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two-body_problem [2013/11/09 20:36] nikolaj |
two-body_problem [2014/03/21 11:11] 127.0.0.1 external edit |
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| ===== Two-body problem ===== | ===== Two-body problem ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#FFBB00: $\langle \mathbb R^{2\times 3}, H\rangle \in \mathrm{it} $ ... Classical Hamiltonian system | | + | | @#FFBB00: definiendum | @#FFBB00: $\langle \mathbb R^{2\times 3}, H\rangle \in \mathrm{it} $ ... Classical Hamiltonian system | |
| - | | @#55EE55: $ H({\bf r}_1,{\bf r}_2,{\bf p}_1,{\bf p}_2) = \frac{1}{m_1}\frac{1}{2}{\bf p}_1^2 + \frac{1}{m_2}\frac{1}{2}{\bf p}_2^2 + V(|{\bf r}_1-{\bf r}_2|) $ | | + | | @#55EE55: postulate | @#55EE55: $ H({\bf r}_1,{\bf r}_2,{\bf p}_1,{\bf p}_2) = \frac{1}{m_1}\frac{1}{2}{\bf p}_1^2 + \frac{1}{m_2}\frac{1}{2}{\bf p}_2^2 + V(|{\bf r}_1-{\bf r}_2|) $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| == Equations of motion in suitable coordinates == | == Equations of motion in suitable coordinates == | ||
| - | | @#DDDDDD: $ M \equiv m_1+m_2 $ | | + | | @#DDDDDD: range | @#DDDDDD: $ M \equiv m_1+m_2 $ | |
| - | | @#DDDDDD: $ \mu \equiv m_1 m_2/M $ | | + | | @#DDDDDD: range | @#DDDDDD: $ \mu \equiv m_1 m_2/M $ | |
| The following choice of coordinates eliminates the singles out the center of mass, which the Hamiltonian is independent of. | The following choice of coordinates eliminates the singles out the center of mass, which the Hamiltonian is independent of. | ||
| - | | @#DDDDDD: $ {\bf r} \equiv {\bf r}_2 - {\bf r}_1 $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf r} \equiv {\bf r}_2 - {\bf r}_1 $ | |
| - | | @#DDDDDD: $ {\bf R} \equiv (m_1\ {\bf r}_1 + m_2\ {\bf r}_2)/M $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf R} \equiv (m_1\ {\bf r}_1 + m_2\ {\bf r}_2)/M $ | |
| - | | @#DDDDDD: $ {\bf p} \equiv (m_1\ {\bf p}_2 - m_2\ {\bf p}_1)/M $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf p} \equiv (m_1\ {\bf p}_2 - m_2\ {\bf p}_1)/M $ | |
| - | | @#DDDDDD: $ {\bf P} \equiv {\bf p}_1 + {\bf p}_2 $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf P} \equiv {\bf p}_1 + {\bf p}_2 $ | |
| - | | @#DDDDDD: $ r \equiv |{\bf r}| $ | | + | | @#DDDDDD: range | @#DDDDDD: $ r \equiv |{\bf r}| $ | |
| ^ $ H({\bf r},{\bf R},{\bf p},{\bf R}) = \frac{1}{M}\frac{1}{2}{\bf P}^2 + \frac{1}{\mu}\frac{1}{2}{\bf p}^2 + V(r) $ ^ | ^ $ H({\bf r},{\bf R},{\bf p},{\bf R}) = \frac{1}{M}\frac{1}{2}{\bf P}^2 + \frac{1}{\mu}\frac{1}{2}{\bf p}^2 + V(r) $ ^ | ||
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| == Scattering process == | == Scattering process == | ||
| - | | @#DDDDDD: $ {\bf g} \equiv \frac{\partial}{\partial t}{\bf r} $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf g} \equiv \frac{\partial}{\partial t}{\bf r} $ | |
| - | | @#DDDDDD: $ {\bf g}_{-\infty} \equiv \lim_{t\to-\infty}{\bf g} $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf g}_{-\infty} \equiv \lim_{t\to-\infty}{\bf g} $ | |
| - | | @#DDDDDD: $ {\bf g}_{+\infty} \equiv \lim_{t\to+\infty}{\bf g} $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf g}_{+\infty} \equiv \lim_{t\to+\infty}{\bf g} $ | |
| Geometric considerations lead to the conclusion that there is a unit vector ${\bf \alpha}^V$, depending on the particle interaction potential $V$, with | Geometric considerations lead to the conclusion that there is a unit vector ${\bf \alpha}^V$, depending on the particle interaction potential $V$, with | ||
| - | | @#DDDDDD: $ S_{ij} \equiv \delta_{ij}-2\alpha_i^V\alpha_j^V $ | | + | | @#DDDDDD: range | @#DDDDDD: $ S_{ij} \equiv \delta_{ij}-2\alpha_i^V\alpha_j^V $ | |
| ^ $ {\bf g}_{+\infty} = S\ {\bf g}_{-\infty} $ ^ | ^ $ {\bf g}_{+\infty} = S\ {\bf g}_{-\infty} $ ^ | ||
