PN Hamiltonians, Lagrangians, and Contact Transformations

Post-Newtonian dynamics can be encoded in equations of motion, a reduced Lagrangian, or a Hamiltonian. These are representations of the same conservative physics only after their variables, perturbative order, boundary conditions, and allowed transformations are matched. A term-by-term difference between two coordinate Hamiltonians is not evidence of a physical disagreement; failure of invariant relations to agree is.

Required background. Weak Metric Perturbations and Gauge Freedom supplies field redefinitions and coordinate gauge; Presymplectic Potentials and Currents supplies boundary variations and canonical equivalence. Helpful background. The Einstein Equation supplies the parent field theory; Gravitational Radiation and Waveforms clarifies the distinction between conservative and retarded dynamics.

A common 1PN dynamics in two languages

Restore $c$ for PN bookkeeping and retain $G$. In the center-of-mass frame let $\mathbf x$, $r$, $\mathbf n$, $\mathbf v$, $M$, $\mu$, and $\nu$ have the definitions used on Conservative PN Equations of Motion. An ordinary reduced Lagrangian through 1PN is

$$\begin{aligned} \frac{L}{\mu}={}& \frac{v^2}{2}+\frac{GM}{r}\\ &+\frac{1}{c^2}\bigg[ \frac18(1-3\nu)v^4 +\frac{GM}{2r} \bigg((3+\nu)v^2+\nu(\mathbf n\cdot\mathbf v)^2-\frac{GM}{r}\bigg) \bigg] +O(c^{-4}). \end{aligned}$$

The Euler–Lagrange equation

$$\frac{\partial L}{\partial\mathbf x} -\frac{d}{dt}\frac{\partial L}{\partial\mathbf v}=0$$

reproduces the 1PN relative acceleration. Define the reduced canonical momentum $\mathbf p=\mathbf P/\mu$ by $\mathbf P=\partial L/\partial\mathbf v$ and perform the Legendre transform $H=\mathbf P\cdot\mathbf v-L$. Expressing $\mathbf v$ perturbatively in terms of $\mathbf p$ gives

$$\begin{aligned} \frac{H}{\mu}={}& \frac{p^2}{2}-\frac{GM}{r}\\ &+\frac{1}{c^2}\bigg[ \frac18(3\nu-1)p^4 -\frac{GM}{2r}\Big((3+\nu)p^2+\nu(\mathbf n\cdot\mathbf p)^2\Big) +\frac{G^2M^2}{2r^2} \bigg] +O(c^{-4}). \end{aligned}$$

Hamilton’s equations, $\dot{\mathbf x}=\partial H/\partial\mathbf P$ and $\dot{\mathbf P}=-\partial H/\partial\mathbf x$, generate the same trajectories after momentum and velocity are related consistently. The reduced ADM-type Hamiltonians through 4PN, including the exact 1PN expression in rescaled variables, are collected in Schäfer and Jaranowski 2024, § 6.1, eqs. (6.6)–(6.13).

The comparison already has several axes. A Lagrangian uses positions and velocities and makes spacetime covariance comparatively transparent in harmonic coordinates. A Hamiltonian uses canonical positions and momenta, exposes the symplectic structure, and is convenient for conserved generators, action-angle variables, and effective-one-body mappings. Neither set of phase-space coordinates is itself observable. Their utility depends on the question.

Generalized Lagrangians and order reduction

At 2PN and above, a manifestly Lorentz-covariant harmonic-coordinate description requires a generalized Lagrangian,

$$L=L(\mathbf y_A,\mathbf v_A,\mathbf a_A),$$

which can be chosen linear in the accelerations. Its variational derivative is

$$\frac{\delta L}{\delta \mathbf y_A} =\frac{\partial L}{\partial\mathbf y_A} -\frac{d}{dt}\frac{\partial L}{\partial\mathbf v_A} +\frac{d^2}{dt^2}\frac{\partial L}{\partial\mathbf a_A}.$$

The acceleration dependence does not add physical degrees of freedom: it appears only perturbatively and is removed by order reduction. One first varies the generalized action. Only then may lower-order equations replace higher derivatives in the resulting equations or conserved quantities. Substituting $\mathbf a_A$ directly into the generalized Lagrangian before variation is generally inequivalent because the discarded variation of $\mathbf a_A$ contributes at the working order. The precise harmonic-coordinate result and this ordering rule are described in Blanchet 2024, § 3.2.3, eqs. (343)–(344).

