Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S7 / 2

For the canonical Poisson bracket, direct substitution of $F=q_i$ and $F=p_i$ givesThereforeThis expresses the bracket as a first-order derivative acting on $F$, with coefficients determined by $G$; it is the phase-space chain-rule form of the definition.

Hamilton's equations are ${\dot{q}}_i=H_{p_i}$ and ${\dot{p}}_i=-H_{q_i}$. Using the elementary brackets from the previous part givesA constant of motion is a function that stays unchanged along each physical trajectory. For a function $C(\mathbf{q},\mathbf{p})$ with no explicit time dependence, the chain rule and Hamilton's equations giveThus $\boxed{C\text{ is conserved}⟺\{C,H\}=0}$ on the accessible phase-space domain. If a function explicitly depends on time, the criterion is instead ${\partial }_{t}C+\{C,H\}=0$; the simpler statement uses the source's time-independent $C$ assumption.

Put $\ell =p_{\varphi }$. The Hamiltonian has no explicit time or azimuthal dependence, so $\dot{H}=\{H,H\}=0$ and $\dot{\ell }=-H_{\varphi }=0$. Its equations of motion areWrite $C=p_R\ell \sin \varphi +({\ell }^2/R+A)\cos \varphi$. Differentiating along these equations givesEquivalently, the same cancellation evaluates $\{C,H\}$. Hence $\boxed{H,p_{\varphi },C\text{ are all constants of motion}.}$ The extra conserved component is associated with the Runge-Lenz symmetry of this inverse-distance central potential, rather than with conservation of radial momentum.

The Jacobi identity and the two brackets $\{C,H\}=\{p_{\varphi },H\}=0$ implyThus $\boxed{D=\{C,p_{\varphi }\}\text{ is conserved}}$ without re-solving the motion. Explicitly $D={\partial }_{\varphi }C=p_R{p}_{\varphi }\cos \varphi -(p_{\varphi }^2/R+A)\sin \varphi$. Squaring and adding the two rotated components givesConsequently the four quantities are not functionally independent: generically there are three independent constants. One may choose, locally, $H,p_{\varphi }$ and the orientation encoded by $C,D$; their magnitude is fixed by the displayed relation. This also matches the maximum of three independent trajectory-labeling constants for a nonstationary autonomous flow in four-dimensional phase space. On singular subsets the local functional rank can be lower: for example, at $A=0$ and $p_{\varphi }=0$, the differentials of $C,D$ are proportional to $d{p}_{\varphi }$, leaving at most two independent differentials together with $dH$; “three” is the generic count, not an assertion about every singular configuration.