What symmetry causes the Runge-Lenz vector to be conserved?

Question

Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the conservation of this vector?

Answer 1

1. Hamiltonian Problem. The Kepler problem has Hamiltonian $$\begin{align} H~=~&T+V, \cr T~:=~& \frac{p^2}{2m}, \cr V~:=~& -\frac{k}{q}, \end{align}\tag{1} $$ where $m$ is the 2-body reduced mass. The Laplace–Runge–Lenz vector is (up to an irrelevant normalization) $$\begin{align} A^j ~:=~&a^j + km\frac{q^j}{q}, \cr a^j~:=~&({\bf L} \times {\bf p})^j\cr ~=~&{\bf q}\cdot{\bf p}~p^j- p^2~q^j, \cr \cr {\bf L}~:=~& {\bf q} \times {\bf p}.\end{align} \tag{2}$$

2. Action. The Hamiltonian Lagrangian is $$ L_H~:=~ \dot{\bf q}\cdot{\bf p} - H,\tag{3} $$ and the action is $$ S[{\bf q},{\bf p}]~=~ \int {\rm d}t~L_H .\tag{4}$$ The non-zero fundamental canonical Poisson brackets are $$ \{ q^i , p^j\}~=~ \delta^{ij}. \tag{5}$$

3. Inverse Noether's Theorem. Quite generally in the Hamiltonian formulation, given a constant of motion $Q$, then the infinitesimal variation $$\delta ~=~ -\varepsilon \{Q,\cdot\}\tag{6}$$ is a global off-shell symmetry of the action $S$ (modulo boundary terms). Here $\varepsilon$ is an infinitesimal global parameter, and $X_Q=\{Q,\cdot\}$ is a Hamiltonian vector field with Hamiltonian generator $Q$. The full Noether charge is $Q$, see e.g. my answer to this question. (The words on-shell and off-shell refer to whether the equations of motion are satisfied or not. The minus is conventional.)

4. Variation. Let us check that the three Laplace–Runge–Lenz components $A^j$ are Hamiltonian generators of three continuous global off-shell symmetries of the action $S$. In detail, the infinitesimal variations $\delta= \varepsilon_j \{A^j,\cdot\}$ read $$\begin{align} \delta q^i ~=~& \varepsilon_j \{A^j,q^i\} , \cr \{A^j,q^i\} ~=~& 2 p^i q^j - q^i p^j - {\bf q}\cdot{\bf p}~\delta^{ij}, \cr \delta p^i ~=~& \varepsilon_j \{A^j,p^i\} , \cr \{A^j,p^i\}~=~& p^i p^j - p^2~\delta^{ij} +km\left(\frac{\delta^{ij}}{q}- \frac{q^i q^j}{q^3}\right), \cr \delta t ~=~&0,\end{align} \tag{7}\label{eq:7}$$ where $\varepsilon_j$ are three infinitesimal parameters.

5. Notice for later that $$ {\bf q}\cdot\delta {\bf q}~=~\varepsilon_j({\bf q}\cdot{\bf p}~q^j - q^2~p^j), \tag{8} $$ $$\begin{align} {\bf p}\cdot\delta {\bf p} ~=~&\varepsilon_j km \left(\frac{p^j}{q}-\frac{{\bf q}\cdot{\bf p}~q^j}{q^3}\right)\cr ~=~& -\frac{km}{q^3}{\bf q}\cdot\delta {\bf q},\end{align} \tag{9} $$ $$\begin{align} {\bf q}\cdot\delta {\bf p}~=~&\varepsilon_j({\bf q}\cdot{\bf p}~p^j - p^2~q^j )\cr ~=~&\varepsilon_j a^j, \end{align} \tag{10} $$ $$\begin{align} {\bf p}\cdot\delta {\bf q}~=~&2\varepsilon_j( p^2~q^j - {\bf q}\cdot{\bf p}~p^j)\cr ~=~&-2\varepsilon_j a^j~.\end{align} \tag{11} $$

6. The Hamiltonian is invariant $$ \delta H ~=~ \frac{1}{m}{\bf p}\cdot\delta {\bf p} + \frac{k}{q^3}{\bf q}\cdot\delta {\bf q}~=~0, \tag{12}$$ showing that the Laplace–Runge–Lenz vector $A^j$ is classically a constant of motion $$\frac{\mathrm{d}A^j}{\mathrm{d}t} ~\approx~ \{ A^j, H\}+\frac{\partial A^j}{\partial t} ~=~ 0.\tag{13}$$
   (We will use the $\approx$ sign to stress that an equation is an on-shell equation.)

