Well-posedness of linear ODE problem on vector bundle
$begingroup$
Let $gamma colon [0,alpha] to mathbb{R}^{3}$ be a smooth regular curve and let $Ngamma$ denote its normal bundle.
Recall that $Ngamma$ is a smooth vector bundle, whose fiber at $gamma(t)$ is the orthogonal complement $dot{gamma}(t)^{perp}$ in $T_{gamma(t)}mathbb{R}^{3} cong mathbb{R}^{3}$ of the tangent vector $dot{gamma}(t)$.
For any smooth section $V$ of $Ngamma$, we define $D_{t}V$ to be the orthogonal projection onto $Ngamma$ of the Euclidean acceleration $overline{D}_{t}V equiv dot{V}$.
Question: Given a smooth function $f colon I to mathbb{R}$, does the linear ODE system on $Ngamma$
$$
begin{cases}
D_{t}V = f W\
D_{t}W = -fV
end{cases}
$$
have unique global solution for any initial condition $(v,w)$?
ordinary-differential-equations differential-geometry riemannian-geometry smooth-manifolds vector-bundles
asked Jan 11 at 14:03
MK7MK7
310210
$endgroup$
add a comment |
$begingroup$
Let $gamma colon [0,alpha] to mathbb{R}^{3}$ be a smooth regular curve and let $Ngamma$ denote its normal bundle.
Recall that $Ngamma$ is a smooth vector bundle, whose fiber at $gamma(t)$ is the orthogonal complement $dot{gamma}(t)^{perp}$ in $T_{gamma(t)}mathbb{R}^{3} cong mathbb{R}^{3}$ of the tangent vector $dot{gamma}(t)$.
For any smooth section $V$ of $Ngamma$, we define $D_{t}V$ to be the orthogonal projection onto $Ngamma$ of the Euclidean acceleration $overline{D}_{t}V equiv dot{V}$.
Question: Given a smooth function $f colon I to mathbb{R}$, does the linear ODE system on $Ngamma$
$$
begin{cases}
D_{t}V = f W\
D_{t}W = -fV
end{cases}
$$
have unique global solution for any initial condition $(v,w)$?
ordinary-differential-equations differential-geometry riemannian-geometry smooth-manifolds vector-bundles
asked Jan 11 at 14:03
MK7MK7
310210
$endgroup$
1
$begingroup$
I would say yes. However, it would help if you provided more background. In particular, what did you try? What do you know about your vector bundle? What do you know about vector bundles in general?
$endgroup$
– Amitai Yuval
Jan 11 at 19:34
$begingroup$
@AmitaiYuval I thought about looking at the system of ODEs in a smooth local trivialization of $Ngamma$. In fact, my vector bundle admits a smooth global trivialization. Is this the correct strategy?
$endgroup$
– MK7
Jan 12 at 9:22
1
$begingroup$
Sounds like a good strategy, yeah. Take a global trivialization and consider the space of double sections, that is, the space of pairs $(v,w)$, where $v$ and $w$ are sections. In terms of a trivialization, this space looks like the space of functions from interval to $mathbb{R}^4$, or equivalently, the space of paths in $mathbb{R}^4$. This makes your ODE look like the flow of a time-dependent linear vector field. This flow exists for all time and for every initial condition (you don't need the assumption $vcdot w=0$.
$endgroup$
– Amitai Yuval
Jan 12 at 11:32
$begingroup$
@AmitaiYuval OK, thanks for the help! I have edited the question and removed the assumption on the initial condition.
$endgroup$
– MK7
Jan 12 at 17:22
add a comment |
$begingroup$
Let $gamma colon [0,alpha] to mathbb{R}^{3}$ be a smooth regular curve and let $Ngamma$ denote its normal bundle.
Recall that $Ngamma$ is a smooth vector bundle, whose fiber at $gamma(t)$ is the orthogonal complement $dot{gamma}(t)^{perp}$ in $T_{gamma(t)}mathbb{R}^{3} cong mathbb{R}^{3}$ of the tangent vector $dot{gamma}(t)$.
