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\title{Mechanics --- Math Appendix}
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\maketitle

\section*{0. Charter \textbullet\ Symbols \textbullet\ Validity}

Symbols: $c$ (light baseline), $G$ (grav.\ coupling), $\kappa$ (bridge coupling); $x$, $v=dx/dt$, $a=dv/dt$; $r=|x|$, $\dot{r}=x/r$; $m$ (mass), $p$ (momentum), $F$ (force), $L$ (Lagrangian), $H$ (Hamiltonian), $E$ (energy); $a$ (semi-major axis), $e$ (ecc.), $\ell$ (angular momentum), $b$ (impact); $\Phi$ (Newtonian potential), $h_{\mu\nu}$ (metric pert.), $\Gamma^\mu_{\ \nu\rho}$ (connection), $\tau$ (proper time).

Validity: §1--2 nonrelativistic ($v\ll c$), conservative $V(x)$; §3 relativistic free particle; §§4--7 weak-field $|2\Phi/c^2|\ll 1$, linearized metric; geometric-optics rays for bending/delay; PN where noted.

\section*{1. Newtonian Mechanics from Variational Principle}

Eq.\ (§1.1) \quad $L(x, v) = \frac{1}{2} m \, |v|^2 - V(x)$

We posit kinetic minus potential so that stationary action reproduces Newton's law for conservative forces.

Eq.\ (§1.2) \quad $S[x] = \int L \, dt$ (Action)

Extremizing $S$ with fixed endpoints ensures endpoint variations vanish, leaving only interior terms.

Eq.\ (§1.3a) \quad $\delta S = 0 \Rightarrow \int \left[ \left(\frac{\partial L}{\partial x_i} - \frac{d}{dt} \frac{\partial L}{\partial v_i} \right) \delta x_i \right] dt = 0$

Eq.\ (§1.3b) \quad $\Rightarrow \frac{d}{dt}\left(\frac{\partial L}{\partial v_i}\right) - \frac{\partial L}{\partial x_i} = 0$ \quad (Euler--Lagrange)

Because $\delta x_i$ are arbitrary functions (zero at endpoints), the bracket must vanish pointwise.

Eq.\ (§1.3c) \quad $\frac{\partial L}{\partial v_i} = m v_i \, , \quad \frac{\partial L}{\partial x_i} = -\frac{\partial V}{\partial x_i} \Rightarrow \frac{d}{dt}\left(m v_i\right) + \frac{\partial V}{\partial x_i} = 0$

Eq.\ (§1.4) \quad $m a_i = - \frac{\partial V}{\partial x_i} = F_i(x)$

Recovered Newton's 2nd law with $F = -\nabla V$. Energy conservation follows by dotting with $v$.

\subsection*{1A. Worked Example \textbullet\ 1D Harmonic Oscillator}

Eq.\ (§1A.1) \quad $V=\frac{1}{2} k x^2 \, , \quad L=\frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2 \Rightarrow \frac{\partial L}{\partial x} = m x \, , \quad \frac{\partial L}{\partial \dot{x}} = -k x$

Eq.\ (§1A.2) \quad $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = m \ddot{x} + k x = 0 \quad \Rightarrow \quad \ddot{x} + (k/m)x=0$

Linear ODE with constant coefficients; sinusoidal solutions exhaust the solution space.

Eq.\ (§1A.3) \quad $x(t)=A \cos(\omega t+\varphi) \, , \quad \omega^2=k/m \, ; \quad E=\frac{1}{2} m \dot{x}^2+\frac{1}{2} k x^2=\frac{1}{2} k A^2$

Matches classical oscillator energy and frequency; basis for small-oscillation limits elsewhere.

\section*{2. Central Forces \textbullet\ Effective Potential}

Eq.\ (§2.1) \quad $T=\frac{1}{2} m(\dot{r}^2 + r^2 \dot{\varphi}^2) \, , \quad V=V(r) \, ; \quad \ell=\frac{\partial L}{\partial \dot{\varphi}} = m r^2 \dot{\varphi} = \text{const.}$

Rotational symmetry $\Rightarrow$ Noether $\rightarrow$ conserved $\ell$; this reduction makes the problem 1D in $r$ with $V_\text{eff}$.

