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By the shell theorem, only the mass inside the particle's radius contributes:The conserved Newtonian energy per unit particle mass isWriting $r=a(t)r_0$ and using conservation of enclosed mass, so that $\rho a^3$ is constant, givesThis is the first Friedmann equation without a cosmological constant.

The derivation has two central defects. It uses instantaneous Newtonian gravity and omits the relativistic gravitational effect of pressure, so it is not valid for general cosmological matter. It also starts from a finite ball with a preferred centre and boundary, contrary to exact homogeneity and isotropy; the assumption $r\ll L$ only hides that defect locally.

Order the phase-space coordinates asThen Hamilton's equations areThe Poisson bracket is

For $X=X(x)$ with Jacobian $J_{ab}=\partial X_a/\partial x_b$, the chain rule givesThis has Hamiltonian form with the same canonical matrix for every $H$ exactly whenTaking determinants gives $(\det J{)}^2=1$; the symplectic condition fixes the orientation, so $\det J=1$. Hence the change-of-variables formula shows that the phase-space volume element is invariant. This is Liouville theorem in Hamiltonian mechanics.

Choose paths from $0$ to $z$ in the cut plane and deform them continuously with the endpoint, never crossing the specified cut. With $0\le \arg z<2\pi$, use the sheet reached from the positive real axis; when an endpoint is moved into the left half-plane, the path passes around the upper endpoint $i$ of the cut. The cut makes all such admissible paths homotopic with fixed endpoints, so the integral is single valued and its endpoint derivative is $(1+z^2{)}^{-1/2}$; hence it is analytic.

For $z=-\sinh u$, continue the endpoint counterclockwise from the positive real axis through the upper half-plane. Inthe continued square root equals $-\cosh u$ at $z=-\sinh u$. The logarithm is reached with argument $\pi$, and therefore

The identitiesshow directly why continuation without an argument restriction is multivalued. The complete set of values of the multivalued inverse hyperbolic sine at $\sinh u$ is