Appendix/Ramblings/ForCohlHoward: Difference between revisions
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==Understanding the Dimensionality of EFE Index Symbols== | ==Understanding the Dimensionality of EFE Index Symbols== | ||
Howard put together a Mathematica script intended to provide — for any specification of the semi-axis length triplet <math>(a, b, c)</math> — very high-precision, numerical evaluations of any of the index symbols, <math>A_{ijk\ldots}</math> and <math>B_{ijk\ldots}</math> as defined by Eqs. (103 - 104) in §21 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]. Originally I suggested that, without loss of generality, he should only need to specify the ''pair'' of length ratios, <math>(1, b/a, c/a)</math>. In response, Howard pointed out that evaluation of | Howard put together a Mathematica script intended to provide — for any specification of the semi-axis length triplet <math>(a, b, c)</math> — very high-precision, numerical evaluations of any of the index symbols, <math>A_{ijk\ldots}</math> and <math>B_{ijk\ldots}</math> as defined by Eqs. (103 - 104) in §21 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]. Originally I suggested that, without loss of generality, he should only need to specify the ''pair'' of length ratios, <math>(1, b/a, c/a)</math>. In response, Howard pointed out that evaluation of all but a few of the lowest-numbered index symbols — as defined by [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] — ''does'' explicitly depend on specification of (various powers of) the semi-axis length, <math>a</math>. | ||
<font color="red">Joel's response:</font> Howard is correct! He should leave the explicit dependence of <math>a</math> — to various powers — in his Mathematica notebook's determination of all the EFE index symbols. | <font color="red">Joel's response:</font> Howard is correct! He should leave the explicit dependence of <math>a</math> — to various powers — in his Mathematica notebook's determination of all the EFE index symbols. | ||
Instead, what we should expect is that the evaluation of various ''physically relevant'' parameters will produce results that are independent of the semi-axis length, <math>a</math>; these evaluations should involve combining various index symbols in such a way that the dependence on <math>a</math> disappears. Consider, for example, our [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Virial_Equilibrium|accompanying discussion]] of the virial-equilibrium-based determination of the frequency ratio, <math>f \equiv \zeta/\Omega_f</math>, in equilibrium S-Type Riemann Ellipsoids. Although most of the required index symbols, <math>A_1, A_2, A_3</math> and <math>B_{12}</math>, are dimensionless parameters, the index symbol <math>A_{12}</math> has the unit of inverse-length-squared. Notice, however, that when <math>A_{12}</math> appears along with any of these other ''dimensionless'' parameters in the definition of | Instead, what we should expect is that the evaluation of various ''physically relevant'' parameters will produce results that are independent of the semi-axis length, <math>a</math>; these evaluations should involve combining various index symbols in such a way that the dependence on <math>a</math> disappears. Consider, for example, our [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Virial_Equilibrium|accompanying discussion]] (click to see relevant expressions) of the virial-equilibrium-based determination of the frequency ratio, <math>f \equiv \zeta/\Omega_f</math>, in equilibrium S-Type Riemann Ellipsoids. Although most of the required index symbols, <math>A_1, A_2, A_3</math> and <math>B_{12}</math>, are dimensionless parameters, the index symbol <math>A_{12}</math> has the unit of inverse-length-squared. Notice, however, that when <math>A_{12}</math> appears along with any of these other ''dimensionless'' parameters in the definition of <math>f</math>, it is accompanied by an extra "length-squared" factor, such as <math>a^2</math>. Hence, although I strongly agree that Howard should continue to include various powers of <math>a</math> (etc.) in his Mathematica notebook expressions, I suspect that, without loss of generality, in the end we will always be able to set <math>a=1</math> and only need to specify the ''pair'' of length ratios, <math>(1, b/a, c/a)</math>. | ||
==Evaluation of Index Symbols== | ==Evaluation of Index Symbols== | ||
Revision as of 14:32, 11 January 2022
Discussions With Howard Cohl
These discussions began in late 2021, when Howard asked if I would be interested in working with him on establishing a better understanding of the stability of Riemann S-Type Ellipsoids.
Understanding the Dimensionality of EFE Index Symbols
Howard put together a Mathematica script intended to provide — for any specification of the semi-axis length triplet — very high-precision, numerical evaluations of any of the index symbols, and as defined by Eqs. (103 - 104) in §21 of [EFE]. Originally I suggested that, without loss of generality, he should only need to specify the pair of length ratios, . In response, Howard pointed out that evaluation of all but a few of the lowest-numbered index symbols — as defined by [EFE] — does explicitly depend on specification of (various powers of) the semi-axis length, .
Joel's response: Howard is correct! He should leave the explicit dependence of — to various powers — in his Mathematica notebook's determination of all the EFE index symbols.
Instead, what we should expect is that the evaluation of various physically relevant parameters will produce results that are independent of the semi-axis length, ; these evaluations should involve combining various index symbols in such a way that the dependence on disappears. Consider, for example, our accompanying discussion (click to see relevant expressions) of the virial-equilibrium-based determination of the frequency ratio, , in equilibrium S-Type Riemann Ellipsoids. Although most of the required index symbols, and , are dimensionless parameters, the index symbol has the unit of inverse-length-squared. Notice, however, that when appears along with any of these other dimensionless parameters in the definition of , it is accompanied by an extra "length-squared" factor, such as . Hence, although I strongly agree that Howard should continue to include various powers of (etc.) in his Mathematica notebook expressions, I suspect that, without loss of generality, in the end we will always be able to set and only need to specify the pair of length ratios, .
Evaluation of Index Symbols
Howard's Mathematica notebook performs brute-force integrations of various index-symbol definitions. Why doesn't he lean, instead, on the definitions of in terms of incomplete elliptic integrals of the first and second kind, then rely on recurrence relations to evaluate index symbols with a larger number of indexes?
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