Angular Momentum Conservation for Additively Separable Hamiltonians

The conservation of angular momentum for a mass moving in an invariant plane under the influence of a spherically symmetric potential can be stated as

$m r^{2} \overset{\cdot}{φ} = L$

where the angle φ measures the position of the radial vector relative to some fixed direction in the plane. This conservation statement can also be expressed in terms of an angle that measures the position of the momentum vector relative to some fixed direction. By working with a general additively separable Hamiltonian, instead of the usual choice of kinetic energy quadratic in momenta, the two statements are seen to be symmetrically identical under interchange of variables.

Begin with an additively separable spherically symmetric n-dimensional Hamiltonian

$H (r, p) = T (p) + V (r)$

that is a function of the squared dynamic variables

$r = \sqrt{r_{k} r_{k}} p = \sqrt{p_{k} p_{k}}$

with repeated indices summed over the arbitrary number of dimensions here and in the sequel. The equations of motion for the individual dynamic variables are

${\overset{\cdot}{r}}_{i} = \frac{\partial H}{\partial p_{i}} = \frac{d T}{d p} \frac{p_{i}}{p} {\overset{\cdot}{p}}_{i} = - \frac{\partial H}{\partial r_{i}} = - \frac{d V}{d r} \frac{r_{i}}{r}$

and the equations of motion for the squared variables are

$\begin{array}{l} \frac{d r}{d t} = \frac{r_{k}}{r} {\overset{\cdot}{r}}_{k} = \frac{(r p)}{r p} \frac{d T}{d p} \\ \frac{d p}{d t} = \frac{p_{k}}{p} {\overset{\cdot}{p}}_{k} = - \frac{(r p)}{r p} \frac{d V}{d r} \end{array} (r p) \equiv r_{k} p_{k}$

For clarity of method, first derive the statements of conservation in two dimensions. The polar angle in the plane is given by an inverse tangent of the ratio of the two spatial variables. Its temporal derivative is

$\begin{array}{l} \overset{\cdot}{φ} = \frac{d}{d t} {tan}^{- 1} (\frac{r_{2}}{r_{1}}) = \frac{1}{1 + (\frac{r_{2}}{r_{1}})^{2}} [\frac{{\overset{\cdot}{r}}_{2}}{r_{1}} - \frac{r_{2} {\overset{\cdot}{r}}_{1}}{r_{1}^{2}}] \\ = \frac{r_{1} {\overset{\cdot}{r}}_{2} - r_{2} {\overset{\cdot}{r}}_{1}}{r_{1}^{2} + r_{2}^{2}} = \frac{1}{p r^{2}} \frac{d T}{d p} (r_{1} p_{2} - r_{2} p_{1}) \end{array}$

so that the statement of conservation in terms of the spatial angle is

$p (\frac{d T}{d p})^{- 1} r^{2} \overset{\cdot}{φ} = L$

For a kinetic energy quadratic in momenta, this becomes the statement at the outset.

Now consider an angle in the plane defined by the inverse tangent of the two components of linear momentum. Its temporal derivative is

$\begin{array}{l} \overset{\cdot}{α} = \frac{d}{d t} {tan}^{- 1} (\frac{p_{2}}{p_{1}}) = \frac{1}{1 + (\frac{p_{2}}{p_{1}})^{2}} [\frac{{\overset{\cdot}{p}}_{2}}{p_{1}} - \frac{p_{2} {\overset{\cdot}{p}}_{1}}{p_{1}^{2}}] \\ = \frac{p_{1} {\overset{\cdot}{p}}_{2} - p_{2} {\overset{\cdot}{p}}_{1}}{p_{1}^{2} + p_{2}^{2}} = \frac{1}{r p^{2}} \frac{d V}{d r} (r_{1} p_{2} - r_{2} p_{1}) \end{array}$

so that the statement of conservation in terms of the angle between the momentum vector and a fixed direction is

$r (\frac{d V}{d r})^{- 1} p^{2} \overset{\cdot}{α} = L$

This statement has precisely the same structure as that with the spatial angle. They can be transformed into each other by interchanging radial and momentum variables along with kinetic and potential energies.

The analog of a kinetic energy quadratic in momenta is the simple harmonic oscillator potential, for which the statement of conservation is structurally identical to that at the outset with coupling constant replacing mass.

Now repeat both derivations in an arbitrary number of dimensions. Define an angle in the invariant plane by a dot product of a constant vector of direction cosines with the radial vector,

$cos φ \equiv \frac{c_{k} r_{k}}{r} = \frac{(c r)}{r} c_{k} c_{k} = c^{2} = 1$

where the shorthand notation for the dot product helps to avoid errors in what follows. The other combination of spatial variables in the invariant plane is

