Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S7 / 1

Hamilton's principle makes the action $S[q]={\int }_{t_1}^{t_2}L(q,\dot{q},t)dt$ stationary under variations vanishing at the endpoints. It need not be a strict minimum. Varying the action and integrating the velocity variation by parts givesThe endpoint term vanishes. Since the interior variations are arbitrary, the fundamental lemma of the calculus of variations yields

Write $u=\pi x^2/l_0^2$ and assume $V_0>0$. The derivative of the potential is $V^{\prime }(x)=-2\pi V_{0}x\sin u/l_0^2$. Its equilibrium positions are zero and $x=\pm l_0\sqrt{n}$ for positive integers $n$. At a nonzero equilibrium,Thus the stable equilibria and their small-oscillation normal mode frequencies areEven positive $n$ give unstable maxima. At zero the quadratic term vanishes but $V(x)=V_0-V_0{\pi }^2{x}^4/(2l_0^4)+\cdots$, so the equilibrium is still unstable; a zero linearized frequency does not prove stability. If the unspecified sign of $V_0$ were chosen negative, even-$n$ positions would be stable instead, together with a degenerate quartic minimum at zero whose frequency tends to zero with oscillation amplitude.

For an infinitesimal continuous symmetry $\delta q_i=\epsilon {\xi }_i(q,t)$, allow the Lagrangian to change by a total derivative, $\delta L=\epsilon dF/dt$. Direct variation givesOn a solution of the Euler-Lagrange equations, the first term vanishes. This proves Noether's theorem in the fixed-time form:Spatial translations yield total momentum, and rotations yield angular momentum. Time translations yield energy when the Lagrangian has no explicit time dependence; this follows either from the extended time-transformation form of the theorem or by differentiating $H={\sum }_i{p}_i{\dot{q}}_i-L$ to obtain $\dot{H}=-\partial L/\partial t$.

The central pair potential is invariant under simultaneous spatial translations, simultaneous rotations and time translations. The corresponding time-independent Noether charges are total momentum $P=p_1+p_2$, total angular momentum about a fixed origin, and total energy $H$.

In one dimension this supplies $P$ and $H$: two charges. In two dimensions it supplies $P_x,P_y,L_{xy},H$: four charges. In three dimensions it supplies three momentum components, three angular-momentum components and $H$: seven charges. For generic states these listed charges are independent in those dimensions. Galilean boosts can also be expressed by explicitly time-dependent conserved combinations, but are not part of the requested translation-rotation-energy count.

In $D$ dimensions the antisymmetric angular momentum tensor has componentsThere are $D(D-1)/2$ independent rotation generators. Together with $D$ translation generators and one time-translation generator, the standard Noether charge enumeration is thereforeThe word independent here counts symmetry generators and tensor components; it needs qualification if it is meant as functional independence of all charges for only two particles in arbitrarily high dimension. In centre-of-mass coordinates, $L=R\wedge P+r\wedge p$. Consequently $P\wedge L\wedge L=0$, a nontrivial identity from dimension five onwards, since $(r\wedge p)\wedge (r\wedge p)=0$. Thus the displayed number is the symmetry-charge count; it is not an assertion that these functions have unrestricted values for a two-particle realization in every $D$.