For any smooth section $V$ of $Ngamma$, we define $D_{t}V$ to be the orthogonal projection onto $Ngamma$ of the Euclidean acceleration $overline{D}_{t}V equiv dot{V}$.
Question: Given a smooth function $f colon I to mathbb{R}$, does the linear ODE system on $Ngamma$
$$
begin{cases}
D_{t}V = f W\
D_{t}W = -fV
end{cases}
$$
have unique global solution for any initial condition $(v,w)$?
ordinary-differential-equations differential-geometry riemannian-geometry smooth-manifolds vector-bundles
asked Jan 11 at 14:03
MK7MK7
310210
$endgroup$
Let $gamma colon [0,alpha] to mathbb{R}^{3}$ be a smooth regular curve and let $Ngamma$ denote its normal bundle.
Recall that $Ngamma$ is a smooth vector bundle, whose fiber at $gamma(t)$ is the orthogonal complement $dot{gamma}(t)^{perp}$ in $T_{gamma(t)}mathbb{R}^{3} cong mathbb{R}^{3}$ of the tangent vector $dot{gamma}(t)$.
For any smooth section $V$ of $Ngamma$, we define $D_{t}V$ to be the orthogonal projection onto $Ngamma$ of the Euclidean acceleration $overline{D}_{t}V equiv dot{V}$.
Question: Given a smooth function $f colon I to mathbb{R}$, does the linear ODE system on $Ngamma$
$$
begin{cases}
D_{t}V = f W\
D_{t}W = -fV
end{cases}
$$
have unique global solution for any initial condition $(v,w)$?
ordinary-differential-equations differential-geometry riemannian-geometry smooth-manifolds vector-bundles
ordinary-differential-equations differential-geometry riemannian-geometry smooth-manifolds vector-bundles
asked Jan 11 at 14:03
MK7MK7
310210
asked Jan 11 at 14:03
MK7MK7
310210
edited Jan 12 at 17:22
MK7
asked Jan 11 at 14:03
MK7MK7
310210
asked Jan 11 at 14:03
MK7MK7
310210
asked Jan 11 at 14:03
MK7MK7
310210
310210
1
$begingroup$
I would say yes. However, it would help if you provided more background. In particular, what did you try? What do you know about your vector bundle? What do you know about vector bundles in general?
$endgroup$
– Amitai Yuval
Jan 11 at 19:34
$begingroup$
@AmitaiYuval I thought about looking at the system of ODEs in a smooth local trivialization of $Ngamma$. In fact, my vector bundle admits a smooth global trivialization. Is this the correct strategy?
$endgroup$
– MK7
Jan 12 at 9:22
1
$begingroup$
Sounds like a good strategy, yeah. Take a global trivialization and consider the space of double sections, that is, the space of pairs $(v,w)$, where $v$ and $w$ are sections. In terms of a trivialization, this space looks like the space of functions from interval to $mathbb{R}^4$, or equivalently, the space of paths in $mathbb{R}^4$. This makes your ODE look like the flow of a time-dependent linear vector field. This flow exists for all time and for every initial condition (you don't need the assumption $vcdot w=0$.
$endgroup$
– Amitai Yuval
Jan 12 at 11:32
$begingroup$
@AmitaiYuval OK, thanks for the help! I have edited the question and removed the assumption on the initial condition.
$endgroup$
– MK7
Jan 12 at 17:22
add a comment |
1
$begingroup$
I would say yes. However, it would help if you provided more background. In particular, what did you try? What do you know about your vector bundle? What do you know about vector bundles in general?
$endgroup$
– Amitai Yuval
Jan 11 at 19:34
$begingroup$
@AmitaiYuval I thought about looking at the system of ODEs in a smooth local trivialization of $Ngamma$. In fact, my vector bundle admits a smooth global trivialization. Is this the correct strategy?
$endgroup$
– MK7
Jan 12 at 9:22
1
$begingroup$
Sounds like a good strategy, yeah. Take a global trivialization and consider the space of double sections, that is, the space of pairs $(v,w)$, where $v$ and $w$ are sections. In terms of a trivialization, this space looks like the space of functions from interval to $mathbb{R}^4$, or equivalently, the space of paths in $mathbb{R}^4$. This makes your ODE look like the flow of a time-dependent linear vector field. This flow exists for all time and for every initial condition (you don't need the assumption $vcdot w=0$.