Eq.\ (§2.2) \quad $E=\frac{1}{2} m \dot{r}^2 + \frac{\ell^2}{(2 m r^2)} + V(r) \equiv \frac{1}{2} m \dot{r}^2 + V_\text{eff}(r)$

Eq.\ (§2.3) \quad $V_\text{eff}(r)=V(r)+\frac{\ell^2}{(2 m r^2)} \quad \Rightarrow \quad \text{turning points where } E=V_\text{eff}$

The centrifugal term erects an $r^{-2}$ barrier; bound orbits correspond to wells in $V_\text{eff}$. $\frac{d}{dt}(m\dot{r}) - m \cdot r \dot{\varphi}^2 = - \frac{dV}{dr}$

\subsection*{2A. Worked Example \textbullet\ Orbit via Binet}

Eq.\ (§2A.1) \quad $u=1/r \, , \quad \varphi=\ell/(m r^2)=\ell u^2/m \, ; \quad \dot{r}=-({\ell^2}/{m}) u' \cdot u''$

Eq.\ (§2A.2) \quad Radial eqn: $m \ddot{r} - m r \dot{\varphi}^2 = F_r = - \frac{dV}{dr} \quad \Rightarrow \quad u''+u = -(m/\ell^2)(1/u^2)\frac{dV}{dr}$

Eq.\ (§2A.3) \quad $V=-k/r \Rightarrow \frac{dV}{dr} = k/r^2 = k u^2 \quad \Rightarrow \quad u'' + u = (m k)/\ell^2$

Eq.\ (§2A.4) \quad $u(\varphi) = (m k)/\ell^2 \, [1 + e \cos(\varphi-\varphi_0)] \quad \Rightarrow \quad r(\varphi) = p / [1+e \cos(\cdots)] \, , \quad p=\ell^2/(m k)$

This reproduces the conic-section orbits of Kepler; ellipse for $0<e<1$ with pericenter/apocenter at turning points. Radial Euler--Lagrange: $\frac{d}{dt}(m\dot{r}) - m \cdot r \dot{\varphi}^2 = - \frac{dV}{dr}$

Eq.\ (§2A.1) \quad Define $u = 1/r$. From angular momentum conservation, $\dot{\varphi} = \ell / (m r^2) = \ell u^2/m$.

Rotational symmetry (Noether) $\Rightarrow$ angular momentum $\ell$ is constant. Introducing $u$ simplifies radial motion to $\varphi$-dependence.

Eq.\ (§2A.2) \quad $\dot{r} = dr/dt = (dr/d\varphi) \dot{\varphi} = - (\ell/m) du/d\varphi$

Eq.\ (§2A.3) \quad $\ddot{r} = d \dot{r}/dt = - (\ell^2/m^2) u^2 d^2u/d\varphi^2$

Expressing $\ddot{r}$ in $\varphi$ derivatives allows substitution into the radial equation.

Eq.\ (§2A.4) \quad Radial eqn: $m \ddot{r} - m r \dot{\varphi}^2 = F_r = - dV/dr$

Substitute $r=1/u$ and $\dot{\varphi}$, $\ddot{r}$:

Eq.\ (§2A.5) \quad $u'' + u = - (m/\ell^2)(1/u^2) dV/dr$

For $V(r)=-k/r \Rightarrow dV/dr = k/r^2 = k u^2$:

Eq.\ (§2A.6) \quad $u'' + u = (m k)/\ell^2$

Eq.\ (§2A.7) \quad Solution: $u(\varphi) = (m k)/\ell^2 \, [1 + e \cos(\varphi-\varphi_0)]$

This is Kepler's conic orbit equation; ellipse for $0<e<1$, parabola for $e=1$, hyperbola for $e>1$.

\section*{3. Relativistic Point Particle}

\subsection*{Relativistic Energy--Momentum Relation}

Eq.\ (§3.1) \quad $E^2 = p^2c^2 + m^2c^4$

This is the exact relation between energy, momentum, and rest mass in special relativity. It reduces correctly to both the Newtonian limit and the rest-energy definition.

\vspace{1em}

Eq.\ (§3.2) \quad $E = \gamma mc^2 \, , \quad p = \gamma mv$

These definitions follow from time-translation invariance (Noether's theorem) and Lorentz invariance. They automatically satisfy Eq.\ (§3.1).

\vspace{1em}

\subsection*{Low-velocity expansion}

For $v \ll c$, expand the Lorentz factor:
\begin{equation*}
\gamma = (1 - v^2/c^2)^{-1/2} \approx 1 + \frac{1}{2}\frac{v^2}{c^2} + \frac{3}{8}\frac{v^4}{c^4} + \cdots
\end{equation*}

Substitute into $E = \gamma mc^2$:
\begin{equation*}
E \approx mc^2 + \frac{1}{2}mv^2 + \frac{3}{8}\frac{mv^4}{c^2} + \cdots
\end{equation*}

\subsection*{Interpretation}

\begin{itemize}
\item The first term $mc^2$ is rest energy.
\item The second term $\frac{1}{2}mv^2$ is the Newtonian kinetic energy.
\item The higher-order correction $\frac{3}{8}mv^4/c^2$ shows the first relativistic adjustment.
\end{itemize}

Thus, the Newtonian limit is smoothly recovered, while relativity predicts specific corrections that grow with velocity.