7. The variation of the Hamiltonian Lagrangian $L_H$ is a total time derivative $$\begin{align} \delta L_H ~=~& \delta (\dot{\bf q}\cdot{\bf p}) \cr ~=~& \dot{\bf q}\cdot\delta {\bf p} - \dot{\bf p}\cdot\delta {\bf q} + \frac{\mathrm{d}({\bf p}\cdot\delta {\bf q})}{\mathrm{d}t} \cr ~=~& \varepsilon_j\left( \dot{\bf q}\cdot{\bf p}~p^j - p^2~\dot{q}^j + km\left( \frac{\dot{q}^j}{q} - \frac{{\bf q} \cdot \dot{\bf q}~q^j}{q^3}\right)\right) \cr ~-~&\varepsilon_j\left(2 \dot{\bf p}\cdot{\bf p}~q^j - \dot{\bf p}\cdot{\bf q}~p^j- {\bf p}\cdot{\bf q}~\dot{p}^j \right) - 2\varepsilon_j\frac{\mathrm{d}a^j}{\mathrm{d}t}\cr ~=~&\varepsilon_j\frac{\mathrm{d}f^j}{\mathrm{d}t}, \cr f^j ~:=~& A^j-2a^j, \end{align} \tag{14}$$ and hence the action $S$ is invariant off-shell up to boundary terms.

8. Noether charge. The bare Noether charge $Q_{(0)}^j$ is $$\begin{align} Q_{(0)}^j~:=~& \frac{\partial L_H}{\partial \dot{q}^i} \{A^j,q^i\}+\frac{\partial L_H}{\partial \dot{p}^i} \{A^j,p^i\} \cr ~=~& p^i\{A^j,q^i\}\cr ~=~& -2a^j.\end{align} \tag{15}$$ The full Noether charge $Q^j$ (which takes the total time-derivative into account) becomes (minus) the Laplace–Runge–Lenz vector $$\begin{align} Q^j~:=~&Q_{(0)}^j-f^j\cr ~=~& -2a^j-(A^j-2a^j)\cr ~=~& -A^j.\end{align}\tag{16}\label{eq:16}$$ $Q^j$ is conserved on-shell $$\frac{\mathrm{d}Q^j}{\mathrm{d}t} ~\approx~ 0,\tag{17}$$
   due to Noether's first Theorem. Here $j$ is an index that labels the three symmetries.

9. Lagrangian Problem. The Kepler problem has Lagrangian $$\begin{align} L~=~&T-V, \cr T~:=~& \frac{m}{2}\dot{q}^2, \cr V~:=~& -\frac{k}{q}. \end{align} \tag{18} $$ The Lagrangian momentum is $$ {\bf p}~:=~\frac{\partial L}{\partial \dot{\bf q}}~=~m\dot{\bf q} \tag{19} . $$ Let us project the infinitesimal symmetry transformation $\eqref{eq:7}$ to the Lagrangian configuration space $$\begin{align} \delta q^i ~=~& \varepsilon_j m \left( 2 \dot{q}^i q^j - q^i \dot{q}^j - {\bf q}\cdot\dot{\bf q}~\delta^{ij}\right), \cr \delta t ~=~&0.\end{align}\tag{20}\label{eq:20}$$ It would have been difficult to guess the infinitesimal symmetry transformation $\eqref{eq:20}$ without using the corresponding Hamiltonian formulation $\eqref{eq:7}$. But once we know it we can proceed within the Lagrangian formalism. The variation of the Lagrangian is a total time derivative $$\begin{align} \delta L~=~&\varepsilon_j\frac{\mathrm{d}f^j}{\mathrm{d}t}, \cr f_j~:=~& m\left(m\dot{q}^2q^j- m{\bf q}\cdot\dot{\bf q}~\dot{q}^j +k \frac{q^j}{q}\right)\cr ~=~&A^j-2 a^j . \end{align}\tag{21}$$ The bare Noether charge $Q_{(0)}^j$ is again $$Q_{(0)}^j~:=~2m^2\left(\dot{q}^2q^j- {\bf q}\cdot\dot{\bf q}~\dot{q}^j\right) ~=~-2a^j . \tag{22}$$ The full Noether charge $Q^j$ becomes (minus) the Laplace–Runge–Lenz vector $$\begin{align} Q^j~:=~&Q_{(0)}^j-f^j \cr ~=~& -2a^j-(A^j-2a^j)\cr ~=~& -A^j,\end{align}\tag{23}$$ similar to the Hamiltonian formulation $\eqref{eq:16}$.