Field elimination introduces another layer. If particle variables are Legendre transformed but gravitational field variables remain Lagrangian, the mixed object is a Routhian. Solving the field equations and evaluating the Routhian on that solution produces a particle Hamiltonian. Boundary terms must be retained consistently: a total time derivative does not alter the Lagrangian equations with fixed endpoint data, but it changes canonical momenta and therefore participates in the map between phase-space variables.

Order reduction has a controlled meaning. If a term is already multiplied by $c^{-4}$, replacing an acceleration inside it by the Newtonian acceleration changes the result only at 3PN and higher. Using a more accurate substitution is harmless only if all induced terms are kept to a common order. Partial substitutions can violate energy conservation or Poincaré symmetry even when every individual formula looks dimensionally correct.

Contact and canonical transformations

A contact transformation is a perturbative change of particle variables that may depend on velocities,

$$\mathbf Y_A =\mathbf y_A+\delta\mathbf y_A(\mathbf y_B,\mathbf v_B), \qquad \delta\mathbf y_A=O(c^{-4})$$

for the first acceleration-removing transformation in this problem. Together with a total derivative $dF/dt$, it can transform the generalized harmonic-coordinate Lagrangian into an ordinary ADM-type Lagrangian. At 3PN the transformation also removes harmonic-coordinate logarithms involving arbitrary gauge scales. The explicit construction and the uniqueness after choosing $F$ are given in Blanchet 2024, § 3.2.3, eqs. (344)–(345).

In Hamiltonian language, fix the direction of the transformation explicitly. Use the type-two generating function $F_2(\mathbf q,\mathbf P,t)=\mathbf q\cdot\mathbf P+g(\mathbf q,\mathbf P,t)$, where $g$ is small. The new coordinates obey

$$\mathbf Q=\mathbf q+\partial_{\mathbf p}g, \qquad \mathbf P=\mathbf p-\partial_{\mathbf q}g.$$

Writing the old Hamiltonian in these new coordinates requires the inverse map, so the new Hamiltonian is

$$K(\mathbf Q,\mathbf P,t) =\left[H-\{H,g\}+\partial_tg\right](\mathbf Q,\mathbf P,t) +O(g^2),$$

where $\{f,g\}=\partial_{\mathbf q}f\cdot\partial_{\mathbf p}g -\partial_{\mathbf p}f\cdot\partial_{\mathbf q}g$. The minus sign in front of $\{H,g\}$ is essential with this direction of the coordinate map. An active pullback of functions is also often written with a plus sign; it must not be combined with the inverse coordinate convention.

For a time-independent $g=c^{-2}g_{\rm 1PN}+\cdots$, the 1PN coefficient therefore becomes $K_{\rm 1PN}=H_{\rm 1PN}-\{H_{\rm N},g_{\rm 1PN}\}$. A canonical transformation preserves Poisson brackets by construction; an arbitrary change of coordinates in phase space does not.

This distinction prevents two common mistakes. First, agreement after inserting only circular-orbit conditions does not prove full canonical equivalence; the generator must work on the claimed phase-space domain. Second, a map that contains poles, violates exchange symmetry, or changes asymptotic normalization may not be admissible even if it removes selected terms.

What must agree

The strongest comparisons use quantities invariant under the proposed transformation. For circular nonspinning motion these include the binding energy $E(\Omega)$ and angular momentum $J(\Omega)$. For eccentric bound motion one can compare the radial action

$$I_r(E,J)=\frac{1}{2\pi}\oint p_r\,dr$$

and the frequency ratio $K=\Omega_\phi/\Omega_r$. For unbound motion the scattering angle $\chi(E,J)$ plays the analogous role. The first law of binary mechanics, redshift invariants in the small-mass-ratio limit, and the global Poincaré algebra give additional checks.

