Timeline for General systems of linear differential equations with variable coefficients
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
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| Feb 21, 2014 at 13:58 | comment | added | Robert Bryant | Well, that depends on whether you are willing to count each application of $IP_{\frak{g}}$ as some 'finite number of steps' and whether you regard this as 'finding the solution analytically in closed form'. Even for the first simple Lie algebra ${\frak{sl}}(2,\mathbb{R})$, which is the theory of the Riccati equation, many people (including some of the other commenters) do not regard $IP_{\frak{sl}(2,\mathbb{R})}$ as giving a 'closed-form' solution to the Riccati equation, since the resulting functions aren't 'elementary'. The point is that it's a fundamentally new operation beyond quadrature. | |
| Feb 21, 2014 at 13:37 | comment | added | PLG | Alright so, if I understand you well, allowing for integration, it is possible to find $M(t)$ from $A(t)$ analytically and in a \emph{finite} number of steps (say 1 integral = 1 step), for all $A(t)$ (finite and smooth)? I am asking because it seems to me that what you said gives rise to the Magnus series for $M(t)$, which is still not (in general) computable exactly in a finite number of steps. In any case thanks for your enlightning answer! I am going straight away to read more on this. | |
| Feb 21, 2014 at 12:32 | history | answered | Robert Bryant | CC BY-SA 3.0 |