\subsection*{§3 Relativistic Particle --- Energy Expansion}

Eq.\ (§3.3) \quad $E^2 = p^2c^2 + m^2c^4$

This is the exact relativistic energy--momentum relation. It unifies kinetic and rest energy into one invariant expression. Any valid definition of $E$ and $p$ must satisfy this equation.

\vspace{1em}

Eq.\ (§3.4) \quad $\gamma = \frac{1}{\sqrt{1-v^2/c^2}} \approx 1 + \frac{1}{2}\frac{v^2}{c^2} + \frac{3}{8}\frac{v^4}{c^4} + \cdots$

The Lorentz factor expansion in powers of $v/c$. This Taylor series provides the systematic Newtonian limit for small velocities.

\vspace{1em}

Eq.\ (§3.5) \quad $E = \gamma mc^2 \approx mc^2 + \frac{1}{2}mv^2 + \frac{3}{8}\frac{mv^4}{c^2} + \cdots$

Substituting the expansion of $\gamma$ into $E = \gamma mc^2$ gives the hierarchy of terms:

\begin{itemize}
\item Rest energy $mc^2$.
\item Newtonian kinetic energy $\frac{1}{2}mv^2$.
\item Higher-order relativistic corrections such as $\frac{3}{8}mv^4/c^2$.
\end{itemize}

\subsection*{Interpretation}

The Newtonian kinetic energy appears naturally as the first correction to rest energy in the expansion. This shows explicitly that classical mechanics is contained as a limiting case of relativity, with higher-order terms quantifying the deviation as velocity increases.

\section*{4. Gravity Bridge (Reference Import)}

Eq.\ (§4.1) \quad $d^2x^\mu/d\tau^2 + \Gamma^\mu_{\ \nu\rho} (dx^\nu/d\tau) (dx^\rho/d\tau)=0$ \quad (geodesic)

Eq.\ (§4.2) \quad $g_{00}\approx-(1+2\Phi/c^2) \, , \quad \nabla^2\Phi=4\pi G\rho$ \quad (linearized field)

Eq.\ (§4.3) \quad $a=-\nabla\Phi \Rightarrow a(r)=-GM \, r^{-2} \, \hat{r}$ \quad (Newtonian limit)

We import these from the Bridge Appendix; Mechanics only applies them. No duplication here.

Cross-reference: Bridge Appendix § Linearized Field $\rightarrow$ § Geodesics $\rightarrow$ § Newtonian Limit.

\section*{5. Light Bending (Weak Field, Eikonal)}

\textbf{Convention.} $\Phi < 0$near mass; $\phi \equiv \Phi/c^2$; $n = 1 + \eta_0\phi$ with $|\eta_0| = 2$fixed by solar deflection. Bending is toward the mass.

\textbf{Setup (straight path, small deflection).} Parametrize the unperturbed ray by $z$with impact parameter $b$; then $ds \approx dz$.

\textbf{One-line road map.}

\begin{equation*}
\Delta\theta = \int \frac{d\theta}{dz} dz \approx - \frac{2}{c^2} \int \frac{\partial \Phi}{\partial b} dz = - \frac{2}{c^2} \int \frac{\partial}{\partial b}\left(\frac{-GM}{\sqrt{b^2 + z^2}}\right) dz = \frac{4GM}{bc^2} .
\end{equation*}

\textbf{Derivation steps (numbered).}

\begin{itemize}
\item Eq.\ (§5.1) \quad $n(x) = 1 - \frac{2\Phi}{c^2} \, , \quad \ln n \approx - \frac{2\Phi}{c^2} \, , \quad \frac{d\theta}{ds} \approx \nabla_\perp \ln n$.

Why: geometric optics; linearizing $\ln n$ makes the transverse gradient explicit.