An efficient equivalence test proceeds in layers. Verify the Newtonian and test-mass limits; construct the transformation through the required order; check that the transformed Hamiltonian differs only beyond that order; and compare at least one generic-orbit invariant, not only a circular specialization. Arbitrary regularization or gauge scales must cancel from every invariant.

The complete 4PN conservative dynamics adds a genuine qualification. Its tail contribution is nonlocal in time:

$$H_{\rm 4PN} =H_{\rm 4PN}^{\rm local}(\mathbf q,\mathbf p) +H_{\rm 4PN}^{\rm nonlocal}[\mathbf q,\mathbf p].$$

A local Hamiltonian obtained by orbit averaging or by introducing auxiliary variables is a representation with stated restrictions, not literally the original functional on all histories. Schäfer and Jaranowski give this split in Schäfer and Jaranowski 2024, §§ 6.1 and 6.3.2, eqs. (6.11) and (6.38)–(6.40). Canonical-equivalence claims at 4PN must say whether they concern the full nonlocal action, its localized form, circular invariants, or an expansion in eccentricity.

Limits of the formulation

The point-mass Hamiltonians here omit spin and finite-size response unless those operators are explicitly added. They represent the conservative sector and do not generate the secular inspiral caused by outgoing flux. Their PN truncation assumes slow motion and weak interbody gravity; canonical rearrangement cannot extend the underlying asymptotic regime by itself. A resummation may improve behavior, but it changes uncontrolled higher-order terms and requires independent validation.

Finally, “Hamiltonian” does not automatically mean unique, local, or gauge invariant. It means that a chosen set of variables carries a specified symplectic structure and generates time evolution. The physical content lies in the equivalence class together with its boundary conditions and observables.

Exercise

Let

$$H=H_0+\varepsilon H_1+O(\varepsilon^2), \qquad g=\varepsilon g_1.$$

Take $g_1$ to be time independent. Show that the new-coordinate Hamiltonian $K=H-\{H,g\}$ describes the same energy at corresponding phase-space points $Z=z+\varepsilon\{z,g_1\}$. Check the sign using a one-dimensional free particle and $g_1=\alpha qp$, where $\alpha$ is constant.

Solution

To first order,

$$K(Z)=H_0(Z)+\varepsilon\left(H_1(Z)-\{H_0,g_1\}(Z)\right) +O(\varepsilon^2).$$

Taylor expanding $H_0(Z)$ around $z$ gives

$$H_0(Z)=H_0(z)+\varepsilon\{H_0,g_1\}(z)+O(\varepsilon^2),$$

so $K(Z)=H(z)+O(\varepsilon^2)$. Because the map preserves the symplectic form, Hamilton’s equations transform with it.

For $H=p^2/(2m)$, the dilation gives $Q=(1+\varepsilon\alpha)q$ and $P=(1-\varepsilon\alpha)p$. Since $\{H,g_1\}=-\alpha p^2/m$,

$$K(Q,P)=\frac{P^2}{2m}(1+2\varepsilon\alpha)+O(\varepsilon^2).$$

Then $\dot Q=(1+2\varepsilon\alpha)P/m =(1+\varepsilon\alpha)\dot q+O(\varepsilon^2)$, exactly as the coordinate map requires. Using the opposite Hamiltonian sign would fail this elementary check. The two expressions for the energy look different at the same numerical momentum because $p$ and $P$ label different phase-space coordinates.

References

• Blanchet, Luc. 2024. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4. DOI.
• Schäfer, Gerhard, and Piotr Jaranowski. 2024. “Hamiltonian Formulation of General Relativity and Post-Newtonian Dynamics of Compact Binaries.” Living Reviews in Relativity 27, 2. DOI.