Timeline for Global first integral for certain $3$ dimensional system
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2019 at 19:26 | comment | added | Michael Engelhardt | Indeed, the trajectories $(x(t),y(t),z(t))$ I describe are by construction equipotential lines, i.e., lines of constant $f$, to the extent that $f$ exists. So they at least provide a scaffolding for a more complete construction of $f$. | |
| Sep 9, 2019 at 23:47 | comment | added | Michael Engelhardt | Oh - looks like we talked past each other then ... this is what I was trying to clarify with my initial question, but it seems we still didn't quite match minds. Still, maybe this helps anyway. I'll update if I think of anything else. | |
| Sep 9, 2019 at 21:17 | comment | added | Ali Taghavi | Thank you very much for your answer. I will forward your answer to him. by global first integral I mean a function $f:\mathbb{R}63 \to \mathbb{R}$ with $f_x sin(y)+f_y sin(z)+f_z sin(x0=0$. may be your answer can help to find this globaly defined first integral. Thanks again for consideration of the question. | |
| Sep 8, 2019 at 14:20 | comment | added | Michael Engelhardt | In particular, for large $x$, $y$, $z$, one is always very close to one of the analytic solutions. In combination with the form of solutions for small $x$, $y$, $z$ in comments above, there isn't that much wiggle room left. | |
| Sep 8, 2019 at 14:11 | comment | added | Michael Engelhardt | The latter symmetries presumably provide useful guardrails if one ultimately wants to resort to a numerical solution - one only has to interpolate between two adjacent analytic solutions bounding the direction of the vector $(x,y,z)$. | |
| Sep 8, 2019 at 4:12 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
added 170 characters in body
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| Sep 8, 2019 at 3:39 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |