_p,,dot p=-H_q$ is per construction symplectic for $omega=sum_i dp_iwedge dq_i$.
$endgroup$
– LutzL
Dec 20 '18 at 15:27
$begingroup$
@LutzL: It's not really clear, but maybe it's supposed to refer to Hamiltonian systems on Poisson manifolds (as opposed to symplectic manifolds)?
$endgroup$
– Hans Lundmark
Dec 20 '18 at 20:21
add a comment |
1
$begingroup$
Please tell more about these "'non-symplectic' Hamiltonian systems". $dot q=H_p,,dot p=-H_q$ is per construction symplectic for $omega=sum_i dp_iwedge dq_i$.
$endgroup$
– LutzL
Dec 20 '18 at 15:27
$begingroup$
@LutzL: It's not really clear, but maybe it's supposed to refer to Hamiltonian systems on Poisson manifolds (as opposed to symplectic manifolds)?
$endgroup$
– Hans Lundmark
Dec 20 '18 at 20:21
1
1
$begingroup$
Please tell more about these "'non-symplectic' Hamiltonian systems". $dot q=H_p,,dot p=-H_q$ is per construction symplectic for $omega=sum_i dp_iwedge dq_i$.
$endgroup$
– LutzL
Dec 20 '18 at 15:27
$begingroup$
Please tell more about these "'non-symplectic' Hamiltonian systems". $dot q=H_p,,dot p=-H_q$ is per construction symplectic for $omega=sum_i dp_iwedge dq_i$.
$endgroup$
– LutzL
Dec 20 '18 at 15:27
$begingroup$
@LutzL: It's not really clear, but maybe it's supposed to refer to Hamiltonian systems on Poisson manifolds (as opposed to symplectic manifolds)?
$endgroup$
– Hans Lundmark
Dec 20 '18 at 20:21
$begingroup$
@LutzL: It's not really clear, but maybe it's supposed to refer to Hamiltonian systems on Poisson manifolds (as opposed to symplectic manifolds)?
$endgroup$
– Hans Lundmark
Dec 20 '18 at 20:21
add a comment |
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Please tell more about these "'non-symplectic' Hamiltonian systems". $dot q=H_p,,dot p=-H_q$ is per construction symplectic for $omega=sum_i dp_iwedge dq_i$.
$endgroup$
– LutzL
Dec 20 '18 at 15:27
$begingroup$
@LutzL: It's not really clear, but maybe it's supposed to refer to Hamiltonian systems on Poisson manifolds (as opposed to symplectic manifolds)?
$endgroup$
– Hans Lundmark
Dec 20 '18 at 20:21