Orbital Dynamics · Action, Conservation and Precession

01How planetary motion enters the spatial background

As a planet repeatedly approaches and recedes from its central body, why does the direction of periapsis shift from one orbit to the next? In ASD, this question connects two spatial responses: the central source establishes a static, nonuniform longitudinal stretching, while the moving body and its surrounding accompanying field develop a directional inertial response. An orbit must describe both the actual path traversed by the body and how this combined structure changes its momentum and energy.

The planet is approximated here as a test body whose internal state remains fixed. Its mass parameter is $m_0$; self-gravity, deformation and backreaction on the central source are neglected. ASD interprets matter and inertia through localized spatial structures. This page uses the macroscopic momentum and coupling relations specified in Chapter 2, without treating a candidate particle picture as an already obtained stable solution.

The central source has a static, spherical, uncharged background. Let its mass be $M$, the fixed gravitational coupling be $G_0$, and the gravitational length be $\ell_g=G_0M/c^2$. The radial coordinate $r$ is defined by the sphere's area $4\pi r^2$: it is an areal radius, distinct from actual length accumulated radially. Outside the material source and for $r>2\ell_g$, adopt $X=1-2\ell_g/r$, the linear expansion factor $\Omega=X^{-1/2}$, and its inverse $\Phi=\sqrt X$. On this page, $\Phi$ is a dimensionless spatial state variable.

The upstream dynamics starts from the directional length relation of LVE and separately uses an inertial constitutive relation to specify how each direction contributes to total momentum. It gives $\mathbf p=m_0\gamma\mathbf v$, where $\gamma=(1-v^2/c^2)^{-1/2}$ represents the inertial gain of the moving spatial structure. Combining the work relation with the rest-energy normalization $E_0=m_0c^2$, then integrating $\mathbf p=\partial\mathcal L_0/\partial\mathbf v$, gives the free Lagrangian

The inertial weighting, rest energy and universal mass normalization each play a specified role in this construction; see dynamics and energy. The orbital calculation places this free-motion relation in the spatial background established by the central source.

02Actual path, local speed and action

The central source's gradient selects the radial direction, while spherical symmetry relates the two tangential directions. The manuscript adopts the actual radial length $d\ell_r=\Omega\,dr$. Areal radius already determines the spherical arc $r\,d\varphi$, so tangential length does not receive another factor of $\Omega$. In the orbital plane, with azimuth $\varphi$,

Spatial stretching also changes the characteristic wavelength and reference frequency of a standard process. A far-field process establishes the common counting parameter $t$; during $dt$, a local process of the same type accumulates $dN=f_0\Phi\,dt$ cycles, where $f_0$ is the far-field reference frequency. Local speed, defined through actual length and cycle count, is therefore $v_*=f_0d\ell_s/dN=\Phi^{-1}d\ell_s/dt$, or

A dot denotes differentiation with respect to $t$. Local speed $v_*$ and reference-coordinate speed use different calibrations; the conversion must be made before applying the free-motion relation.

For a fixed internal state, Chapter 2 separately adopts universal static energy coupling: local structural energy is converted to the common reference by $\Phi$, so the action increment is $\mathcal L\,dt=\mathcal L_0(v_*)\Phi\,dt$. This is a constitutive input consistent with the standard-process response, not a consequence of cycle counting alone. Substitution gives

The action $S=\int\mathcal L\,dt$ assembles the dynamical relations along the entire trajectory; its stationarity condition determines the orbit. The expression combines the effect of actual path length on speed with the effect of position on reference energy. As $X\to1$, free motion is recovered. For a stationary body, the reference energy is $m_0c^2\Phi$, whose weak-field increment is $m_0\Psi+O(m_0\Psi^2/c^2)$, with $\Psi=c^2(X-1)/2=-G_0M/r$. Newtonian potential energy thus uses the same mass and energy normalization as the upstream inertia.

03Which motion terms belong to the same order

To see how this action approaches Newtonian motion, set $U=G_0M/r>0$, so $\Psi=-U$. Also define the reference-coordinate radial speed $v_r=\dot r$ and $v^2=\dot r^2+r^2\dot\varphi^2$. Here $v$ specifically means the coordinate-speed magnitude used in this expansion, distinct from $v_*$ in the preceding section. For $U/c^2\ll1$ and $v^2/c^2\ll1$, retaining the complete corrections one order beyond the Newtonian terms gives the first post-Newtonian expansion:

This retained website expansion is calculated from final equation (2.27). Final equation (2.28) now gives the two canonical momenta, rather than this 1PN series.

The expansion uses $X^{-1}=1+2U/c^2+O(c^{-4})$ and $\sqrt{1-z}=1-z/2-z^2/8+\cdots$. The term $v^4/8$ continues the inertial correction of a free structure; $Uv^2/2$ and $Uv_r^2$ couple motion to spatial stretching; $U^2/2$ comes from the next-order change in static energy. They belong to the same level of approximation. Selecting just one of them to modify a Newtonian orbit would omit contributions of the same order.

The calculation below obtains conserved quantities and orbital shape directly from the untruncated action, then makes a first-order approximation when extracting periapsis advance. This keeps the background, motion and approximation order consistent throughout.

04Energy and angular momentum from symmetry

First denote the quantity under the action's square root by

The condition $D=X(1-v_*^2/c^2)>0$ corresponds to locally subluminal motion and makes the action real. The background has no preferred azimuth, so $\mathcal L$ has no explicit dependence on $\varphi$ and the corresponding canonical angular momentum is conserved. Here “canonical” means momentum defined by differentiating the Lagrangian with respect to velocity. Dividing by $m_0$ gives the specific angular momentum

The dimensions of $j$ are length squared divided by time. Since $\sqrt D$ enters the conversion, it cannot simply be replaced by $r^2\dot\varphi$ in a strong-field or high-speed calculation.

