Kinetic energy of incompressible fluid as quadratic form on tangent space.
$begingroup$
I have started working through "Topological Methods in Hydrodynamics" by V.I. Arnold and I am confused by the following located on page 2:
"The kinetic energy of an [incompressible] fluid [with density 1] is...
begin{align*}
E = frac{1}{2} int_{M} v^{2} dx
end{align*}
where v is the velocity field of the fluid: $v(x,t) = frac{partial}{partial t} g^t(y), x = g^t(y)$ (y is the initial position of the particle whose position is x at the moment t.
Suppose that the configuration $g$ has velocity $dot{g}$. The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$. The kinetic energy is a quadratic form on this vector space of velocities"
Some definitions of the objects used:
M is the manifold that the fluid lives in
SDiff(M) is the connected component of the Lie group of diffeomorphisms: $M rightarrow M$ that preserve the volume form (i.e $f^{*}w = w$, where $f^{*}w$ is the pullback of $w$ by $f in SDiff(M)$ where $w$ is the volume form).
x and y are elements of M
My confusion lies with the statement: "The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$"
I fail to see how $v = frac{partial}{partial t} g^t(y)$ satisfies the definition of a tangent vector that I was taught:
My def:
Let $gamma$ be a curve:
begin{align*}
R & rightarrow SDiff(M)\
t & mapsto g^t
end{align*}
with $gamma (0) = g$
Then $D_{gamma}$ is the derivation of $gamma$ at $g$ defined by:
$D_{gamma}(h) = (h circ gamma)'(0)$
where $hin C^{infty}(M)$
These $D_{gamma}$ are the elements of $T_{g}G$ where $G = SDiff(M)$
I do not see how to interpret $frac{partial}{partial t} g^t(y)$ as this type of object. For one thing $g^t$ is a map $M rightarrow M$ and I am unsure of how to take the partial derivative of such a thing. Furthermore I am not sure of how I can interpret $frac{partial}{partial t} g^t(y)$ as a map from $C^{infty}(M)$ to $R$.
differential-geometry fluid-dynamics
asked Dec 24 '18 at 22:43
GooseGoose
63
$endgroup$
add a comment |
$begingroup$
I have started working through "Topological Methods in Hydrodynamics" by V.I. Arnold and I am confused by the following located on page 2:
"The kinetic energy of an [incompressible] fluid [with density 1] is...
begin{align*}
E = frac{1}{2} int_{M} v^{2} dx
end{align*}
where v is the velocity field of the fluid: $v(x,t) = frac{partial}{partial t} g^t(y), x = g^t(y)$ (y is the initial position of the particle whose position is x at the moment t.
Suppose that the configuration $g$ has velocity $dot{g}$. The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$. The kinetic energy is a quadratic form on this vector space of velocities"
Some definitions of the objects used:
M is the manifold that the fluid lives in
SDiff(M) is the connected component of the Lie group of diffeomorphisms: $M rightarrow M$ that preserve the volume form (i.e $f^{*}w = w$, where $f^{*}w$ is the pullback of $w$ by $f in SDiff(M)$ where $w$ is the volume form).
x and y are elements of M
My confusion lies with the statement: "The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$"
I fail to see how $v = frac{partial}{partial t} g^t(y)$ satisfies the definition of a tangent vector that I was taught:
My def:
Let $gamma$ be a curve:
begin{align*}
R & rightarrow SDiff(M)\
t & mapsto g^t
end{align*}
with $gamma (0) = g$
Then $D_{gamma}$ is the derivation of $gamma$ at $g$ defined by:
$D_{gamma}(h) = (h circ gamma)'(0)$
where $hin C^{infty}(M)$
These $D_{gamma}$ are the elements of $T_{g}G$ where $G = SDiff(M)$
I do not see how to interpret $frac{partial}{partial t} g^t(y)$ as this type of object. For one thing $g^t$ is a map $M rightarrow M$ and I am unsure of how to take the partial derivative of such a thing. Furthermore I am not sure of how I can interpret $frac{partial}{partial t} g^t(y)$ as a map from $C^{infty}(M)$ to $R$.
differential-geometry fluid-dynamics
asked Dec 24 '18 at 22:43
GooseGoose
63
$endgroup$
add a comment |
$begingroup$
I have started working through "Topological Methods in Hydrodynamics" by V.I. Arnold and I am confused by the following located on page 2:
"The kinetic energy of an [incompressible] fluid [with density 1] is...
begin{align*}
E = frac{1}{2} int_{M} v^{2} dx
end{align*}
where v is the velocity field of the fluid: $v(x,t) = frac{partial}{partial t} g^t(y), x = g^t(y)$ (y is the initial position of the particle whose position is x at the moment t.
