"케플러의 법칙, 행성운동과 타원"의 두 판 사이의 차이

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==리뷰, 에세이, 강의노트==
 
==리뷰, 에세이, 강의노트==
 
* Hsiang, Wu-Yi, and Eldar Straume. “Revisiting the Mathematical Synthesis of the Laws of Kepler and Galileo Leading to Newton’s Law of Universal Gravitation.” arXiv:1408.6758 [math], August 28, 2014. http://arxiv.org/abs/1408.6758.
 
* Hsiang, Wu-Yi, and Eldar Straume. “Revisiting the Mathematical Synthesis of the Laws of Kepler and Galileo Leading to Newton’s Law of Universal Gravitation.” arXiv:1408.6758 [math], August 28, 2014. http://arxiv.org/abs/1408.6758.
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* Colwell, Peter. 1992. Bessel Functions and Kepler's Equation. The American Mathematical Monthly 99, no. 1 (January 1): 45-48. doi:[http://dx.doi.org/10.2307/2324547 10.2307/2324547].
 
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* Wilson, Curtis. 1994. Newton's Orbit Problem: A Historian's Response. The College Mathematics Journal 25, no. 3 (May 1): 193-200. doi:[http://dx.doi.org/10.2307/2687647 10.2307/2687647].
 
 
* Haandel, Maris, and Gert Heckman. 2009. Teaching the Kepler Laws for Freshmen. The Mathematical Intelligencer 31, no. 2 (3): 40-44. doi:[http://dx.doi.org/10.1007/s00283-008-9022-x 10.1007/s00283-008-9022-x].  
 
* Haandel, Maris, and Gert Heckman. 2009. Teaching the Kepler Laws for Freshmen. The Mathematical Intelligencer 31, no. 2 (3): 40-44. doi:[http://dx.doi.org/10.1007/s00283-008-9022-x 10.1007/s00283-008-9022-x].  
 
* Osler, Thomas J. “An Unusual Approach to Kepler’s First Law.” American Journal of Physics 69, no. 10 (October 1, 2001): 1036–38. doi:10.1119/1.1379735. http://www.rowan.edu/colleges/las/departments/math/facultystaff/osler/ELLIPSE2.pdf
 
* Osler, Thomas J. “An Unusual Approach to Kepler’s First Law.” American Journal of Physics 69, no. 10 (October 1, 2001): 1036–38. doi:10.1119/1.1379735. http://www.rowan.edu/colleges/las/departments/math/facultystaff/osler/ELLIPSE2.pdf
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* [http://www.jstor.org/stable/2691148 Central Force Laws, Hodographs, and Polar Reciprocals,] Don Chakerian, <cite>Mathematics Magazine</cite>, Vol. 74, No. 1 (Feb., 2001), pp. 3-18
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* Chakerian, Don. ‘Central Force Laws, Hodographs, and Polar Reciprocals’. Mathematics Magazine 74, no. 1 (1 February 2001): 3–18. doi:[http://www.jstor.org/stable/2691148 10.2307/2691148].
−
* [http://www.jstor.org/stable/2687254 Computation of Planetary Orbits,] Donald A. Teets and Karen Whitehead, <cite>The College Mathematics Journal</cite>, Vol. 29, No. 5 (Nov., 1998), pp. 397-404
+
* Wilson, Curtis. 1994. Newton's Orbit Problem: A Historian's Response. The College Mathematics Journal 25, no. 3 (May 1): 193-200. doi:[http://dx.doi.org/10.2307/2687647 10.2307/2687647].
−
* [http://www.jstor.org/stable/3616881 How Kepler Discovered the Elliptical Orbit,] Eric J. Aiton, <cite>The Mathematical Gazette</cite>, Vol. 59, No. 410 (Dec., 1975), pp. 250-260
+
* Colwell, Peter. 1992. Bessel Functions and Kepler's Equation. The American Mathematical Monthly 99, no. 1 (January 1): 45-48. doi:[http://dx.doi.org/10.2307/2324547 10.2307/2324547].
−
 
+
* Teets, Donald A., and Karen Whitehead. ‘Computation of Planetary Orbits’. The College Mathematics Journal 29, no. 5 (1 November 1998): 397–404. doi:[ http://www.jstor.org/stable/2687254 10.2307/2687254].
−
+
* Aiton, Eric J. ‘How Kepler Discovered the Elliptical Orbit’. The Mathematical Gazette 59, no. 410 (1 December 1975): 250–60. doi:http://www.jstor.org/stable/3616881 10.2307/3616881].
  