\item Eq.\ (§5.2) \quad $\Phi(z) = \frac{-GM}{\sqrt{b^2 + z^2}} \Rightarrow \frac{\partial\Phi}{\partial b} = GM \frac{b}{(b^2 + z^2)^{3/2}} \, .$

\item Eq.\ (§5.3) \quad $\frac{d\theta}{dz} \approx - \frac{2}{c^2} \frac{\partial \Phi}{\partial b} = - \frac{2GM}{c^2} \frac{b}{(b^2 + z^2)^{3/2}} \, .$

\item Eq.\ (§5.4) \quad $\int_{-\infty}^{+\infty} \frac{b \, dz}{(b^2 + z^2)^{3/2}} = \frac{2}{b} \, .$

\item Eq.\ (§5.5) \quad $\alpha = |\Delta\theta| = \frac{4GM}{bc^2}$

\end{itemize}

\textbf{Domain.} Weak field $|2\Phi/c^2|\ll 1$; small deflection; straight-line zeroth order.

\textbf{Worked numeric (solar grazing).}

$G = 6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}$, $M_{\odot} = 1.9885 \times 10^{30}$ kg, $R_{\odot} = 6.9634 \times 10^8$ m, $c = 2.998 \times 10^8$ m/s.

\begin{equation*}
\alpha = \frac{4GM_{\odot}}{R_{\odot}c^2} \approx 1.75 \, \text{arcsec}.
\end{equation*}

Matches the 1919 eclipse measurements.

Worked numeric (solar limb): $\alpha \approx 1.75$ arcsec.

\section*{6. Travel-Time (Shapiro) Delay}

Eq.\ (§6.1) \quad $t=(1/c) \int n \, ds \, , \quad n=1-2\Phi/c^2 \Rightarrow \Delta t = (2/c^3) \int \Phi \, ds$

Treat the path as straight at zeroth-order; curvature enters only via $\Phi$ in the integrand to first order. $\Delta t \approx (2GM/c^3)$

Eq.\ (§6.2) \quad For $\Phi=-GM/r$, $\int dz/r = \ln[(z+r)/b] \approx \Delta t=(2GM/c^3) \ln \Lambda$

The logarithm captures how delay grows with alignment (smaller $b$) and long endpoints.

-- Numeric (Earth--Mars near conjunction): $\Delta t \approx 123.6$ $\mu$s.

Eq.\ (§6.2) \quad $\Delta t = (2GM/c^3) \ln \left( (4 \, r_1 r_2)/b^2 \right)$

Worked Example (Earth--Mars conjunction):

Take $r_1=1$ AU, $r_2=1.52$ AU, $b=R_{\odot}$.

Computation: $\Delta t \approx 123.6$ $\mu$s.

Matches radio ranging tests, confirming GR time delay.

Note: logarithm argument is dimensionless; far-field reduces to $\ln[(4r_1r_2)/b^2]$.

\section*{7. Gravitational Redshift (Clock Drift)}

Eq.\ (§7.1) \quad $d\tau=\sqrt{-g_{00}} \, dt \approx (1+\Phi/c^2) \, dt \Rightarrow \nu_{\text{obs}}/\nu_{\text{src}} \approx 1-\Phi/c^2$

Eq.\ (§7.2) \quad $\Delta\nu/\nu=\Delta\Phi/c^2 \, ; \quad \text{near Earth } \Phi\approx g h \Rightarrow \Delta\nu/\nu\approx \Delta h / c^2$

$\nu_{\text{obs}} / \nu_{\text{emit}} = \sqrt{1 + 2\Phi/c^4}$. Time runs slower deeper in a potential well. This formula shows photons lose frequency climbing out. Verified in Pound--Rebka.

This links time dilation directly to potential difference; GPS and tower tests validate it.

-- Numeric ($\Delta h=22.5$ m): $\Delta\nu/\nu \approx 2.455e-15$.

Eq.\ (§7.2) \quad $\Delta\nu/\nu \approx g \, \Delta h / c^2$

Worked Example (Pound--Rebka Tower):

For $\Delta h=22.5$ m: $\Delta\nu/\nu = 2.455e-15$.

This fractional shift matches laboratory M\"ossbauer experiments.

\section*{8. Perihelion (Periapsis) Precession --- Sketch}

Eq.\ (§8.1) \quad $u''+u = GM/\ell^2 + 3 GM \, u^2 / c^2$ \quad (leading PN correction)

The extra $3GM \, u^2/c^2$ arises from relativistic corrections to the effective potential.

Eq.\ (§8.2) \quad $\Delta \omega = 6\pi GM / (a (1-e^2) c^2)$ \quad per orbit

Worked Example (Mercury): $a=5.79\times10^{10}$ m, $e=0.206$

Computation: $\Delta \omega = 43.0$ arcsec/century.

Observed anomaly $43''$/century, precisely matched by this correction.

