Newtonian Motion Primitives

• Newtonian motion primitives are a systematic framework that encapsulates both classical and relativistic orbital dynamics using closed-form expressions.
• They enable precise calculation of orbital parameters such as energy, angular momentum, and frequencies through efficient mappings from observable trajectories.
• The methodology supports both astrophysical simulations and laboratory experiments by quantifying orbital precession, epicyclic frequencies, and error bounds.

Newtonian motion primitives are a systematic set of analytical and computational tools that encapsulate the essential properties of Newtonian and generalized Newtonian orbital motion. They enable precise yet computationally efficient modeling of particle dynamics in both classical gravitational potentials and generalized potentials derived from the low-energy limit of spherically symmetric relativistic spacetimes. These primitives consist of closed-form expressions and algorithmic procedures for evaluating equations of motion, specific energies, angular momentum, orbital frequencies, pericentre precession, and the mapping between observable trajectory data and parameters such as conic-section elements and orbital anomalies. Their relevance spans astrophysical modeling, laboratory analog experiments, and direct numerical integration for simulation codes (Tejeda et al., 2014, Bhambhu et al., 2023).

1. Low-Energy Limit and Metric Ansatz

The construction of Newtonian motion primitives begins with the general static, spherically symmetric metric in Schwarzschild-like coordinates:

$$ds^2 = -\alpha(r)c^2 dt^2 + \alpha(r)^{-1} dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2), \quad \alpha(r) \equiv 1 + \frac{2\Phi(r)}{c^2}$$

where $\Phi(r)$ asymptotically matches the Newtonian gravitational potential. The low-energy/geodesic limit ($|\mathcal{E}| \rightarrow c^2$) yields a Newtonian-like Lagrangian for a test mass $m$:

$$L(r, \dot{r}, \phi, \dot{\phi}) = \frac{1}{2} \left( \frac{\dot{r}^2}{\alpha^2} + \frac{r^2 \dot{\phi}^2}{\alpha} \right) - \Phi(r)$$

The generalized potential structure allows direct mapping between Newtonian and relativistic orbital properties up to leading-order corrections (Tejeda et al., 2014).

2. Effective Potentials and Canonical Quantities

From the Lagrangian, canonical momenta ($p_r$, $h_*$) and a form of conserved “Newtonian” energy ($E_*$) are defined:

$$p_r = \frac{\dot{r}}{\alpha^2}, \qquad h_* = \frac{r^2 \dot{\phi}}{\alpha}$$

$$E_* = \frac{1}{2} \left( \frac{\dot{r}^2}{\alpha^2} + \frac{r^2 \dot{\phi}^2}{\alpha} \right) + \Phi(r)$$

Upon eliminating $\dot{\phi}$ in favor of $h_*$, the energy equation reduces to:

$$\frac{1}{2} \frac{\dot{r}^2}{\alpha^2} + V_{\mathrm{eff}}(r;h_*) = E_*$$

$$V_{\mathrm{eff}}(r;h_*) = \Phi(r) + \frac{\alpha(r) h_*^2}{2 r^2}$$

This formulation allows the analysis of radial motion, capture of pericentre advance, and direct calculation of orbital characteristics—mirroring the energy-potential formalism of Newtonian dynamics with incorporated relativistic corrections for $\alpha(r)$ (Tejeda et al., 2014).

3. Equations of Motion and Characteristic Orbits

The Euler–Lagrange equations derived from the generalized Lagrangian yield the full equations of motion in both spherical and Cartesian coordinates. These equations admit exact expressions for special orbital radii:

• Photon circular orbit: $$2\alpha(r_{\mathrm{ph}}) - r_{\mathrm{ph}} \alpha'(r_{\mathrm{ph}}) = 0$$
• Marginally bound orbit: $E_*^c(r_{\mathrm{mb}}) = 0$
• Marginally stable orbit (ISCO): Smallest positive root of $$\mathcal{A}(r) = 3\alpha(r) - 2r\alpha'(r) + r\alpha(r)\frac{\alpha''(r)}{\alpha'(r)} = 0$$

Notably, these results hold exactly for $\alpha(r)$ corresponding to Schwarzschild, Reissner–Nordström, Schwarzschild–de Sitter, Ayón–Beato–García, and Kehagias–Sfetsos metrics (Tejeda et al., 2014).

