# Proving that $\vec{v}\times\sum_i\dfrac{dm_i}{dt}\vec{r}_i=t(\vec{v}\times\sum_i\vec{F}_i)$ when no external torque

There is this idea of relativity in Classical Mechanics:

The laws of mechanics valid in an inertial frame must also be valid in any frame moving uniformly with respect to it.

I was just trying to apply these to the case of the law of conservation of momentum and the law of conservation of angular momentum.

Let there be an inertial frame S and another frame S' moving with velocity $$\mathbf{\vec{v}}$$ w.r.t to S with:

$$\mathbf{\vec{r}}'_i = \mathbf{\vec{r}}_i - \mathbf{\vec{v}}t$$

$$\mathbf{\vec{v}}'_i = \mathbf{\vec{v}}_i - \mathbf{\vec{v}}$$

For momentum conservation: In frame S', putting $$\dfrac{d}{dt} \sum_i \mathbf{\vec{p}}'_i = \mathbf{0}$$ and substituting $$\dfrac{d}{dt} \sum_i \mathbf{\vec{p}}_i = \mathbf{0}$$ of frame S in it:

$$\dfrac{d}{dt} \sum_i \mathbf{\vec{p}}'_i = \dfrac{d}{dt} \sum_i \mathbf{\vec{p}}_i - \dfrac{d}{dt} \sum_i m_i \mathbf{\vec{v}} = \mathbf{0} - \mathbf{\vec{v}} \dfrac{d}{dt} \sum_i m_i$$

If this has to be $$\mathbf{0}$$, then $$\sum_i m_i = 0$$

Now, on to angular momentum. In frame S:

$$\dfrac{d}{dt} \sum_i \mathbf{\vec{L}}_i = \dfrac{d}{dt} \sum_i (\mathbf{\vec{r}}_i \times m_i\mathbf{\vec{v}}_i) = \mathbf{0}$$

Am trying to prove the law in frame S' from the law in S:

$$\dfrac{d}{dt} \sum_i \mathbf{\vec{L}}'_i = \dfrac{d}{dt} \sum_i \mathbf{\vec{L}}_i - \dfrac{d}{dt} \sum_i (\mathbf{\vec{r}}_i \times m_i\mathbf{\vec{v}}) - \dfrac{d}{dt} \sum_i (\mathbf{\vec{v}}t \times m_i \mathbf{\vec{v}}_i)$$

$$= \mathbf{0} - \dfrac{d}{dt} \sum_i (\mathbf{\vec{r}}_i \times m_i\mathbf{\vec{v}}) - \dfrac{d}{dt} \sum_i (\mathbf{\vec{v}}t \times m_i \mathbf{\vec{v}}_i)$$

$$= - \sum_i m_i (\mathbf{\vec{v}}_i \times \mathbf{\vec{v}}) - \sum_i \dfrac{dm_i}{dt} (\mathbf{\vec{r}}_i \times \mathbf{\vec{v}}) + \sum_i m_i (\mathbf{\vec{v}}_i \times \mathbf{\vec{v}}) - \sum_i m_i (\mathbf{\vec{v}}t \times \mathbf{\vec{a}}_i) - \sum_i \dfrac{dm_i}{dt} (\mathbf{\vec{v}}t \times \mathbf{\vec{v}}_i)$$

$$= - \sum_i \dfrac{dm_i}{dt} (\mathbf{\vec{r}}_i \times \mathbf{\vec{v}}) - \sum_i m_i (\mathbf{\vec{v}}t \times \mathbf{\vec{a}}_i) - \sum_i \dfrac{dm_i}{dt} (\mathbf{\vec{v}}t \times \mathbf{\vec{v}}_i)$$

$$= \mathbf{\vec{v}} \times \sum_i \dfrac{dm_i}{dt} \mathbf{\vec{r}}_i - \mathbf{\vec{v}}t \times \sum_i \mathbf{\vec{F}}_i$$

But this is what I wanted to prove to be $$\mathbf{0}$$. I stil have to prove the following:

For a system of particles at $$\mathbf{\vec{r}}_i$$ with mass $$m_i$$, which have forces $$\mathbf{\vec{F}}_i$$ acting on them such that $$\sum_i \mathbf{\vec{r}}_i \times \mathbf{\vec{F}}_i = \mathbf{0}$$, given $$\sum_i \dfrac{dm_i}{dt} = 0$$; how do I prove:

$$\mathbf{\vec{v}} \times \sum_i \dfrac{dm_i}{dt} \mathbf{\vec{r}}_i = \mathbf{\vec{v}}t \times \sum_i \mathbf{\vec{F}}_i$$

for any arbitrary $$\mathbf{\vec{v}}$$ and for all time $$t$$.

