Geodesic Flow and Hamilton-Jacobi Structure

The inverse metric defines a quadratic Hamiltonian on the cotangent bundle whose Hamiltonian characteristics project to affinely parametrized geodesics. The Hamilton–Jacobi equation replaces the momentum by the gradient of a generating function, organizing wavefronts, separability, and conserved quantities. The mass shell and signs must remain explicit: with signature $(-,+,+,+)$, timelike motion lies on $H=-m^2/2$, while null motion lies on $H=0$.

Required background. Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework; Affine and Variational Geodesics supplies the affine geodesic equation. The Riemann Curvature Tensor fixes the curvature convention used when neighboring trajectories are compared.

Helpful background. Calculus and Linear Algebra for Relativity supplies multivariable differentiation and bilinear forms; canonical coordinates and the first-order Hamilton–Jacobi equation are developed here. Presymplectic Potentials and Currents develops an advanced covariant field-space symplectic language, not the finite-dimensional geodesic-flow construction assumed on this page.

The metric Hamiltonian

Use canonical coordinates $(x^\mu,p_\mu)$ on $T^*M$ with symplectic form

$$\omega=dx^\mu\wedge dp_\mu.$$

Choose the convention $\iota_{X_H}\omega=dH$. Then Hamilton’s equations are $\dot x^\mu=\partial H/\partial p_\mu$ and $\dot p_\mu=-\partial H/\partial x^\mu$. For

$$H(x,p)=\frac12g^{\mu\nu}(x)p_\mu p_\nu,$$

they become

$$\dot x^\mu=g^{\mu\nu}p_\nu, \qquad \dot p_\mu=-\frac12\partial_\mu g^{\alpha\beta}p_\alpha p_\beta.$$

The first equation gives $p_\mu=g_{\mu\nu}\dot x^\nu$. Differentiate it, insert the second equation and $\partial_\mu g^{\alpha\beta}=-g^{\alpha\rho}g^{\beta\sigma}\partial_\mu g_{\rho\sigma}$, and obtain

$$\ddot x^\rho+\Gamma^\rho{}_{\mu\nu}\dot x^\mu\dot x^\nu=0.$$

Thus the projected Hamiltonian trajectories are affine geodesics. The canonical cotangent-bundle construction, metric Hamiltonian, and affine normalization are developed in Frolov, Krtouš, and Kubizňák 2017, §2.1, pp. 14–15, especially eqs. (2.10)–(2.16), PDF.

Mass shells and affine normalization

The Hamiltonian is conserved because it has no explicit parameter dependence:

$$2H=g^{\mu\nu}p_\mu p_\nu=g_{\mu\nu}\dot x^\mu\dot x^\nu.$$

For a massive particle parametrized by $\lambda=\tau/m$ or, equivalently, with $p^\mu=mu^\mu$ and $u^2=-1$,

$$H=-\frac{m^2}{2}.$$

For a null geodesic, $H=0$. A spacelike normalization would have $H>0$. Rescaling an affine parameter rescales $p$ and the numerical Hamiltonian level, so the mass-shell interpretation also fixes normalization. Null curves retain an arbitrary constant affine rescaling; $H=0$ alone does not remove it.

In Minkowski space, $p_\mu$ is constant and $x^\mu=x_0^\mu+p^\mu\lambda$. This independently reproduces the connection equation and checks the timelike sign.

Hamilton–Jacobi equation

A generating function $S(x)$ defines a Lagrangian graph $p_\mu=\partial_\mu S$. Requiring it to lie on a Hamiltonian level $\kappa$ gives

$$\frac12g^{\mu\nu}\partial_\mu S\,\partial_\nu S=\kappa.$$

For massive timelike geodesics, $\kappa=-m^2/2$; for null geodesics, $\kappa=0$. The characteristic curves of this first-order PDE are precisely the Hamiltonian trajectories. Locally, a complete integral depending on enough constants can generate a congruence of geodesics. When symmetries permit an additive ansatz, separation reduces the equations to quadratures.