$r sin φ = r \sqrt{1 - \frac{(c r)^{2}}{r^{2}}} = \sqrt{r^{2} - (c r)^{2}}$

The temporal derivative of the defined angle is

$\begin{array}{l} \overset{\cdot}{φ} = \frac{d}{d t} {tan}^{- 1} [\frac{\sqrt{r^{2} - (c r)^{2}}}{(c r)}] \\ = \frac{1}{1 + (\frac{\sqrt{r^{2} - (c r)^{2}}}{(c r)})^{2}} [\frac{r \overset{\cdot}{r} - (c r) (c \overset{\cdot}{r})}{(c r) \sqrt{r^{2} - (c r)^{2}}} - \frac{(c \overset{\cdot}{r}) \sqrt{r^{2} - (c r)^{2}}}{(c r)^{2}}] \\ \overset{\cdot}{φ} = \frac{1}{p r^{2}} \frac{d T}{d p} [(c r) \frac{(r p) - (c r) (c p)}{\sqrt{r^{2} - (c r)^{2}}} - (c p) \sqrt{r^{2} - (c r)^{2}}] \end{array}$

where the additional dot products

$(c \overset{\cdot}{r}) = c_{k} {\overset{\cdot}{r}}_{k} (c p) = c_{k} p_{k}$

have been introduced. The quantity in brackets is composed of products of combinations with the dimensions of the radial and momentum variables. It is the single component of the angular momentum tensor with respect to variables in the invariant plane if pairs of combinations are canonical conjugates. The canonicity of the linear pair is trivial,

$[(c r), (c p)] = [c_{k} r_{k}, c_{l} p_{l}] = c_{k} c_{l} δ_{k l} = 1$

and while the other pair is more complicated it too is canonical:

$\begin{array}{l} [\sqrt{r^{2} - (c r)^{2}}, \frac{(r p) - (c r) (c p)}{\sqrt{r^{2} - (c r)^{2}}}] \\ = \frac{1}{\sqrt{r^{2} - (c r)^{2}}} [\sqrt{r^{2} - (c r)^{2}}, [r_{l} - (c r) c_{l}] p_{l}] \\ = \frac{1}{r^{2} - (c r)^{2}} [r_{k} - (c r) c_{k}] [r_{l} - (c r) c_{l}] δ_{k l} \\ = \frac{1}{r^{2} - (c r)^{2}} [r^{2} - 2 (c r)^{2} + c^{2} (c r)^{2}] = 1 \end{array}$

This means that the quantity in brackets is indeed the magnitude of the angular momentum tensor, resulting in the same statement of conservation in terms of the spatial angle as for two dimensions.

Now define an angle in the invariant plane by a dot product of the same constant vector of direction cosines with the momentum vector:

$cos α \equiv \frac{c_{k} p_{k}}{p} = \frac{(c p)}{p} c_{k} c_{k} = c^{2} = 1$

The other combination of momenta in the invariant plane is

$p sin α = p \sqrt{1 - \frac{(c p)^{2}}{p^{2}}} = \sqrt{p^{2} - (c p)^{2}}$

The temporal derivative of this second defined angle is

$\begin{array}{l} \overset{\cdot}{α} = \frac{d}{d t} {tan}^{- 1} [\frac{\sqrt{p^{2} - (c p)^{2}}}{(c p)}] \\ = \frac{1}{1 + (\frac{\sqrt{p^{2} - (c p)^{2}}}{(c p)})^{2}} [\frac{p \overset{\cdot}{p} - (c p) (c \overset{\cdot}{p})}{(c p) \sqrt{p^{2} - (c p)^{2}}} - \frac{(c \overset{\cdot}{p}) \sqrt{p^{2} - (c p)^{2}}}{(c p)^{2}}] \\ \overset{\cdot}{α} = \frac{1}{r p^{2}} \frac{d V}{d r} [(c r) \sqrt{p^{2} - (c p)^{2}} - (c p) \frac{(r p) - (c r) (c p)}{\sqrt{p^{2} - (c p)^{2}}}] \end{array}$

The quantity in brackets is again composed of products of combinations with the dimensions of the radial and momentum variables. The linear pair is the same as before and so canonical. The more complicated pair is again canonical as well:

$\begin{array}{l} [\frac{(r p) - (c r) (c p)}{\sqrt{p^{2} - (c p)^{2}}}, \sqrt{p^{2} - (c p)^{2}}] \\ = \frac{1}{\sqrt{p^{2} - (c p)^{2}}} [[p_{k} - (c p) c_{k}] r_{k}, \sqrt{p^{2} - (c p)^{2}}] \\ = \frac{1}{p^{2} - (c p)^{2}} [p_{k} - (c p) c_{k}] [p_{l} - (c p) c_{l}] δ_{k l} \\ = \frac{1}{p^{2} - (c p)^{2}} [p^{2} - 2 (c p)^{2} + c^{2} (c p)^{2}] = 1 \end{array}$

The quantity in brackets is again the magnitude of the angular momentum tensor, resulting in the same statement of conservation in terms of the angle between the momentum vector and a fixed direction as for two dimensions.

In summary the two statements

$p (\frac{d T}{d p})^{- 1} r^{2} \overset{\cdot}{φ} = L r (\frac{d V}{d r})^{- 1} p^{2} \overset{\cdot}{α} = L$

both represent the conservation of angular momentum for a mass moving in an invariant plane under the influence of a spherically symmetric potential. Integration of these two statements with inclusion of temporal derivatives above leads to the integrals appearing in a generalized Runge vector.

One can further interpret these statements of conservation in terms Sundman transformations that produce a temporal variable for which angles advance at unit rate as the mass moves along its orbit:

$t^{'} = \frac{d t}{d s} = \frac{p}{L} (\frac{d T}{d p})^{- 1} r^{2} = \frac{r}{L} (\frac{d V}{d r})^{- 1} p^{2}$

The transformation can be written in either coordinate or momentum space, with the two transformations related symmetrically under interchange of radial and momentum variables along with kinetic and potential energies.

Uploaded 2015.03.31 — Updated 2015.08.17 analyticphysics.com