Quartic Integral in Rigid Body-Gyrostat Dynamics

<table class="table-group" id="tab3"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left">No.</td><td class="align_center">Authors</td><td class="align_center">Conditions of the parameters</td><td class="align_center">References</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left">1</td><td class="align_center">Yehia and Elmandouh [<a href="/journals/aa/2020/6651277/#B6" target="_blank">6</a>]</td><td class="align_center"><span style="width: 27.2105ptpx;"><svg height="8.58131pt" id="M148" style="vertical-align:-0.2324905pt" version="1.1" viewbox="-0.0498162 -8.34882 27.2105 8.58131" width="27.2105pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.534,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.797,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg></span></td><td class="align_center">[<a href="/journals/aa/2020/6651277/#B6" target="_blank">6</a>]</td></tr><tr><td class="align_left">2</td><td class="align_center">Goriatchev [<a href="/journals/aa/2020/6651277/#B55" target="_blank">55</a>]</td><td class="align_center"><span style="width: 73.9422ptpx;"><svg height="12.0653pt" id="M149" style="vertical-align:-3.4294pt" version="1.1" viewbox="-0.0498162 -8.6359 73.9422 12.0653" width="73.9422pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.449,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.712,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.246,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,45.509,0)"><path d="M516 277C516 378 460 448 376 448C335 448 288 434 240 401C158 345 127 277 107 159C53 -161 38 -221 23 -247L28 -261C43 -260 95 -250 114 -226C122 -216 129 -199 146 -75C152 -29 157 0 164 31C182 9 206 -12 237 -12C267 -12 325 8 376 46C451 102 516 179 516 277ZM432 259C432 134 366 40 259 40C236 40 204 48 171 76L195 203C219 327 253 400 326 400C398 400 432 336 432 259Z"></path></g><g transform="matrix(.013,0,0,-0.013,56.151,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,67.414,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg></span></td><td class="align_center">[<a href="/journals/aa/2020/6651277/#B55" target="_blank">55</a>]</td></tr><tr><td class="align_left">3</td><td class="align_center">Chaplygin [<a href="/journals/aa/2020/6651277/#B56" target="_blank">56</a>]</td><td class="align_center"><span style="width: 96.0685ptpx;"><svg height="12.7178pt" id="M150" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 96.0685 12.7178" width="96.0685pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.449,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.712,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.246,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,45.509,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 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href="/journals/aa/2020/6651277/#B56" target="_blank">56</a>]</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Comparison the second new integrable case with previous results.</div>

Advances in Astronomy

tab3

Table 3

Table 3: Quartic Integral in Rigid Body-Gyrostat Dynamics 