# 헤논-헤일스 방정식(Hénon-Heiles Equation)

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## 개요

• 자유도가 2인 해밀토니안 계의 대표적인 모델
• 해밀토니안의 파라메터에 따라서, 적분가능한 경우와 카오스 인 경우가 존재

## 해밀토니안

$$H(x,y,\dot{x},\dot{y})=\frac{1}{2} \left(\dot{x}^2+\dot{y}^2+A x^2+B y^2\right)+x y^2+\frac{C x^3}{3}$$

• A=B =1 and C= −1 인 경우는 대표적인 카오스의 예

## 적분가능한 경우

• [Bountis1982] 에서 Painleve analysis에 의해 분석
• 세 가지 적분 가능한 경우 (i) C =1 and A=B (known to be separable in the variables s =x +y, d =x −y). (ii) C =6 and any A and B. (iii) C =16 and B =16A.

## 리뷰, 에세이, 강의노트

• Ford, Joseph. 1992. The Fermi-Pasta-Ulam problem: Paradox turns discovery. Physics Reports 213, no. 5 (May): 271-310. doi:10.1016/0370-1573(92)90116-H.

## 관련논문

• Hiroki Takahasi, Removal of phase transition of the Chebyshev quadratic and thermodynamics of Hénon-like maps near the first bifurcation, http://arxiv.org/abs/1603.00591v1
• Ballesteros, Angel, Alfonso Blasco, and Francisco J. Herranz. ‘A Curved Henon-Heiles System and Its Integrable Perturbations’. arXiv:1503.09187 [math-Ph, Physics:nlin], 31 March 2015. http://arxiv.org/abs/1503.09187.
• Tsiganov, A. V. “On Auto and Hetero Backlund Transformations for the Henon-Heiles Systems.” arXiv:1501.06695 [math-Ph, Physics:nlin], January 27, 2015. http://arxiv.org/abs/1501.06695.
• Ballesteros, Angel, Alfonso Blasco, Francisco J. Herranz, and Fabio Musso. ‘An Integrable Henon-Heiles System on the Sphere and the Hyperbolic Plane’. arXiv:1411.2033 [math-Ph, Physics:nlin], 7 November 2014. http://arxiv.org/abs/1411.2033.
• Grammaticos, B., B. Dorizzi, and R. Padjen. 1982. Painleve property and integrals of motion for the Henon-Heiles system. Physics Letters A 89, no. 3 (May 3): 111-113. doi:10.1016/0375-9601(82)90868-4.
• [Bountis1982]Bountis, Tassos, Harvey Segur, and Franco Vivaldi. 1982. Integrable Hamiltonian systems and the Painlevé property. Physical Review A 25, no. 3 (March 1): 1257. doi:10.1103/PhysRevA.25.1257.
• Branching of solutions and the nonexistence of first integrals in Hamiltonian mechanics
• Hénon, M. & Heiles, C. 1964. The applicability of the third integral of motion: some numerical experiments, The Astronomical Journal, 69(1): 73–99

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'carl'}, {'LOWER': 'e.'}, {'LEMMA': 'Heiles'}]