For Minkowski space, $S=p_\mu x^\mu$ solves the equation when $\eta^{\mu\nu}p_\mu p_\nu=2\kappa$. If $dS\ne0$ and $g^{\mu\nu}\partial_\mu S\partial_\nu S=0$, the regular level sets are null wavefronts. Their normal $k_\mu=\partial_\mu S$ is also tangent after raising its index, since $k^\mu\partial_\mu S=k^2=0$. Moreover, symmetry of the scalar Hessian gives

$$k^\nu\nabla_\nu k_\mu =k^\nu\nabla_\mu k_\nu =\frac12\nabla_\mu(k^2)=0.$$

Thus the gradient itself fixes an affine tangent, an independent check connecting the eikonal PDE to the geodesic equation. A systematic pseudo-Riemannian treatment of orthogonal separation states the required integrability hypotheses in Rajaratnam, McLenaghan, and Valero 2016, §2, Theorem 2.1, pp. 2–3, PDF.

Conserved quantities and phase-space meaning

With the canonical bracket $\{F,G\}=\partial_{x^\mu}F\,\partial_{p_\mu}G-\partial_{p_\mu}F\,\partial_{x^\mu}G$, Hamiltonian evolution is $\dot F=\{F,H\}$. If $\xi$ is a Killing vector, $C_\xi=\xi^\mu p_\mu$ Poisson-commutes with $H$:

$$\{C_\xi,H\}=0.$$

The result follows either from the canonical bracket or from differentiating $\xi_\mu u^\mu$ along a geodesic. Rank-two Killing tensors similarly give quadratic integrals $K^{\mu\nu}p_\mu p_\nu$. Enough independent commuting integrals make the geodesic flow integrable in the Liouville sense, but a separated Hamilton–Jacobi ansatz should never be assumed without verifying those structures.

This finite-dimensional construction is genuinely symplectic because $\omega$ on $T^*M$ is nondegenerate. Constrained field theories and gauge reduction require presymplectic methods, boundary conditions, and quotient hypotheses. Physical particle motion also requires a choice of action and coupling; this page establishes only the metric geodesic flow.

The Hamiltonian vector field is tangent to every mass shell because $X_H(H)=\{H,H\}=0$. On a nonzero shell the restriction of $\omega$ to the hypersurface is presymplectic: its kernel is generated by $X_H$. For the null shell the zero section must be excluded if one wants a regular hypersurface, since $dH$ vanishes at $p=0$. Projectivizing nonzero null covectors removes positive scale but still specifies a point and a null direction at that point. To obtain the space of oriented, unparametrized null geodesics one must also identify points along the geodesic flow. Such a quotient is a smooth manifold only under additional regularity and global hypotheses; projectivization alone does not perform it.

Hamilton–Jacobi solutions are generally local. Characteristics can cross, at which point a single smooth function $S$ may cease to represent the whole Lagrangian submanifold even though the Hamiltonian flow remains smooth. In geometric optics this is a caustic of the projected wavefront, not necessarily a singularity of phase space or spacetime curvature. Multiple branches of $S$ or a generating family are then required.

Sign conventions in symplectic geometry vary. Had we chosen $\omega=dp_\mu\wedge dx^\mu$ while keeping $\iota_{X_H}\omega=dH$, both Hamilton equations would reverse. The projected unparametrized trajectories would be the same after reversing the flow parameter, but intermediate Poisson-bracket signs would differ. Declaring $\omega$ and the contraction equation together removes this ambiguity.

As an independent check, the Legendre transform of $L=g_{\mu\nu}\dot x^\mu\dot x^\nu/2$ gives $p_\mu=g_{\mu\nu}\dot x^\nu$ and $p_\mu\dot x^\mu-L=H$. This reproduces the same quadratic Hamiltonian and verifies that no extra factor of two enters either the mass shell or Hamilton’s first equation.

Exercises

Verify the mass shell for the Minkowski principal function $S=-Et+p_ix^i$.

Solution

With $\eta^{\mu\nu}=\operatorname{diag}(-1,1,1,1)$,

$$\frac12\eta^{\mu\nu}\partial_\mu S\partial_\nu S =\frac12(-E^2+\mathbf p^2).$$

For a particle of mass $m$, $E^2=\mathbf p^2+m^2$, giving $H=-m^2/2$. For a null ray, $E^2=\mathbf p^2$, giving $H=0$.

References

• Frolov, Valeri P., Pavel Krtouš, and David Kubizňák. “Black Holes, Hidden Symmetries, and Complete Integrability.” Living Reviews in Relativity 20, 6 (2017). DOI. Open PDF.
• Rajaratnam, Krishan, Raymond G. McLenaghan, and Carlos Valero. “Orthogonal Separation of the Hamilton–Jacobi Equation on Spaces of Constant Curvature.” SIGMA 12, 117 (2016). DOI. Open PDF.