Understanding ODE and phase portraits on Manifolds
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I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$.
Firstly, let $X: Omega subset mathbb{R}^{n} rightarrow mathbb{R}^{n} $ a vector field and consider the equation
$dfrac{d varphi(t)}{dt}= X(varphi(t))$,
a solution for this ODE is a path $varphi(t): I subset mathbb{R} rightarrow Omega$, such that $frac{d varphi(t)}{dt}= X(varphi(t))$, for all $t in I$.
Then, I have the following doubts
1- If $M$ is a $m$-dimensional manifold $(mneq n)$, then, is the vector field defined by $X: M rightarrow TM$, which associes each $p in M$ to the vector $X(p) in T_{p}M$? This make sense for me, once that, the space $T_{p}M$ has more algebraic and topologic properties. However, take $M= S^{1}$ for example, for each $p in S^{1}$ the tangent space $T_{p}M$ is equal to the orthogonal complement of $p$, that is, $T_{p}M = [p]^{perp}$. How to construct a vector field with this property? How to construct vector fields for more general manifolds?
2- Has the solution $varphi(t)$ lie in the manifold $M$, for all $t$? If yes, then is every solution on $S^{1}$ "like" $(sin (pi t), cos(pi t))$?
3- How to draw phase portraits on manifolds? I would like to see examples of two vector fields $X,Y:S^{1}rightarrow TS^{1}*(?)*$, when $X$ and $Y$ are conjugated topologically and when they are not.
I would like to apologize if something pointed out here is too obvious or standart. Any help will be aprecciated.
differential-equations dynamical-systems vector-fields
edited Nov 27 at 17:52
Federico
4,193512
asked Nov 27 at 17:46
BBVM
17112
add a comment |
up vote
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I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$.
Firstly, let $X: Omega subset mathbb{R}^{n} rightarrow mathbb{R}^{n} $ a vector field and consider the equation
$dfrac{d varphi(t)}{dt}= X(varphi(t))$,
a solution for this ODE is a path $varphi(t): I subset mathbb{R} rightarrow Omega$, such that $frac{d varphi(t)}{dt}= X(varphi(t))$, for all $t in I$.
Then, I have the following doubts
1- If $M$ is a $m$-dimensional manifold $(mneq n)$, then, is the vector field defined by $X: M rightarrow TM$, which associes each $p in M$ to the vector $X(p) in T_{p}M$? This make sense for me, once that, the space $T_{p}M$ has more algebraic and topologic properties. However, take $M= S^{1}$ for example, for each $p in S^{1}$ the tangent space $T_{p}M$ is equal to the orthogonal complement of $p$, that is, $T_{p}M = [p]^{perp}$. How to construct a vector field with this property? How to construct vector fields for more general manifolds?
2- Has the solution $varphi(t)$ lie in the manifold $M$, for all $t$? If yes, then is every solution on $S^{1}$ "like" $(sin (pi t), cos(pi t))$?
3- How to draw phase portraits on manifolds? I would like to see examples of two vector fields $X,Y:S^{1}rightarrow TS^{1}*(?)*$, when $X$ and $Y$ are conjugated topologically and when they are not.
I would like to apologize if something pointed out here is too obvious or standart. Any help will be aprecciated.
differential-equations dynamical-systems vector-fields
edited Nov 27 at 17:52
Federico
4,193512
asked Nov 27 at 17:46
BBVM
17112
1- is answered by studying the definition of $TM$. There is an intrinsic definition that works for abstract manifolds, and then you can compare it with the extrinsic tangent in the case of an embedded manifold, as in your $S^1$ example
– Federico
Nov 27 at 17:54
2- the solution lies on $M$ by definition in the abstract setting. In the case of an embedded manifold and $X$ defined on the ambient space, the solution again lies on $M$ by the definition of $TM$: take for instance the implicit approach $M={f=0}$ and try differentiating $f(varphi(t))$
– Federico
Nov 27 at 17:59
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$.
Firstly, let $X: Omega subset mathbb{R}^{n} rightarrow mathbb{R}^{n} $ a vector field and consider the equation
$dfrac{d varphi(t)}{dt}= X(varphi(t))$,
a solution for this ODE is a path $varphi(t): I subset mathbb{R} rightarrow Omega$, such that $frac{d varphi(t)}{dt}= X(varphi(t))$, for all $t in I$.
Then, I have the following doubts
1- If $M$ is a $m$-dimensional manifold $(mneq n)$, then, is the vector field defined by $X: M rightarrow TM$, which associes each $p in M$ to the vector $X(p) in T_{p}M$? This make sense for me, once that, the space $T_{p}M$ has more algebraic and topologic properties. However, take $M= S^{1}$ for example, for each $p in S^{1}$ the tangent space $T_{p}M$ is equal to the orthogonal complement of $p$, that is, $T_{p}M = [p]^{perp}$. How to construct a vector field with this property? How to construct vector fields for more general manifolds?
2- Has the solution $varphi(t)$ lie in the manifold $M$, for all $t$? If yes, then is every solution on $S^{1}$ "like" $(sin (pi t), cos(pi t))$?
3- How to draw phase portraits on manifolds? I would like to see examples of two vector fields $X,Y:S^{1}rightarrow TS^{1}*(?)*$, when $X$ and $Y$ are conjugated topologically and when they are not.
