- #1

deuteron

- 57

- 13

- Homework Statement
- Find the equation of motion

- Relevant Equations
- ##\mathcal L= T-U##

For the central force ##F=-\nabla U(r_r)## where ##\vec r_r=\vec r_1-\vec r_2##, and ##\vec r_1## and ##\vec r_2## denote the positions of the masses, we get the following kinetic energy using the definition of center of mass ##\vec r_{cm}= \frac{m_1\vec r_1+m_2\vec r_2}{m_1+m_2}##:

$$T= \frac 1 2 (m_1+m_2) \dot{\vec r}_{cm}^2 + \frac 12 \frac{m_1m_2}{m_1+m_2} \dot {\vec r}_r^2$$

We can write the ##\dot {\vec r}## terms in polar coordinates as:

$$\dot{\vec r} = \dot r^2 \vec e_r+ r^2\dot\theta^2\vec e_\theta$$

However, then we get the following equation of mass for the center of mass:

$$\frac{\partial\mathcal L}{\partial r_{cm}}=(m_1+m_2)r_{cm}\dot\theta_{cm}= \frac d {dt}\frac {\partial\mathcal L}{\partial\dot r_{cm}}= M\ddot r_{cm}$$

$$\frac {\partial\mathcal L}{\partial\theta_{cm}}=0=\frac d {dt} \frac {\partial\mathcal L}{\partial \dot \theta_{cm}}= (m_1+m_2) \ddot\theta_{cm}$$

from which we get:

$$\dot\theta_{cm}=\text{const.}$$

$$M\ddot r _{cm} = M r_{cm}\dot\theta_{cm}$$

and if we don't decide ##\vec r_{cm}## to be the origin, which I don't think we *have* to do, then ##\ddot r_{cm}## has a value, which I am not really sure is true. What am I doing wrong above?

$$T= \frac 1 2 (m_1+m_2) \dot{\vec r}_{cm}^2 + \frac 12 \frac{m_1m_2}{m_1+m_2} \dot {\vec r}_r^2$$

We can write the ##\dot {\vec r}## terms in polar coordinates as:

$$\dot{\vec r} = \dot r^2 \vec e_r+ r^2\dot\theta^2\vec e_\theta$$

However, then we get the following equation of mass for the center of mass:

$$\frac{\partial\mathcal L}{\partial r_{cm}}=(m_1+m_2)r_{cm}\dot\theta_{cm}= \frac d {dt}\frac {\partial\mathcal L}{\partial\dot r_{cm}}= M\ddot r_{cm}$$

$$\frac {\partial\mathcal L}{\partial\theta_{cm}}=0=\frac d {dt} \frac {\partial\mathcal L}{\partial \dot \theta_{cm}}= (m_1+m_2) \ddot\theta_{cm}$$

from which we get:

$$\dot\theta_{cm}=\text{const.}$$

$$M\ddot r _{cm} = M r_{cm}\dot\theta_{cm}$$

and if we don't decide ##\vec r_{cm}## to be the origin, which I don't think we *have* to do, then ##\ddot r_{cm}## has a value, which I am not really sure is true. What am I doing wrong above?

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