Suppose that the configuration $g$ has velocity $dot{g}$. The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$. The kinetic energy is a quadratic form on this vector space of velocities"
Some definitions of the objects used:
M is the manifold that the fluid lives in
SDiff(M) is the connected component of the Lie group of diffeomorphisms: $M rightarrow M$ that preserve the volume form (i.e $f^{*}w = w$, where $f^{*}w$ is the pullback of $w$ by $f in SDiff(M)$ where $w$ is the volume form).
x and y are elements of M
My confusion lies with the statement: "The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$"
I fail to see how $v = frac{partial}{partial t} g^t(y)$ satisfies the definition of a tangent vector that I was taught:
My def:
Let $gamma$ be a curve:
begin{align*}
R & rightarrow SDiff(M)\
t & mapsto g^t
end{align*}
with $gamma (0) = g$
Then $D_{gamma}$ is the derivation of $gamma$ at $g$ defined by:
$D_{gamma}(h) = (h circ gamma)'(0)$
where $hin C^{infty}(M)$
These $D_{gamma}$ are the elements of $T_{g}G$ where $G = SDiff(M)$
I do not see how to interpret $frac{partial}{partial t} g^t(y)$ as this type of object. For one thing $g^t$ is a map $M rightarrow M$ and I am unsure of how to take the partial derivative of such a thing. Furthermore I am not sure of how I can interpret $frac{partial}{partial t} g^t(y)$ as a map from $C^{infty}(M)$ to $R$.
differential-geometry fluid-dynamics
asked Dec 24 '18 at 22:43
GooseGoose
63
$endgroup$
I have started working through "Topological Methods in Hydrodynamics" by V.I. Arnold and I am confused by the following located on page 2:
"The kinetic energy of an [incompressible] fluid [with density 1] is...
begin{align*}
E = frac{1}{2} int_{M} v^{2} dx
end{align*}
where v is the velocity field of the fluid: $v(x,t) = frac{partial}{partial t} g^t(y), x = g^t(y)$ (y is the initial position of the particle whose position is x at the moment t.
Suppose that the configuration $g$ has velocity $dot{g}$. The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$. The kinetic energy is a quadratic form on this vector space of velocities"
Some definitions of the objects used:
M is the manifold that the fluid lives in
SDiff(M) is the connected component of the Lie group of diffeomorphisms: $M rightarrow M$ that preserve the volume form (i.e $f^{*}w = w$, where $f^{*}w$ is the pullback of $w$ by $f in SDiff(M)$ where $w$ is the volume form).
x and y are elements of M
My confusion lies with the statement: "The vector $dot{g}$ belongs to the tangent space $T_{g}G$ of the [Lie] group $G = SDiff(M)$"
I fail to see how $v = frac{partial}{partial t} g^t(y)$ satisfies the definition of a tangent vector that I was taught:
My def:
Let $gamma$ be a curve:
begin{align*}
R & rightarrow SDiff(M)\
t & mapsto g^t
end{align*}
with $gamma (0) = g$
Then $D_{gamma}$ is the derivation of $gamma$ at $g$ defined by:
$D_{gamma}(h) = (h circ gamma)'(0)$
where $hin C^{infty}(M)$
These $D_{gamma}$ are the elements of $T_{g}G$ where $G = SDiff(M)$
I do not see how to interpret $frac{partial}{partial t} g^t(y)$ as this type of object. For one thing $g^t$ is a map $M rightarrow M$ and I am unsure of how to take the partial derivative of such a thing. Furthermore I am not sure of how I can interpret $frac{partial}{partial t} g^t(y)$ as a map from $C^{infty}(M)$ to $R$.
differential-geometry fluid-dynamics
differential-geometry fluid-dynamics
asked Dec 24 '18 at 22:43
GooseGoose
63
asked Dec 24 '18 at 22:43
GooseGoose
63
asked Dec 24 '18 at 22:43
GooseGoose
63
asked Dec 24 '18 at 22:43
GooseGoose
63
asked Dec 24 '18 at 22:43
GooseGoose
63
63
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