"측지선"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) (→관련논문) |
Pythagoras0 (토론 | 기여) (→관련논문) |
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| 72번째 줄: | 72번째 줄: | ||
==관련논문== | ==관련논문== | ||
| + | * Radeschi, Marco, and Burkhard Wilking. “On the Berger Conjecture for Manifolds All of Whose Geodesics Are Closed.” arXiv:1511.07852 [math], November 24, 2015. http://arxiv.org/abs/1511.07852. | ||
* Erlandsson, Viveka, and Juan Souto. “Counting Curves in Hyperbolic Surfaces.” arXiv:1508.02265 [math], August 10, 2015. http://arxiv.org/abs/1508.02265. | * Erlandsson, Viveka, and Juan Souto. “Counting Curves in Hyperbolic Surfaces.” arXiv:1508.02265 [math], August 10, 2015. http://arxiv.org/abs/1508.02265. | ||
* Kennard, Lee, and Jordan Rainone. “Characterizations of the Round Two-Dimensional Sphere in Terms of Closed Geodesics.” arXiv:1507.00414 [math], July 1, 2015. http://arxiv.org/abs/1507.00414. | * Kennard, Lee, and Jordan Rainone. “Characterizations of the Round Two-Dimensional Sphere in Terms of Closed Geodesics.” arXiv:1507.00414 [math], July 1, 2015. http://arxiv.org/abs/1507.00414. | ||
2015년 11월 24일 (화) 22:24 판
개요
- n차원 다양체 M의 coordinate chart 에서 \(\alpha(t)=(\alpha_1(t),\alpha_2(t),\cdots, \alpha_n(t))\) 로 표현되는 곡선이 측지선이 될 조건은 크리스토펠 기호를 사용하여 다음 미분방정식으로 쓸 수 있다
\[\frac{d^2\alpha_k }{dt^2} + \sum_{i,j}\Gamma^{k}_{~i j }\frac{d\alpha_i }{dt}\frac{d\alpha_j }{dt} = 0,\quad k=1,2,\cdots, n\]
또는\[\ddot{\alpha_k } + \sum_{i,j}\Gamma^{k}_{~i j }\dot{\alpha_i}\dot{\alpha_j }= 0,\quad k=1,2,\cdots, n\]
곡면의 측지선
- 곡선 (\((x(t),y(t))\) 가 다음의 미분방정식을 만족해야 한다\[x''(t)+\Gamma _{1,1}{}^1 x'(t)^2+\Gamma _{1,2}{}^1 x'(t) y'(t)+\Gamma _{2,1}{}^1 x'(t) y'(t)+\Gamma _{2,2}{}^1 y'(t)^2=0\]\[y''(t)+\Gamma _{1,1}{}^2 x'(t)^2+\Gamma _{1,2}{}^2 x'(t) y'(t)+\Gamma _{2,1}{}^2 x'(t) y'(t)+\Gamma _{2,2}{}^2 y'(t)^2=0\]
예
역사
메모
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/측지선
- http://en.wikipedia.org/wiki/Geodesics
- http://mathworld.wolfram.com/Geodesic.html
- http://www.wolframalpha.com/input/?i=geodesic
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관련논문
- Radeschi, Marco, and Burkhard Wilking. “On the Berger Conjecture for Manifolds All of Whose Geodesics Are Closed.” arXiv:1511.07852 [math], November 24, 2015. http://arxiv.org/abs/1511.07852.
- Erlandsson, Viveka, and Juan Souto. “Counting Curves in Hyperbolic Surfaces.” arXiv:1508.02265 [math], August 10, 2015. http://arxiv.org/abs/1508.02265.
- Kennard, Lee, and Jordan Rainone. “Characterizations of the Round Two-Dimensional Sphere in Terms of Closed Geodesics.” arXiv:1507.00414 [math], July 1, 2015. http://arxiv.org/abs/1507.00414.
- Sapir, Jenya. ‘Lower Bound on the Number of Non-Simple Closed Geodesics on Surfaces’. arXiv:1505.06805 [math], 26 May 2015. http://arxiv.org/abs/1505.06805.