Conservation of momentum

References: Anthony Zee, Einstein Gravity in a Nutshell, (Princeton University Press, 2013) – Chapter I.2, problem 1.

The nature of the dependence of a force or potential on the underlying position coordinates can determine certain conservation laws. In his chapter I.2, Zee shows that a central force (a force that is always directed towards its source, such as the Earth’s gravity or a point charge’s electrostatic field) conserves angular momentum. Actually his derivation is a generalization to any number {D\ge2} dimensions of the more familiar proof in 3-d, which goes like this:

The angular momentum of a mass {m} is defined as

\displaystyle  \mathbf{L}=\mathbf{r}\times\mathbf{p} \ \ \ \ \ (1)

where {\mathbf{p}=m\mathbf{v}=m\dot{\mathbf{r}}} is the linear momentum. Taking the time derivative we get

\displaystyle   \dot{\mathbf{L}} \displaystyle  = \displaystyle  \dot{\mathbf{r}}\times\mathbf{p}+\mathbf{r}\times\dot{\mathbf{p}}\ \ \ \ \ (2)
\displaystyle  \displaystyle  = \displaystyle  m\dot{\mathbf{r}}\times\dot{\mathbf{r}}+\mathbf{r}\times\mathbf{F}\ \ \ \ \ (3)
\displaystyle  \displaystyle  = \displaystyle  0+0\ \ \ \ \ (4)
\displaystyle  \displaystyle  = \displaystyle  0 \ \ \ \ \ (5)

where the third line uses the fact that {\mathbf{r}\parallel\mathbf{F}} for a central force, so their cross product is zero. Thus {\mathbf{L}} doesn’t change with time.

Now suppose that a force {F} is the negative gradient of a potential function {V} so that by Newton’s law:

\displaystyle  m_{a}\frac{d^{2}x_{a}^{i}}{dt^{2}}=-\frac{\partial V\left(x\right)}{\partial x_{a}^{i}} \ \ \ \ \ (6)

where the index {a} refers to particle {a} in a collection of {N} interacting particles, and {i} is the component of the coordinate {x}. Note that the {x} in {V\left(x\right)} represents all {D} components of {x} (if we’re doing the calculation in {D}-dimensional space) and not just the magnitude of the distance.

Now suppose that {V} is a function only of the coordinate differences {x_{a}^{i}-x_{b}^{i}} between particles {a} and {b}, where {a,b=1,\ldots,N} with {a\ne b}. In this case, the total linear momentum is given by

\displaystyle  p^{i}=\sum_{a}m_{a}\frac{dx_{a}^{i}}{dt} \ \ \ \ \ (7)

The time derivative is

\displaystyle   \dot{p}^{i} \displaystyle  = \displaystyle  \sum_{a}m_{a}\frac{d^{2}x_{a}^{i}}{dt^{2}}\ \ \ \ \ (8)
\displaystyle  \displaystyle  = \displaystyle  -\sum_{a}\frac{\partial V\left(x\right)}{\partial x_{a}^{i}} \ \ \ \ \ (9)

Since {V} is a function of the set of all possible differences {x_{a}^{i}-x_{b}^{i}}, the terms in the sum 9 cancel in pairs. For example, for 3 particles if {V=f\left(\left(x_{a}^{i}-x_{b}^{i}\right),\left(x_{a}^{i}-x_{c}^{i}\right),\left(x_{b}^{i}-x_{c}^{i}\right)\right)}, then {\frac{\partial V}{\partial x_{a}^{i}}} will contain a term equivalent to {\frac{\partial V}{\partial\left(x_{a}^{i}-x_{b}^{i}\right)}} (plus another term resulting from {\frac{\partial V}{\partial\left(x_{a}^{i}-x_{c}^{i}\right)}}). However, {\frac{\partial V}{\partial x_{b}^{i}}} will contain a term equivalent to {-\frac{\partial V}{\partial\left(x_{a}^{i}-x_{b}^{i}\right)}} which cancels the first term. That is

\displaystyle   \frac{\partial V}{\partial x_{a}^{i}} \displaystyle  = \displaystyle  \frac{\partial f}{\partial\left(x_{a}^{i}-x_{b}^{i}\right)}+\frac{\partial f}{\partial\left(x_{a}^{i}-x_{c}^{i}\right)}\ \ \ \ \ (10)
\displaystyle  \frac{\partial V}{\partial x_{b}^{i}} \displaystyle  = \displaystyle  -\frac{\partial f}{\partial\left(x_{a}^{i}-x_{b}^{i}\right)}+\frac{\partial f}{\partial\left(x_{b}^{i}-x_{c}^{i}\right)}\ \ \ \ \ (11)
\displaystyle  \frac{\partial V}{\partial x_{c}^{i}} \displaystyle  = \displaystyle  -\frac{\partial f}{\partial\left(x_{a}^{i}-x_{c}^{i}\right)}-\frac{\partial f}{\partial\left(x_{b}^{i}-x_{c}^{i}\right)} \ \ \ \ \ (12)

so adding up all the derivatives causes the terms to cancel in pairs. The argument is fairly easily extended to {N} particles.

Thus {\dot{p}^{i}=0} and linear momentum is conserved. [I realize this isn’t a very elegant or mathematical way of proving it; there is probably a better way of writing it down, but hopefully you get the idea.]

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