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In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.
在物理学和经典力学领域中，三体问题是根据牛顿运动定律和万有引力定律计算三点质量的初始位置和速度（或动量）并求出它们随后的运动的问题。三体是n体问题中的一个特例。与两体不同的是，三体问题不存在通用的封闭形式的解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以需要数值方法。
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.
历史上看，第一个被研究的特定三体问题是月球、地球和太阳构成的“三体”。扩展后三体问题的现代意义是经典力学或量子力学中模拟三个粒子运动的任何问题。
==数学描述==
三体的数学表达式可以用三个质量为<math>m_i</math>相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：
<math>\begin{align}
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.
\end{align}</math>
其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，其中
<math>\mathbf{r_i}</math> <math>\mathbf{p_i}</math>
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In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.
在物理学和经典力学领域中，三体问题是根据牛顿运动定律和万有引力定律计算三点质量的初始位置和速度（或动量）并求出它们随后的运动的问题。三体是n体问题中的一个特例。与两体不同的是，三体问题不存在通用的封闭形式的解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以需要数值方法。
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.
历史上看，第一个被研究的特定三体问题是月球、地球和太阳构成的“三体”。扩展后三体问题的现代意义是经典力学或量子力学中模拟三个粒子运动的任何问题。
==数学描述==
三体的数学表达式可以用三个质量为<math>m_i</math>相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：
<math>\begin{align}
\ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\
\ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\
\ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.
\end{align}</math>
其中<math>G</math>为万有引力常数。这是一组9个二阶微分方程。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，其中
<math>\mathbf{r_i}</math> <math>\mathbf{p_i}</math>