4. Orbital Frequencies, Precession, and Epicyclic Motions

For circular orbits ($\dot{r} = 0$):

$$h_*^c(r) = \sqrt{\frac{2 r^3 \Phi'(r)}{2\alpha(r) - r \alpha'(r)}}$$

$$E_*^c(r) = \Phi(r) + \frac{r \alpha(r) \Phi'(r)}{2\alpha(r) - r\alpha'(r)}$$

The corresponding Keplerian orbital frequency is:

$$\Omega_* = \dot{\phi}^c = \sqrt{\frac{2 \alpha(r)^2 \Phi'(r)}{r [2\alpha(r) - r\alpha'(r)]}}$$

Pericentre precession and spatial trajectory integrals are formally identical to the general relativistic case. Perturbative analysis yields the vertical and radial epicyclic frequencies:

$\Omega_*^\perp = \Omega_*$

$$\Omega_*^\parallel = \sqrt{\mathcal{A}(r)}\,\Omega_*$$

The ratios $\Omega_*^\perp/\Omega_*$ and $\Omega_*^\parallel/\Omega_*$ exactly reproduce those in general relativity (Tejeda et al., 2014).

5. Pipeline Implementation for Experimental and Computational Contexts

For laboratory analogues (e.g., marble motion on a 3D-printed “inverse-r” surface), Newtonian motion primitives furnish a data analysis pipeline (Bhambhu et al., 2023):

1. Surface calibration: $h(r,\theta) = C_1 + C_2 / r$
2. Video extraction: Obtain $(x_i, y_i)$; compute $z_i = h(r_i)$
3. Plane fitting and projection via SVD/PCA to yield planar coordinates $(x_{p,i}, y_{p,i})$
4. Velocity approximation and construction of the state vector $(\mathbf{r}, \mathbf{v})$
5. Conic-section fitting for extraction of $(a, b, e)$
6. Computation of anomalies ($M$, $E$, $\nu$), orbital period $T$
7. Conservation checks on $E_{\mathrm{mech}}, \mathbf{L}, \mathbf{A}$
8. Error analysis: includes surface profile uncertainties, tracking errors, pixel quantization, frictional dissipation, and plane-fit deviations

This protocol allows mapping of raw trajectories to Newton–Kepler primitives, quantifying orbital parameters and demonstrating the limitations of the analogy (friction, loss of spherical symmetry, measurement precision).

6. Application to Spherically Symmetric Spacetimes and Error Estimates

Explicit parameterizations of $\alpha(r)$ for various spacetimes demonstrate the flexibility and limits of Newtonian motion primitives. The maximum fractional error in reproducing Keplerian and epicyclic frequencies never exceeds 10% for all stable circular orbits in the following cases:

Spacetime	$\alpha(r)$ form	$\max[(\Omega - \Omega_*)/\Omega]$
Schwarzschild	$1 - 2M/r$	$\leq 5.7$\%
Schwarzschild–de Sitter	$1 - 2M/r - \Lambda r^2/3$	$\leq 7.1$\%
Reissner–Nordström	$1 - 2M/r + q^2/r^2$	$\leq 8.2$\%
Ayón–Beato–García	$$1-\frac{2Mr^2}{(r^2+q^2)^{3/2}}+\frac{q^2r^2}{(r^2+q^2)^2}$$	$\leq 7.5$\%
Kehagias–Sfetsos	$1 + \omega r^2 [1 - \sqrt{1+4M/(\omega r^3)}]$	$\leq 6.3$\%

All critical radii ($r_{\mathrm{ph}}, r_{\mathrm{mb}}, r_{\mathrm{ms}}$), circular-orbit $E(r), L(r)$, precession, and spatial paths are retrieved exactly in the formalism (Tejeda et al., 2014).

7. Broader Context and Significance

Newtonian motion primitives facilitate interoperability between Newtonian and relativistic orbital dynamics, offering structured methods that are implementable in classical integrators with minimal modification. In computational laboratory experiments, such as warped-surface analogues of gravitational motion, they provide an explicit mapping from experimentally accessible trajectories to formal Newtonian–Keplerian orbital parameters. A notable insight is that, with a suitable projection procedure, laboratory trajectories on 3D-printed “inverse-r” surfaces can closely mimic Keplerian orbits and approximately respect conservation laws, subject to quantified experimental errors (Bhambhu et al., 2023).

The generalization, as established by Tejeda and Rosswog, provides a powerful framework for orbital dynamics in astrophysical modeling where relativistic effects are non-negligible but a full solution of the Einstein field equations is computationally infeasible or unwarranted (Tejeda et al., 2014). The methodology is extensible to a wide range of potentials and has explicit, tabulated error bounds for key observables. As the operational core of many computational pipelines, Newtonian motion primitives have become central to both undergraduate laboratory training and the development of astrophysical simulation codes.