• Can't find $r_i\times m_i dv/dt$, have you made simplifications somewhere or is it missing?
– Emil
Mar 12, 2018 at 17:35
• I haven't looked at this in detail, but note that Newton's Laws are applicable only for constant mass systems, suggesting that $\mathrm{d}m_i/\mathrm{d}t = 0$. I suppose care must be taken if you are allowing particles to break up but stay within the system. Mar 12, 2018 at 17:40
• Are rotations not considered here? Are all the particles moving with the same velocity or only the same direction or what? Mar 12, 2018 at 19:40
• Something is way wrong. The units are inconsistent in the expressions and ${\rm d}m_i /{\rm d}t=0$ by definition. Mar 12, 2018 at 20:41
• @Emil $dv/dt = 0$ since $v$ is the relative speed between inertial frames Mar 13, 2018 at 13:40

$$\boldsymbol{r}_{i}' =\boldsymbol{r}_{i}-\boldsymbol{v}\,t \\ \boldsymbol{v}_{i}' =\boldsymbol{v}_{i}-\boldsymbol{v}$$

and with the assumption that $\dot{\boldsymbol{v}}=0$ (uniform motion of frame S') form the linear and angular momentum expressions on the S frame.

$$\boldsymbol{p} =\sum_{i}m_{i}\boldsymbol{v}_{i} \\ \boldsymbol{L} =\sum_{i}\left(\boldsymbol{r}_{i}\times m_{i}\boldsymbol{v}_{i}\right)$$

Now look at linear and angular momentum in the S' frame and relate them to the ones from S.

\require{cancel} \begin{aligned} \boldsymbol{p}'&=\sum_{i}m_{i}\boldsymbol{v}_{i}'=\sum_{i}m_{i}\left(\boldsymbol{v}_{i}-\boldsymbol{v}\right)=\boldsymbol{p}-\left(\sum_{i}m_{i}\right)\boldsymbol{v}=\boldsymbol{p}-m\,\boldsymbol{v}\\\boldsymbol{L}'&=\sum_{i}\left(\boldsymbol{r}_{i}'\times m_{i}\boldsymbol{v}_{i}'\right)=\sum_{i}\left(\boldsymbol{r}_{i}-\boldsymbol{v}\,t\right)\times m_{i}\left(\boldsymbol{v}_{i}-\boldsymbol{v}\right)\\&=\sum_{i}\left(\boldsymbol{r}_{i}\times m_{i}\boldsymbol{v}_{i}\right)-\sum_{i}(\boldsymbol{r}_{i}\times m_{i}\boldsymbol{v})-\boldsymbol{v}\,t\times\left(\sum_{i}m_{i}\boldsymbol{v}_{i}\right)+\cancel{\boldsymbol{v}\,t\times\left(\sum_{i}m_{i}\right)\boldsymbol{v}}\\&=\boldsymbol{L}+\boldsymbol{v}\times\sum_{i}\left(m_{i}\boldsymbol{r}_{i}\right)+\sum_{i}\left(m_{i}\boldsymbol{v}_{i}\right)\times\boldsymbol{v}\,t \end{aligned}

To show that these quantities are conserved, take the derivative (assuming that $\frac{{\rm d}\boldsymbol{p}}{{\rm d}t}=0$ and that $\frac{{\rm d}\boldsymbol{L}}{{\rm d}t}=0$)

\begin{aligned} \frac{{\rm d}}{{\rm d}t}\boldsymbol{p}'&=\cancel{\frac{{\rm d}}{{\rm d}t}\boldsymbol{p}}-m\,\cancel{\frac{{\rm d}}{{\rm d}t}\boldsymbol{v}}=0\\\frac{{\rm d}}{{\rm d}t}\boldsymbol{L}'&=\cancel{\frac{{\rm d}}{{\rm d}t}\boldsymbol{L}}+\frac{{\rm d}}{{\rm d}t}\left[\boldsymbol{v}\times\sum_{i}\left(m_{i}\boldsymbol{r}_{i}\right)\right]+\frac{{\rm d}}{{\rm d}t}\left[\sum_{i}\left(m_{i}\boldsymbol{v}_{i}\right)\times\boldsymbol{v}\,t\right]\\&=\boldsymbol{v}\times\sum_{i}\left(m_{i}\frac{{\rm d}}{{\rm d}t}\boldsymbol{r}_{i}\right)+\cancel{\frac{{\rm d}}{{\rm d}t}\boldsymbol{p}}\times\boldsymbol{v}\,t+\sum_{i}\left(m_{i}\boldsymbol{v}_{i}\right)\times\boldsymbol{v}\\&=\boldsymbol{v}\times\sum_{i}\left(m_{i}\boldsymbol{v}_{i}\right)+\boldsymbol{p}\times\boldsymbol{v}\\&=\boldsymbol{v}\times\boldsymbol{p}+\boldsymbol{p}\times\boldsymbol{v}=0 \end{aligned}

• But why assume rigid body? I tried for a system of particles that may or may not together form a rigid body Mar 13, 2018 at 13:48
• If you find something wrong with what I have done, please point them out; so that I may rectify them Mar 13, 2018 at 14:18
• You need to show your initial work, where you define momentum in each frame. Work out linear momentum first, before tackling angular momentum. Mar 13, 2018 at 18:05
• I edited my post to reflect more of your notation. I hope you can compare your work with my answer to figure out what went wrong. Mar 13, 2018 at 18:50
• But this matches exactly with what I have done if you put $\sum_i \mathbf{\vec{F}}_i = 0$ and $\dfrac{dm_i}{dt} = 0$ and thus shows what I did is correct? Mar 13, 2018 at 23:48