Answer 2

While Kepler second law is simply a statement of the conservation of angular momentum (and as such it holds for all systems described by central forces), the first and the third laws are special and are linked with the unique form of the newtonian potential $-k/r$. In particular, Bertrand theorem assures that only the newtonian potential and the harmonic potential $kr^2$ give rise to closed orbits (no precession). It is natural to think that this must be due to some kind of symmetry of the problem. In fact, the particular symmetry of the newtonian potential is described exactly by the conservation of the RL vector (it can be shown that the RL vector is conserved iff the potential is central and newtonian). This, in turn, is due to a more general symmetry: if conservation of angular momentum is linked to the group of special orthogonal transformations in 3-dimensional space $SO(3)$, conservation of the RL vector must be linked to a 6-dimensional group of symmetries, since in this case there are apparently six conserved quantities (3 components of $L$ and 3 components of $\mathcal A$). In the case of bound orbits, this group is $SO(4)$, the group of rotations in 4-dimensional space.

Just to fix the notation, the RL vector is:

\begin{equation} \mathcal{A}=\textbf{p}\times\textbf{L}-\frac{km}{r}\textbf{x} \end{equation}

Calculate its total derivative:

\begin{equation}\frac{d\mathcal{A}}{dt}=-\nabla U\times(\textbf{x}\times\textbf{p})+\textbf{p}\times\frac{d\textbf{L}}{dt}-\frac{k\textbf{p}}{r}+\frac{k(\textbf{p}\cdot \textbf{x})}{r^3}\textbf{x} \end{equation}

Make use of Levi-Civita symbol to develop the cross terms:

\begin{equation}\epsilon_{sjk}\epsilon_{sil}=\delta_{ji}\delta_{kl}-\delta_{jl}\delta_{ki} \end{equation}

Finally:

\begin{equation} \frac{d\mathcal{A}}{dt}=\left(\textbf{x}\cdot\nabla U-\frac{k}{r}\right)\textbf{p}+\left[(\textbf{p}\cdot\textbf{x})\frac{k}{r^3}-2\textbf{p}\cdot\nabla U\right]\textbf{x}+(\textbf{p}\cdot\textbf{x})\nabla U \end{equation}

Now, if the potential $U=U(r)$ is central:

\begin{equation} (\nabla U)_j=\frac{\partial U}{\partial x_j}=\frac{dU}{dr}\frac{\partial r}{\partial x_j}=\frac{dU}{dr}\frac{x_j}{r} \end{equation}

so

\begin{equation} \nabla U=\frac{dU}{dr}\frac{\textbf{x}}{r}\end{equation}

Substituting back:

\begin{equation}\frac{d\mathcal A}{dt}=\frac{1}{r}\left(\frac{dU}{dr}-\frac{k}{r^2}\right)[r^2\textbf{p}-(\textbf{x}\cdot\textbf{p})\textbf{x}]\end{equation}

Now, you see that if $U$ has exactly the newtonian form then the first parenthesis is zero and so the RL vector is conserved.

Maybe there's some slicker way to see it (Poisson brackets?), but this works anyway.

Answer 3

The symmetry is an example of an open symmetry, i.e. a symmetry group which varies from group action orbit to orbit. For bound trajectories, it's SO(4). For parabolic ones, it's SE(3). For hyperbolic ones, it's SO(3,1). Such cases are better handled by groupoids.

Answer 4

Conservation of the Runge-Lenz vector does not correspond to a symmetry of the Lagrangian itself. It arises from an invariance of the integral of the Lagrangian with respect to time, the classical action integral. Some time ago I wrote up a derivation of the conserved vector for any spherically symmetric potential:

http://analyticphysics.com/Runge Vector/The Symmetry Corresponding to the Runge Vector.htm

The derivation is at the level of Goldstein and is meant to fill in the gap left by its omission from graduate-level classical mechanics texts.

Answer 5

Looking at https://arxiv.org/abs/1207.5001 one gets a very nice solution. If one is not very keen into mathematics, their basic idea is to use the infinitesimal transformation $$\delta x^i=\epsilon L^{ik}$$ where $L^{ik}=\dot{x}^ix^k-\dot{x}^k x^i$. Since angular momentum is conserved, kinetic energy won't change. On the other hand, the potential changes up to order $\epsilon^2$ like $$\frac{k}{r+\delta r}=\frac{k}{((x^i+\delta x^i)(x_i+\delta x_i))^{1/2}}=\frac{k}{r}\left(1-\frac{x_i\delta x^i}{r^2}\right)=\frac{k}{r}-\epsilon\frac{kx_iL^{ik}}{r^3}=\frac{k}{r}-\epsilon\frac{d}{dt}\left(\frac{kx^k}{r}\right).$$

Therefore, the change in the action is $$\epsilon\left[m\dot{x}_iL^{ik}\right]=[m\dot{x}_i\delta x^i]_{t_1}^{t_2}=\delta S=\epsilon\left[\frac{kx^k}{r}\right]_{t_1}^{t_2}.$$ This gives the conservation of the vector $$m\dot{x}_iL^{ik}-\frac{kx^k}{r},$$ which can be easily shown to be the Runge-Lenz vector.

