- #1
Lambda96
- 189
- 65
- Homework Statement
- Show that ##L_j^{(1)}+L_j^{(2)}## is preserved
- Relevant Equations
- See screenshot
Hi,
Results from the previous task, which we may use
I am unfortunately stuck with the following task
Hi,
I have first started to rewrite the Hamiltonian and the angular momentum from vector notation to scalar notation:
$$H=\frac{1}{2m}\vec{p_1}^2+\frac{1}{2m}\vec{p_2}^2-\alpha|\vec{q_1}- \vec{q_2}|^2= \sum\limits_{i=1}^{3}\frac{1}{2m}p_{1i}^2+ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{2i}^2-\alpha \sum\limits_{i=1}^{3}(q_{1i}-q_{2i})^2$$
$$L_j^{(1)}+L_j^{(2)}=\sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}+\sum\limits_{k,l}\epsilon_{jkl}q_k^{(2)}p_l^{(2)}$$
Before inserting the above values into the Poisson bracket, I have rewritten the Poisson bracket as follows
$$\{ H,L_j^{(1)}+L_j^{(2)} \}=\{ H,L_j^{(1)}\}+\{H,L_j^{(2)} \}$$
To save paperwork, I will now only calculate ##\{ H,L_j^{(1)}\}##. the calculation for ##L_j^{(2)## would run analogously
$$\{ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{1i}^2+ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{2i}^2-\alpha \sum\limits_{i=1}^{3}(q_{1i}-q_{2i})^2, \sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}$$
$$\{ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{1i}^2,\sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}+ \{\sum\limits_{i=1}^{3}\frac{1}{2m}p_{2i}^2,\sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}+ \{-\alpha \sum\limits_{i=1}^{3}(q_{1i}-q_{2i})^2, \sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}$$
Results from the previous task, which we may use
Hi,
I have first started to rewrite the Hamiltonian and the angular momentum from vector notation to scalar notation:
$$H=\frac{1}{2m}\vec{p_1}^2+\frac{1}{2m}\vec{p_2}^2-\alpha|\vec{q_1}- \vec{q_2}|^2= \sum\limits_{i=1}^{3}\frac{1}{2m}p_{1i}^2+ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{2i}^2-\alpha \sum\limits_{i=1}^{3}(q_{1i}-q_{2i})^2$$
$$L_j^{(1)}+L_j^{(2)}=\sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}+\sum\limits_{k,l}\epsilon_{jkl}q_k^{(2)}p_l^{(2)}$$
Before inserting the above values into the Poisson bracket, I have rewritten the Poisson bracket as follows
$$\{ H,L_j^{(1)}+L_j^{(2)} \}=\{ H,L_j^{(1)}\}+\{H,L_j^{(2)} \}$$
To save paperwork, I will now only calculate ##\{ H,L_j^{(1)}\}##. the calculation for ##L_j^{(2)## would run analogously
$$\{ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{1i}^2+ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{2i}^2-\alpha \sum\limits_{i=1}^{3}(q_{1i}-q_{2i})^2, \sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}$$
$$\{ \sum\limits_{i=1}^{3}\frac{1}{2m}p_{1i}^2,\sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}+ \{\sum\limits_{i=1}^{3}\frac{1}{2m}p_{2i}^2,\sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}+ \{-\alpha \sum\limits_{i=1}^{3}(q_{1i}-q_{2i})^2, \sum\limits_{k,l}\epsilon_{jkl}q_k^{(1)}p_l^{(1)}\}$$
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