 
==관련도서==
 
==관련도서==

2015년 1월 2일 (금) 22:28 판

케플러의 법칙

  • 행성은 태양을 하나의 초점으로 하는 타원 궤도를 돌고 있다
  • 태양과 행성을 연결하는 직선은 같은 시간에 같은 면적을 쓸고 지나간다
  • 행성운동의 공전주기의 제곱은 타원 궤도의 장축의 길이의 세제곱에 비례한다


제1법칙

  • 장축의 길이가 $2a$, 단축의 길이가 $2b$인 타원의 이심률 $e$는 다음과 같이 정의된다

\[e=\frac{\sqrt{a^2-b^2}}{a}\]

  • 태양을 원점에 두었을 때, 행성의 극좌표 $(r,\theta)$는 다음을 만족한다

\[r=\frac{a(1-e^2)}{1+e \cos(\theta)}\]


제2법칙

  • 등면적 법칙

케플러의 법칙, 행성운동과 타원1.gif



케플러 방정식


뉴턴 법칙으로부터의 유도

  • \(a_r=\ddot{r} - r\dot{\theta}^2=k/r^2\)
  • \(a_\theta=r\ddot{\theta} + 2\dot{r} \dot{\theta}=0\)
  • 두번째 식으로부터 $r^2 \dot{\theta}$가 상수임을 알 수 있다. 이로부터 케플러의 제2법칙을 얻는다


메모

  • Newton on Abelian functions


관련된 항목들


매스매티카 파일 및 계산 리소스


사전형태의 자료


리뷰, 에세이, 강의노트

  • Hsiang, Wu-Yi, and Eldar Straume. “Revisiting the Mathematical Synthesis of the Laws of Kepler and Galileo Leading to Newton’s Law of Universal Gravitation.” arXiv:1408.6758 [math], August 28, 2014. http://arxiv.org/abs/1408.6758.
  • Haandel, Maris, and Gert Heckman. 2009. Teaching the Kepler Laws for Freshmen. The Mathematical Intelligencer 31, no. 2 (3): 40-44. doi:10.1007/s00283-008-9022-x.
  • Osler, Thomas J. “An Unusual Approach to Kepler’s First Law.” American Journal of Physics 69, no. 10 (October 1, 2001): 1036–38. doi:10.1119/1.1379735. http://www.rowan.edu/colleges/las/departments/math/facultystaff/osler/ELLIPSE2.pdf
  • Chakerian, Don. ‘Central Force Laws, Hodographs, and Polar Reciprocals’. Mathematics Magazine 74, no. 1 (1 February 2001): 3–18. doi:10.2307/2691148.
  • Wilson, Curtis. 1994. Newton's Orbit Problem: A Historian's Response. The College Mathematics Journal 25, no. 3 (May 1): 193-200. doi:10.2307/2687647.
  • Colwell, Peter. 1992. Bessel Functions and Kepler's Equation. The American Mathematical Monthly 99, no. 1 (January 1): 45-48. doi:10.2307/2324547.
  • Teets, Donald A., and Karen Whitehead. ‘Computation of Planetary Orbits’. The College Mathematics Journal 29, no. 5 (1 November 1998): 397–404. doi:[ http://www.jstor.org/stable/2687254 10.2307/2687254].
  • Aiton, Eric J. ‘How Kepler Discovered the Elliptical Orbit’. The Mathematical Gazette 59, no. 410 (1 December 1975): 250–60. doi:http://www.jstor.org/stable/3616881 10.2307/3616881].

관련도서

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