$endgroup$
– Amitai Yuval
Jan 12 at 11:32
$begingroup$
@AmitaiYuval OK, thanks for the help! I have edited the question and removed the assumption on the initial condition.
$endgroup$
– MK7
Jan 12 at 17:22
1
1
$begingroup$
I would say yes. However, it would help if you provided more background. In particular, what did you try? What do you know about your vector bundle? What do you know about vector bundles in general?
$endgroup$
– Amitai Yuval
Jan 11 at 19:34
$begingroup$
I would say yes. However, it would help if you provided more background. In particular, what did you try? What do you know about your vector bundle? What do you know about vector bundles in general?
$endgroup$
– Amitai Yuval
Jan 11 at 19:34
$begingroup$
@AmitaiYuval I thought about looking at the system of ODEs in a smooth local trivialization of $Ngamma$. In fact, my vector bundle admits a smooth global trivialization. Is this the correct strategy?
$endgroup$
– MK7
Jan 12 at 9:22
$begingroup$
@AmitaiYuval I thought about looking at the system of ODEs in a smooth local trivialization of $Ngamma$. In fact, my vector bundle admits a smooth global trivialization. Is this the correct strategy?
$endgroup$
– MK7
Jan 12 at 9:22
1
1
$begingroup$
Sounds like a good strategy, yeah. Take a global trivialization and consider the space of double sections, that is, the space of pairs $(v,w)$, where $v$ and $w$ are sections. In terms of a trivialization, this space looks like the space of functions from interval to $mathbb{R}^4$, or equivalently, the space of paths in $mathbb{R}^4$. This makes your ODE look like the flow of a time-dependent linear vector field. This flow exists for all time and for every initial condition (you don't need the assumption $vcdot w=0$.
$endgroup$
– Amitai Yuval
Jan 12 at 11:32
$begingroup$
Sounds like a good strategy, yeah. Take a global trivialization and consider the space of double sections, that is, the space of pairs $(v,w)$, where $v$ and $w$ are sections. In terms of a trivialization, this space looks like the space of functions from interval to $mathbb{R}^4$, or equivalently, the space of paths in $mathbb{R}^4$. This makes your ODE look like the flow of a time-dependent linear vector field. This flow exists for all time and for every initial condition (you don't need the assumption $vcdot w=0$.
$endgroup$
– Amitai Yuval
Jan 12 at 11:32
$begingroup$
@AmitaiYuval OK, thanks for the help! I have edited the question and removed the assumption on the initial condition.
$endgroup$
– MK7
Jan 12 at 17:22
$begingroup$
@AmitaiYuval OK, thanks for the help! I have edited the question and removed the assumption on the initial condition.
$endgroup$
– MK7
Jan 12 at 17:22
add a comment |
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$begingroup$
I would say yes. However, it would help if you provided more background. In particular, what did you try? What do you know about your vector bundle? What do you know about vector bundles in general?
$endgroup$
– Amitai Yuval
Jan 11 at 19:34
$begingroup$
@AmitaiYuval I thought about looking at the system of ODEs in a smooth local trivialization of $Ngamma$. In fact, my vector bundle admits a smooth global trivialization. Is this the correct strategy?
$endgroup$
– MK7
Jan 12 at 9:22
1
$begingroup$
Sounds like a good strategy, yeah. Take a global trivialization and consider the space of double sections, that is, the space of pairs $(v,w)$, where $v$ and $w$ are sections. In terms of a trivialization, this space looks like the space of functions from interval to $mathbb{R}^4$, or equivalently, the space of paths in $mathbb{R}^4$. This makes your ODE look like the flow of a time-dependent linear vector field. This flow exists for all time and for every initial condition (you don't need the assumption $vcdot w=0$.
$endgroup$
– Amitai Yuval
Jan 12 at 11:32
$begingroup$
@AmitaiYuval OK, thanks for the help! I have edited the question and removed the assumption on the initial condition.
$endgroup$
– MK7
Jan 12 at 17:22