Lie group structure of incompressible deformations?
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1
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In the context of incompressible elasticity, I failed at finding
a Lie group structure, which respects the incopressibility.
Question:
Let $mathcal B, mathcal S$ be Riemannian manifolds and $overline{mathcal B}$ is compact.
Does the smooth manifold of incompressible deformations
$$
mathcal{C} := { varphi: mathcal{B} to mathcal S mid varphi in mathrm{Diff}(mathcal B, mathcal S) ~text{with}~ mathrm{det}(mathrm{D} varphi) = 1}
$$
admit a Lie group structure?
(Here, $mathcal B$ is the reference configuration of the body (fixed in time) and $mathcal S$ is the space in which the body is placed, i.e. $varphi(X)$ is the current position of the material point $X$.)
Is a Lie structure only possible, if for example $mathcal S subseteq mathbb{R}^d$ or if $mathcal S$ is a Lie group?
Recommendations for related literature or a short 'No, there is no Lie group' are very welcome!
My problem:
A reasonable candidate for the Lie group operation is the composition of maps.
But if $mathcal B neq mathcal S$, it is tricky to define $varphi_2 circ varphi_1$ for $varphi_i in mathrm{Diff}(mathcal B, mathcal S)$.
Fluid dynamics:
In fluid dynamics, there seem to exists a Lie group, since typically the Eulerian perspective is used. If $Omega$ denotes the domain of the fluid,
then the displacement of fluid particles is a map $phi in mathrm{Diff}(Omega, Omega)$, which is a Lie group with the composition of two maps as group operation.
But I was not able to transfer this to the setting above.
lie-groups mathematical-physics classical-mechanics
asked Nov 17 at 10:35
Steffen Plunder
498211
add a comment |
up vote
1
down vote
favorite
In the context of incompressible elasticity, I failed at finding
a Lie group structure, which respects the incopressibility.
Question:
Let $mathcal B, mathcal S$ be Riemannian manifolds and $overline{mathcal B}$ is compact.
Does the smooth manifold of incompressible deformations
$$
mathcal{C} := { varphi: mathcal{B} to mathcal S mid varphi in mathrm{Diff}(mathcal B, mathcal S) ~text{with}~ mathrm{det}(mathrm{D} varphi) = 1}
$$
admit a Lie group structure?
(Here, $mathcal B$ is the reference configuration of the body (fixed in time) and $mathcal S$ is the space in which the body is placed, i.e. $varphi(X)$ is the current position of the material point $X$.)
Is a Lie structure only possible, if for example $mathcal S subseteq mathbb{R}^d$ or if $mathcal S$ is a Lie group?
Recommendations for related literature or a short 'No, there is no Lie group' are very welcome!
My problem:
A reasonable candidate for the Lie group operation is the composition of maps.
But if $mathcal B neq mathcal S$, it is tricky to define $varphi_2 circ varphi_1$ for $varphi_i in mathrm{Diff}(mathcal B, mathcal S)$.
Fluid dynamics:
In fluid dynamics, there seem to exists a Lie group, since typically the Eulerian perspective is used. If $Omega$ denotes the domain of the fluid,
then the displacement of fluid particles is a map $phi in mathrm{Diff}(Omega, Omega)$, which is a Lie group with the composition of two maps as group operation.
But I was not able to transfer this to the setting above.
lie-groups mathematical-physics classical-mechanics
asked Nov 17 at 10:35
Steffen Plunder
498211
1
Well, for a 2d manifold, the group is just that of Poisson brackets, namely $SU(infty)$, as you are probably aware from 2d hydrodynamics. In fact, as you can tell from Liouville's theorem in classical mechanics, classical phase-space flows in all dimensions are incompressible. In odd-dimensional manifolds, Nambu brackets are utilized, and the group structure is more moot.
– Cosmas Zachos
Nov 23 at 15:14
1
A modern book on symplectomorphism groups, 2018. For the standard theory, see JMP 1990.
– Cosmas Zachos
Nov 23 at 16:29
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In the context of incompressible elasticity, I failed at finding
a Lie group structure, which respects the incopressibility.
Question:
Let $mathcal B, mathcal S$ be Riemannian manifolds and $overline{mathcal B}$ is compact.
Does the smooth manifold of incompressible deformations
$$
mathcal{C} := { varphi: mathcal{B} to mathcal S mid varphi in mathrm{Diff}(mathcal B, mathcal S) ~text{with}~ mathrm{det}(mathrm{D} varphi) = 1}
$$
admit a Lie group structure?
(Here, $mathcal B$ is the reference configuration of the body (fixed in time) and $mathcal S$ is the space in which the body is placed, i.e. $varphi(X)$ is the current position of the material point $X$.)
Is a Lie structure only possible, if for example $mathcal S subseteq mathbb{R}^d$ or if $mathcal S$ is a Lie group?
Recommendations for related literature or a short 'No, there is no Lie group' are very welcome!
My problem:
A reasonable candidate for the Lie group operation is the composition of maps.
But if $mathcal B neq mathcal S$, it is tricky to define $varphi_2 circ varphi_1$ for $varphi_i in mathrm{Diff}(mathcal B, mathcal S)$.
