The momentum of a closed mechanical system has different values in different (inertial) frames of reference. If a frame \(K'\) moves with velocity \(\boldsymbol{V}\) relative to another frame \(K\), then the velocities \(\boldsymbol{v}_a'\) and \(\boldsymbol{v}_a\) of the particles relative to the two frames are such that \(\boldsymbol{v}_a = \boldsymbol{v}_a' + \boldsymbol{V}\). The momenta \(P\) and \(P'\) in the two frames are therefore related by
\[ \boldsymbol{P} =\sum_a m_a\boldsymbol{v}_a =\sum_a m_a\boldsymbol{v}'_a +\boldsymbol{V}\sum_a m_a \]
or
\[\label{equations/8.1}\tag*{(8.1)}\mathbf{P}=\mathbf{P}'+\mathbf{V}\sum_a m_a\]
In particular, there is always a frame of reference \(K'\) in which the total momentum is zero. Putting \(P' = 0\) in (8.1) \(\mathbf{P}=\mathbf{P}'+\mathbf{V}\sum_a m_a\) , we find the velocity of this frame:
\[\label{equations/8.2}\tag*{(8.2)}\mathbf{V}=\mathbf{P}/\sum_a m_a=\sum_a m_a\mathbf{v}_a/\sum_a m_a\]
If the total momentum of a mechanical system in a given frame of reference is zero, it is said to be at rest relative to that frame. This is a natural generalisation of the term as applied to a particle. Similarly, the velocity V given by (8.2) \(\mathbf{V}=\mathbf{P}/\sum_a m_a=\sum_a m_a\mathbf{v}_a/\sum_a m_a\) is the velocity of the “motion as a whole” of a mechanical system whose momentum is not zero. Thus we see that the law of conservation of momentum makes possible a natural definition of rest and velocity, as applied to a mechanical system as a whole.
Formula (8.2) \(\mathbf{V}=\mathbf{P}/\sum_a m_a=\sum_a m_a\mathbf{v}_a/\sum_a m_a\) shows that the relation between the momentum \(\boldsymbol{P}\) and the velocity \(V\) of the system is the same as that between the momentum and velocity of a single particle of mass \(\mu = \sum m_a\), the sum of the masses of the particles in the system. This result can be regarded as expressing the additivity of mass.
The right-hand side of formula (8.2) \(\mathbf{V}=\mathbf{P}/\sum_a m_a=\sum_a m_a\mathbf{v}_a/\sum_a m_a\) can be written as the total time derivative of the expression
\[\label{equations/8.3}\tag*{(8.3)}\mathbf{R}\equiv\sum_a m_a\mathbf{r}_a/\sum_a m_a\]
We can say that the velocity of the system as a whole is the rate of motion in space of the point whose radius vector is (8.3) \(\mathbf{R}\equiv\sum_a m_a\mathbf{r}_a/\sum_a m_a\) . This point is called the centre of mass of the system.
The law of conservation of momentum for a closed system can be formulated as stating that the centre of mass of the system moves uniformly in a straight line. In this form it generalises the law of inertia derived in §3. Galileo's relativity principle for a single free particle, whose “centre of mass” coincides with the particle itself.
In considering the mechanical properties of a closed system it is natural to use a frame of reference in which the centre of mass is at rest. This eliminates a uniform rectilinear motion of the system as a whole, but such motion is of no interest.
The energy of a mechanical system which is at rest as a whole is usually called its internal energy \(E\). This includes the kinetic energy of the relative motion of the particles in the system and the potential energy of their interaction. The total energy of a system moving as a whole with velocity \(V\) can be written
\[\label{equations/8.4}\tag*{(8.4)}E=\frac{1}{2}\mu V^2+E_i\]
Although this formula is fairly obvious, we may give a direct proof of it. The energies \(E\) and \(E'\) of a mechanical system in two frames of reference \(K\) and \(K'\) are related by
\[\begin{align} E &= \frac{1}{2}\sum_a m_a v_a^2 + U \\ &= \frac{1}{2}\sum_a m_a (\boldsymbol{v}'_a+\boldsymbol{V})^2 + U \\ &= \frac{1}{2}\mu V^2 + \boldsymbol{V}\sum_a m_a\boldsymbol{v}'_a + \frac{1}{2} \sum_a m_a {v'}_a^2 + U \end{align}\]
\[\label{equations/8.5}\tag*{(8.5)}E=E'+\mathbf{V}\cdot\mathbf{P}'+\frac{1}{2}\mu V^2\]
This formula gives the law of transformation of energy from one frame to another, corresponding to formula (8.1) \(\mathbf{P}=\mathbf{P}'+\mathbf{V}\sum_a m_a\) for momentum. If the centre of mass is at rest in \(K'\), then \(\boldsymbol{P}' = 0\), \(E' = E_i\), and we have (8.4) \(E=\frac{1}{2}\mu V^2+E_i\) .