I would like to apologize if something pointed out here is too obvious or standart. Any help will be aprecciated.
differential-equations dynamical-systems vector-fields
edited Nov 27 at 17:52
Federico
4,193512
asked Nov 27 at 17:46
BBVM
17112
I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$.
Firstly, let $X: Omega subset mathbb{R}^{n} rightarrow mathbb{R}^{n} $ a vector field and consider the equation
$dfrac{d varphi(t)}{dt}= X(varphi(t))$,
a solution for this ODE is a path $varphi(t): I subset mathbb{R} rightarrow Omega$, such that $frac{d varphi(t)}{dt}= X(varphi(t))$, for all $t in I$.
Then, I have the following doubts
1- If $M$ is a $m$-dimensional manifold $(mneq n)$, then, is the vector field defined by $X: M rightarrow TM$, which associes each $p in M$ to the vector $X(p) in T_{p}M$? This make sense for me, once that, the space $T_{p}M$ has more algebraic and topologic properties. However, take $M= S^{1}$ for example, for each $p in S^{1}$ the tangent space $T_{p}M$ is equal to the orthogonal complement of $p$, that is, $T_{p}M = [p]^{perp}$. How to construct a vector field with this property? How to construct vector fields for more general manifolds?
2- Has the solution $varphi(t)$ lie in the manifold $M$, for all $t$? If yes, then is every solution on $S^{1}$ "like" $(sin (pi t), cos(pi t))$?
3- How to draw phase portraits on manifolds? I would like to see examples of two vector fields $X,Y:S^{1}rightarrow TS^{1}*(?)*$, when $X$ and $Y$ are conjugated topologically and when they are not.
I would like to apologize if something pointed out here is too obvious or standart. Any help will be aprecciated.
differential-equations dynamical-systems vector-fields
differential-equations dynamical-systems vector-fields
edited Nov 27 at 17:52
Federico
4,193512
asked Nov 27 at 17:46
BBVM
17112
edited Nov 27 at 17:52
Federico
4,193512
asked Nov 27 at 17:46
BBVM
17112
edited Nov 27 at 17:52
Federico
4,193512
edited Nov 27 at 17:52
Federico
4,193512
edited Nov 27 at 17:52
Federico
4,193512
4,193512
asked Nov 27 at 17:46
BBVM
17112
asked Nov 27 at 17:46
BBVM
17112
asked Nov 27 at 17:46
BBVM
17112
17112
1- is answered by studying the definition of $TM$. There is an intrinsic definition that works for abstract manifolds, and then you can compare it with the extrinsic tangent in the case of an embedded manifold, as in your $S^1$ example
– Federico
Nov 27 at 17:54
2- the solution lies on $M$ by definition in the abstract setting. In the case of an embedded manifold and $X$ defined on the ambient space, the solution again lies on $M$ by the definition of $TM$: take for instance the implicit approach $M={f=0}$ and try differentiating $f(varphi(t))$
– Federico
Nov 27 at 17:59
add a comment |
1- is answered by studying the definition of $TM$. There is an intrinsic definition that works for abstract manifolds, and then you can compare it with the extrinsic tangent in the case of an embedded manifold, as in your $S^1$ example
– Federico
Nov 27 at 17:54
2- the solution lies on $M$ by definition in the abstract setting. In the case of an embedded manifold and $X$ defined on the ambient space, the solution again lies on $M$ by the definition of $TM$: take for instance the implicit approach $M={f=0}$ and try differentiating $f(varphi(t))$
– Federico
Nov 27 at 17:59
1- is answered by studying the definition of $TM$. There is an intrinsic definition that works for abstract manifolds, and then you can compare it with the extrinsic tangent in the case of an embedded manifold, as in your $S^1$ example
– Federico
Nov 27 at 17:54
1- is answered by studying the definition of $TM$. There is an intrinsic definition that works for abstract manifolds, and then you can compare it with the extrinsic tangent in the case of an embedded manifold, as in your $S^1$ example
– Federico
Nov 27 at 17:54
2- the solution lies on $M$ by definition in the abstract setting. In the case of an embedded manifold and $X$ defined on the ambient space, the solution again lies on $M$ by the definition of $TM$: take for instance the implicit approach $M={f=0}$ and try differentiating $f(varphi(t))$
– Federico
Nov 27 at 17:59
2- the solution lies on $M$ by definition in the abstract setting. In the case of an embedded manifold and $X$ defined on the ambient space, the solution again lies on $M$ by the definition of $TM$: take for instance the implicit approach $M={f=0}$ and try differentiating $f(varphi(t))$
– Federico
Nov 27 at 17:59
add a comment |
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1- is answered by studying the definition of $TM$. There is an intrinsic definition that works for abstract manifolds, and then you can compare it with the extrinsic tangent in the case of an embedded manifold, as in your $S^1$ example
– Federico
Nov 27 at 17:54
2- the solution lies on $M$ by definition in the abstract setting. In the case of an embedded manifold and $X$ defined on the ambient space, the solution again lies on $M$ by the definition of $TM$: take for instance the implicit approach $M={f=0}$ and try differentiating $f(varphi(t))$
– Federico
Nov 27 at 17:59