Lie group structure of incompressible deformations?
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
478211
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
478211
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
478211
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
478211
asked Nov 17 at 10:35
Steffen Plunder
478211
asked Nov 17 at 10:35
Steffen Plunder
478211
asked Nov 17 at 10:35
Steffen Plunder
478211
asked Nov 17 at 10:35
Steffen Plunder
478211
478211
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 |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002184%2flie-group-structure-of-incompressible-deformations%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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