Staticity means that $\mathcal L$ has no explicit dependence on $t$, giving a conserved reference energy as well. First calculate the radial momentum $p_r=\partial\mathcal L/\partial\dot r$. Multiply each velocity by its conjugate momentum and subtract $\mathcal L$: this is the Legendre transform that gives energy,

Differentiating the same radicand with respect to $\dot r$ supplies the factor $X^{-1}$ in $p_r$, whereas differentiating with respect to $\dot\varphi$ supplies $r^2$ in $p_\varphi$. Thus both momenta in final (2.28) precede the Legendre energy in (2.31).

Here $p_\varphi=m_0j$, $E_\infty$ is energy expressed in the common far-field reference, and $\mathcal E=E_\infty/(m_0c^2)$ is a dimensionless constant. Angular momentum and energy encode azimuthal symmetry and staticity, respectively. They allow orbital shape to be expressed without explicit dependence on the motion's counting parameter.

05Eliminating the counting parameter to obtain orbital shape

Describe shape by inverse radius $u(\varphi)=1/r$, with a prime denoting differentiation with respect to $\varphi$. Specific angular momentum and the chain rule give

Substituting the two velocities into the definition of $D$ gives

Divide by $D$, multiply by $X$, and use $X^2/D=\mathcal E^2$ from energy conservation:

Rearranging yields the first integral of radial motion:

This step assumes $j\ne0$; purely radial motion must be treated using the original action. Differentiate with respect to $\varphi$; where $u'\ne0$, cancel $2u'$ and extend smoothly to periapsis and apoapsis:

Set $\mu=G_0M$, so $X=1-2\ell_gu$ and $dX/du=-2\ell_g$. The three terms on the right become $\mu/j^2$, $-u+2\ell_gu^2$ and $\ell_gu^2$, respectively. Adding them gives

The Newtonian term sets the ellipse's basic scale, while $3\ell_gu^2$ increases as the body approaches the central source. In the present areal-radius representation, the coefficient $3$ combines two position-dependent contributions from the same action; no empirical precession term has been added. This shape equation holds within the adopted static constitutive model. The next step extracts its first-order weak-field advance.

06Why one radial cycle no longer corresponds to one full turn

If the correction is neglected, the orbit is a Kepler ellipse:

Here $a$ is the semimajor axis, $e$ the eccentricity, and $p_{\rm orb}$ the semilatus rectum; $p_{\rm orb}=a(1-e^2)$ holds for the zeroth-order ellipse. Take $\ell_g/p_{\rm orb}\ll1$, write $u=u_0+\delta u$, and use the zeroth-order solution only in the correction term:

Decompose the right-hand side using $\cos^2\varphi=(1+\cos2\varphi)/2$. The constant term adjusts the mean scale, and the double-frequency term produces a bounded deformation. The term $6\ell_ge\cos\varphi/p_{\rm orb}^2$, at the same frequency as the zeroth-order radial cycle, accumulates from orbit to orbit. The identity

gives the accumulating term, which can then be compared with a first-order expansion of a slowly rotating ellipse:

Here $\kappa$ is a dimensionless azimuthal drift coefficient. The angular separation between successive periapses is $2\pi/(1-\kappa)$. At first order, $\Delta\varpi=2\pi\kappa+O(\kappa^2)$, giving the advance per orbit

As the body moves inward from apoapsis into a more strongly stretched region, path scale, local speed and inertial response change together, then adjust continuously as it moves outward again. Completing one radial cycle therefore requires an azimuthal angle greater than $2\pi$: periapsis advances in the orbital direction. The limit $\ell_g/p_{\rm orb}\to0$ restores a closed Kepler orbit. An exactly circular orbit has no distinct periapsis; nearly circular motion can instead be described by comparing radial and azimuthal frequencies.

07Connecting the calculation to observations and its scope

Using the solar gravitational parameter $\mu_\odot\simeq1.3271244\times10^{20}\,\mathrm{m^3s^{-2}}$, Mercury's $a\simeq5.79\times10^{10}\,\mathrm m$ and $e\simeq0.2056$, the manuscript obtains a central-field contribution of about $0.1035$ arcseconds per orbit, or $43.0$ arcseconds per Julian century. An actual ephemeris must jointly fit perturbations from other planets, solar multipoles and the observation chain. This central-field value is not a complete observed residual. Parameter references are provided by JPL; see GRV-EXP-004 for the observational comparison.

The complete static action adopted in the manuscript is mathematically equivalent to the Schwarzschild exterior action. This comparison is made after constructing ASD's spatial length, inertia and energy coupling. Reproducing the same advance under the same conditions therefore does not independently establish either physical interpretation. Nor does calculating higher orders automatically generate an ASD-specific difference.

The term $U^2/c^2$ belongs to the complete first post-Newtonian action; its presence does not make the precession formula a complete second-order result. Simply expanding $2\pi/(1-\kappa)$ further omits orbital corrections of the same order. The orbital-discriminator problem must first establish whether the complete model produces a difference that cannot be absorbed by reparametrization.

This page excludes many-body effects, self-gravity, radiation reaction and rotating strong fields. The S301 case study still lacks complete input data, fits and residuals and remains subject to that data requirement. No fit for this object follows from the formal relations on this page alone.