Fluid dynamics:
In fluid dynamics, there seem to exists a Lie group, since typically the Eulerian perspective is used. If $Omega$ denotes the domain of the fluid,
then the displacement of fluid particles is a map $phi in mathrm{Diff}(Omega, Omega)$, which is a Lie group with the composition of two maps as group operation.
But I was not able to transfer this to the setting above.
lie-groups mathematical-physics classical-mechanics
asked Nov 17 at 10:35
Steffen Plunder
498211
In the context of incompressible elasticity, I failed at finding
a Lie group structure, which respects the incopressibility.
Question:
Let $mathcal B, mathcal S$ be Riemannian manifolds and $overline{mathcal B}$ is compact.
Does the smooth manifold of incompressible deformations
$$
mathcal{C} := { varphi: mathcal{B} to mathcal S mid varphi in mathrm{Diff}(mathcal B, mathcal S) ~text{with}~ mathrm{det}(mathrm{D} varphi) = 1}
$$
admit a Lie group structure?
(Here, $mathcal B$ is the reference configuration of the body (fixed in time) and $mathcal S$ is the space in which the body is placed, i.e. $varphi(X)$ is the current position of the material point $X$.)
Is a Lie structure only possible, if for example $mathcal S subseteq mathbb{R}^d$ or if $mathcal S$ is a Lie group?
Recommendations for related literature or a short 'No, there is no Lie group' are very welcome!
My problem:
A reasonable candidate for the Lie group operation is the composition of maps.
But if $mathcal B neq mathcal S$, it is tricky to define $varphi_2 circ varphi_1$ for $varphi_i in mathrm{Diff}(mathcal B, mathcal S)$.
Fluid dynamics:
In fluid dynamics, there seem to exists a Lie group, since typically the Eulerian perspective is used. If $Omega$ denotes the domain of the fluid,
then the displacement of fluid particles is a map $phi in mathrm{Diff}(Omega, Omega)$, which is a Lie group with the composition of two maps as group operation.
But I was not able to transfer this to the setting above.
lie-groups mathematical-physics classical-mechanics
lie-groups mathematical-physics classical-mechanics
asked Nov 17 at 10:35
Steffen Plunder
498211
asked Nov 17 at 10:35
Steffen Plunder
498211
asked Nov 17 at 10:35
Steffen Plunder
498211
asked Nov 17 at 10:35
Steffen Plunder
498211
asked Nov 17 at 10:35
Steffen Plunder
498211
498211
1
Well, for a 2d manifold, the group is just that of Poisson brackets, namely $SU(infty)$, as you are probably aware from 2d hydrodynamics. In fact, as you can tell from Liouville's theorem in classical mechanics, classical phase-space flows in all dimensions are incompressible. In odd-dimensional manifolds, Nambu brackets are utilized, and the group structure is more moot.
– Cosmas Zachos
Nov 23 at 15:14
1
A modern book on symplectomorphism groups, 2018. For the standard theory, see JMP 1990.
– Cosmas Zachos
Nov 23 at 16:29
add a comment |
1
Well, for a 2d manifold, the group is just that of Poisson brackets, namely $SU(infty)$, as you are probably aware from 2d hydrodynamics. In fact, as you can tell from Liouville's theorem in classical mechanics, classical phase-space flows in all dimensions are incompressible. In odd-dimensional manifolds, Nambu brackets are utilized, and the group structure is more moot.
– Cosmas Zachos
Nov 23 at 15:14
1
A modern book on symplectomorphism groups, 2018. For the standard theory, see JMP 1990.
– Cosmas Zachos
Nov 23 at 16:29
1
1
Well, for a 2d manifold, the group is just that of Poisson brackets, namely $SU(infty)$, as you are probably aware from 2d hydrodynamics. In fact, as you can tell from Liouville's theorem in classical mechanics, classical phase-space flows in all dimensions are incompressible. In odd-dimensional manifolds, Nambu brackets are utilized, and the group structure is more moot.
– Cosmas Zachos
Nov 23 at 15:14
Well, for a 2d manifold, the group is just that of Poisson brackets, namely $SU(infty)$, as you are probably aware from 2d hydrodynamics. In fact, as you can tell from Liouville's theorem in classical mechanics, classical phase-space flows in all dimensions are incompressible. In odd-dimensional manifolds, Nambu brackets are utilized, and the group structure is more moot.
– Cosmas Zachos
Nov 23 at 15:14
1
1
A modern book on symplectomorphism groups, 2018. For the standard theory, see JMP 1990.
– Cosmas Zachos
Nov 23 at 16:29
A modern book on symplectomorphism groups, 2018. For the standard theory, see JMP 1990.
– Cosmas Zachos
Nov 23 at 16:29
add a comment |
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Well, for a 2d manifold, the group is just that of Poisson brackets, namely $SU(infty)$, as you are probably aware from 2d hydrodynamics. In fact, as you can tell from Liouville's theorem in classical mechanics, classical phase-space flows in all dimensions are incompressible. In odd-dimensional manifolds, Nambu brackets are utilized, and the group structure is more moot.
– Cosmas Zachos
Nov 23 at 15:14
1
A modern book on symplectomorphism groups, 2018. For the standard theory, see JMP 1990.
– Cosmas Zachos
Nov 23 at 16:29