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EDIT: In case someone prefers an index free notation, the symmetry is $\delta\vec{r}=\vec{\epsilon}\times\vec{L}$. Allowing for a time dependent $\vec{\epsilon}$, the variation in the action is then $$\delta S=\int dt\left(m\vec{v}\cdot\delta\vec{v}-\frac{k}{\|\vec{r}\|^3}\vec{r}\cdot\delta\vec{r}\right)=\int dt\left(m\vec{v}\cdot(\dot{\vec{\epsilon}}\times\vec{L})+m\vec{v}\cdot(\vec{\epsilon}\times\dot{\vec{L}})-\frac{k}{\|\vec{r}\|^3}\vec{r}\cdot(\vec{\epsilon}\times\vec{L})\right).$$ When the equations of motion are valid, the second term vanishes due to conservation of momentum. Moreover, the third term would be a total derivative if $\vec{\epsilon}$ was constant $$\frac{k}{\|\vec{r}\|^3}\vec{r}\cdot(\vec{\epsilon}\times\vec{L})=\frac{k}{\|\vec{r}\|^3}\vec{\epsilon}\cdot(\vec{L}\times\vec{r})=\frac{km}{\|\vec{r}\|^3}\vec{\epsilon}\cdot((\vec{r}\times\vec{v})\times\vec{r})=\frac{km}{\|\vec{r}\|^3}\vec{\epsilon}\cdot(-\vec{r}(\vec{v}\cdot\vec{r})+\vec{v}\|\vec{r}\|^2)=km\vec{\epsilon}\cdot\frac{d}{dt}\frac{\vec{r}}{\|\vec{r}\|}=km\vec{\epsilon}\cdot\frac{d\hat{r}}{dt}$$. We conclude that when the equations of motion are valid $$\delta S=\int dt\left(m\vec{v}\cdot(\dot{\vec{\epsilon}}\times\vec{L})-\vec{\epsilon}\cdot\frac{d}{dt}(mk\hat{r})\right).$$ We then see that when $\vec{\epsilon}$ is constant, the first term vanishes and the remaining term is a total derivative. We thus have a true symmetry of the system. On the other hand, allowing $\epsilon$ to be an arbitrary function of time which however vanishes at the endpoints, we can integrate by parts the first term to obtain $$\delta S=\int dt\left(m\dot{\vec{\epsilon}}\cdot(\vec{L}\times\vec{v})-\vec{\epsilon}\cdot\frac{d}{dt}(mk\hat{r})\right)=\int dt\left(\dot{\vec{\epsilon}}\cdot(\vec{L}\times\vec{p})-\vec{\epsilon}\cdot\frac{d}{dt}(mk\hat{r})\right)\\=\int dt\vec{\epsilon}\cdot\frac{d}{dt}\left(\vec{p}\times\vec{L}-mk\hat{r}\right).$$ Since this is a variation vanishing at the endpoints, on the equations of motion it must be zero. Since $\vec{\epsilon}$ is otherwise arbitrary, we conclude that the following vector is conserved $$\vec{p}\times\vec{L}-mk\hat{r}.$$ This the Runge-Lenz vector

Answer 6

(This post may be old, but we can add some precisions) The conservation of the RL vector is not trifling, it goes with the fact that you consider a central force, lead here by a Newtonian potential $\frac{1}{r}$ which has the property to be invariant under rotations (as $\frac{1}{r^n}$ but it works only for $n=1$ as shown by @quark1245).

Therefore, the S0(3) which has not 6 conserved quantities as said before but 3, the 3 generators of the symmetry $J_i$, i=1..3 such that the symmetry transformation under an infinitesimal change $x \rightarrow x + \epsilon$ is given in the canonical formalism by $$ \delta_i X = \{X, J_i(\epsilon) \} $$ and the algebra is $$ \{ J_i, J_j \} = \epsilon_{ij}^k J_k. $$ They are conserved because, at least for the Kepler problem, the system is invariant w.r.t a time translation, and the Hamiltonian is also conserved, and the calculations show that $$ \{H,J_i\}= 0. $$

Before their redefinition as shown on Wikipedia to see that the previous algebra is fulfilled, the generators of the rotations are : one is the angular momentum $L$ which shows that the movement is planar, therefore invariant under rotation around $L$, one is the RL vector which is in the plan, therefore perpendicular to $L$ and parallel to the major axis of the ellipse, and the third one has a name I don't remember, but is parallel to the minor axis.

We can see that their are only 3 degrees of freedom if we take place in the referential such that $\vec{J}_1 = \vec{L} = (0,0,L_z)$, then the planar generators are $A = (A_x,0,0)$ and $B = (0,B_y,0)$.

It has been shown that they can be constructed from the Killing-Yano tensors (which mean symmetry), and it works also at dimensions greater than 3. A nice review about the LRL vector derivation can be found in HeckmanVanHaalten