Keeping only the lowest correction term in $u$ produces the secular advance measured for Mercury.

\section*{9. Symbol--Units Table (SI)}

c: speed of light; $\text{m}\cdot\text{s}^{-1}$

G: gravitational constant; $\text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2}$

k: bridge coupling; (var.)

x: position; m

v: velocity; $\text{m}\cdot\text{s}^{-1}$

a: acceleration; $\text{m}\cdot\text{s}^{-2}$

m: mass; kg

p: momentum; $\text{kg} \cdot \text{m} \cdot \text{s}^{-1}$

F: force; N

L: Lagrangian; J

H: Hamiltonian; J

E: total energy; J

a: semi-major axis; m

e: eccentricity; (-)

$\ell$: angular momentum; $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1}$

b: impact parameter; m

$\Phi$: Newtonian potential; $\text{m}^2 \cdot \text{s}^{-2}$

$h_{\mu\nu}$: metric perturbation; (-)

$\Gamma^\mu_{\ \nu\rho}$: connection; $\text{m}^{-1}$

$\tau$: proper time; s

\section*{10. Cross-Checks \& Pointers}

(10.1) \quad Geodesic Eqn: \quad $d^2x^\mu/d\tau^2 + \Gamma^\mu_{\ \nu\rho} \, dx^\nu/d\tau \, dx^\rho/d\tau = 0$

(10.2) \quad Linearized Limit: \quad $g_{00} \approx -(1 + 2\Phi/c^2)$

(10.3) \quad Newtonian Limit: \quad $\nabla^2\Phi = 4\pi Gp$

These are imported from the Bridge Appendix. Here we only reference them to show consistency: the Newtonian and relativistic views of gravity merge smoothly.

Mechanics Walkthrough: each section here extends the matching section there (numbered by §).

Bridge Appendix: import points § Linearized Field $\rightarrow$ § Geodesics $\rightarrow$ § Newtonian Limit (no re-derivation).

\section*{11. Mechanics Checklist (Cross-References)}

\checkmark\ \textbf{Newton's Laws \& Variational Form} --- (Eqs.\ §1.1--§1.5)

$\rightarrow$ Derived in Walkthrough §0 and Appendix §1.

\vspace{0.5em}

\checkmark\ \textbf{Lagrangian \& Euler--Lagrange Equations} --- (Eqs.\ §1.6--§1.10)

$\rightarrow$ Walkthrough §0; explicit radial form inserted (§2.0).

\vspace{0.5em}

\checkmark\ \textbf{Central Force Dynamics (Binet's Equation)} --- (Eqs.\ §2.1--§2.5)

$\rightarrow$ Derived in Walkthrough §2A; expanded in Appendix Insert A.

\vspace{0.5em}

\checkmark\ \textbf{Effective Potential \& Orbital Solutions} --- (Eqs.\ §2.6--§2.10)

$\rightarrow$ Worked in Walkthrough §2B; illustrated with conic orbits.

\vspace{0.5em}

\checkmark\ \textbf{Harmonic Oscillator} --- (Eqs.\ §2.11--§2.15)

$\rightarrow$ Walkthrough §2C; algebra carried into Appendix §2C.

\vspace{0.5em}

\checkmark\ \textbf{Relativistic Particle Energy--Momentum} --- (Eqs.\ §3.1--§3.5)

$\rightarrow$ Derived in Walkthrough §3; expanded in Appendix §3.

\vspace{0.5em}

\checkmark\ \textbf{Gravity Bridge Import} --- (Eq.\ §4.1 pointer)

$\rightarrow$ Imported from Mathematical Bridge Appendix; not rederived here.

\vspace{0.5em}

\checkmark\ \textbf{Light Bending} --- (Eqs.\ §5.1--§5.5)

$\rightarrow$ Derived in Walkthrough §5; fully expanded in Appendix Insert C.

\vspace{0.5em}

\checkmark\ \textbf{Shapiro Time Delay} --- (Eqs.\ §6.1--§6.5)

$\rightarrow$ Derived in Walkthrough §6; full worked integral in Appendix Insert D.

\vspace{0.5em}

\checkmark\ \textbf{Gravitational Redshift} --- (Eqs.\ §7.1--§7.3)

$\rightarrow$ Derived in Walkthrough §7; full expansion in Appendix Insert E.

\vspace{0.5em}

\checkmark\ \textbf{Perihelion Precession} --- (Eqs.\ §8.1--§8.5)

$\rightarrow$ Derived in Walkthrough §8; perturbative expansion in Appendix Insert F.

\end{document}
