Timeline for Proof that a first integral is not a constant function
Current License: CC BY-SA 4.0
23 events
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| S Sep 14, 2023 at 14:07 | history | bounty ended | CommunityBot | ||
| S Sep 14, 2023 at 14:07 | history | notice removed | CommunityBot | ||
| Sep 7, 2023 at 10:41 | comment | added | NicAG | @BenMcKay About the basis functions you are right. Yeah $\theta_i(x_0)$ are the components of $\theta(x_0)$. I just introduced $\theta$ in the beginning to clarify that $\Lambda$ is a linear comnination in the basis functions. | |
| Sep 7, 2023 at 10:38 | comment | added | NicAG | @BenMcKay Sorry, I want to proof that $\Lambda '$ is non constant. Which maps to $\theta_{0}(x_0)$. Not $\Theta$. | |
| Sep 7, 2023 at 10:36 | history | edited | NicAG | CC BY-SA 4.0 |
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| Sep 7, 2023 at 9:05 | comment | added | Ben McKay | You seem to have written that $\Lambda$ is a function that maps an open set $V$ to zero. But you also write that $\Lambda$ is only defined on $V$. So $\Lambda=0$ is constant. | |
| Sep 7, 2023 at 9:02 | comment | added | Ben McKay | Am I correct in understanding that a basis function means precisely a real valued function on $U$ which is differentiable and nonconstant? | |
| Sep 7, 2023 at 8:10 | comment | added | Ben McKay | I think you are using $\theta$ to mean two different things: the components of the expression of $\Lambda$ are called $\theta_i$, while the nonzero vector $\theta(x_0)$ has components with the same names. | |
| Sep 6, 2023 at 19:23 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Sep 6, 2023 at 16:54 | history | edited | NicAG | CC BY-SA 4.0 |
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| Sep 6, 2023 at 16:52 | comment | added | NicAG | @BenMcKay I edited the question and redefined $\Lambda$ | |
| Sep 6, 2023 at 16:51 | comment | added | NicAG | @JochenWengenroth I edited the question to hopefully address all the unclear points. I redefined $U$ to just be an open set on $\mathbb{R}^n$ . | |
| Sep 6, 2023 at 16:49 | history | edited | NicAG | CC BY-SA 4.0 |
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| Sep 6, 2023 at 12:40 | comment | added | Ben McKay | I still don't follow the notation. When you write $\Lambda(x(t))=\dots$, do you mean that there is a function $\Lambda$, defined on $M$, or on $U$, with the property that, for any point $x_0\in U$ and time $t\in\mathbb{R}$, $\Lambda(x(t))$ is equal to the given expression? Also, what is $\theta_0$ defined to be? We suppose that there is a point $\theta(x_0)\in\mathbb{R}^m$, but its components are $\theta_i(x_0)$, $i=1,2,\dots,m$. | |
| Sep 6, 2023 at 12:40 | comment | added | Jochen Wengenroth | There are several things, I don't understand. The flow $\Phi(x,t)$ has values in $M$ (I assume that the vector field $F$ is good enough to have $\Phi:M\times\mathbb R\to M$), hence in order to $\psi_i(\Phi(x_0,t))$ make sense, you would need $M\subseteq \mathbb R^n$. There are several $\theta$: Is $\theta(x_0)\in \mathbb R^{m+1}\setminus\{0\}$ with components $\theta_i(x_0)$? What is $x(t)$ in the definition of $\Lambda$? If it is a solution of the equation $\dot{x}(t)=F(x(t))$, what is the initial condition? Finally, doesn't $\psi_1=-\psi_2$, $\theta_1(x_0)=\theta_2(x_0)=1$ yield $\Lambda=0$? | |
| S Sep 6, 2023 at 12:06 | history | bounty started | NicAG | ||
| S Sep 6, 2023 at 12:06 | history | notice added | NicAG | Draw attention | |
| Aug 31, 2023 at 20:12 | comment | added | NicAG | @BenMcKay A basis function is just a differentiable function of the form I described above. I called it basis function because we expand the other functions in terms of these functions. $\Lambda(x(t))$ is defined for all t. But on a given trajectory $x(t)$ it has a constant value due to the fact that the Lie deriavtive is zero. | |
| Aug 31, 2023 at 18:48 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Aug 31, 2023 at 9:03 | comment | added | Ben McKay | In the definition of $\Lambda$, are we saying that for each point $x$, we write that point as $x=x(t)$ for a point $x_0:=\Phi(x,-t)$? The notation is not clear to me. It looks like you are defining $\Lambda$ in terms of a particular value of $t$. Is that right? | |
| Aug 31, 2023 at 8:57 | comment | added | Ben McKay | What is a basis function? | |
| Aug 31, 2023 at 1:07 | comment | added | NicAG | Cross posted: math.stackexchange.com/questions/4760367/… | |
| Aug 31, 2023 at 1:07 | history | asked | NicAG | CC BY-SA 4.0 |