Dynamics · Energy and Action

01Premises and energy reference · Starting Assumptions

02From momentum to work · Work Integral

Differentiate $p=m_0\gamma v$:

Kinetic energy from rest to speed $v$ is therefore:

$w$ is the integration speed. For a general change of direction, $\mathbf v\cdot d\mathbf p$ retains only the speed-changing component. Under the stated conditions, kinetic energy is a function of speed.

03Essential integration steps · Intermediate Steps

Set $z=1-w^2/c^2$, giving $w\,dw=-c^2dz/2$. The lower limit $w=0$ gives $z=1$, and the upper limit $w=v$ gives $z=1-v^2/c^2$:

This is the unnumbered calculation following manuscript Eq. (2.22); it does not work backwards from a standard energy formula.

Using $\gamma=1+\beta^2/2+3\beta^4/8+O(\beta^6)$ recovers Newtonian kinetic energy and its next correction. Both coefficients come from the expansion of one function.

04Rest energy and total energy · Rest and Total Energy

The work integral determines energy differences and cannot fix $E_0$ by itself. With this normalization:

Thus $E=m_0\gamma c^2$ combines the kinetic-energy result and A2. A microscopic theory must compute $E_0$ and $m_0$ from the same stable structural solution to test the normalization; it is not a fact established by this integral alone.

05Energy–momentum consistency · Algebraic Consistency

This is an algebraic consistency relation after eliminating $v$. In this chapter it follows $p$ and $E$; a presumed Minkowski invariant does not replace ASD’s constitutive derivation.

06Free-motion action · Free Lagrangian

Require the free translational Lagrangian to satisfy $\partial\mathcal L_0/\partial\mathbf v=\mathbf p$. Integrate over speed:

Fix the constant with $\mathcal L_0(0)=-E_0=-m_0c^2$, giving:

Direct differentiation recovers $m_0\gamma\mathbf v$. The Legendre transform gives

The website uses $\mathcal L_0$ to distinguish the Lagrangian from length $L$; it corresponds to $L_0$ in manuscript Eq. (2.24b). Its low-speed expansion is

Chapter 2’s later orbital calculation must retain the $v^4$ term alongside gravitational terms of the same order. This module supplies that interface without developing the orbital approximation here.

07Physical meaning, scope and review · Interpretation and Review

Kinetic energy describes reconfiguration of the moving structure; rest energy describes the static core and accompanying structure. Matching mathematical forms do not remove these distinct physical roles.

For $0\leq v<c$ and positive fixed $m_0$, $E_k\geq0$, $E(0)=E_0$ and $dE_k/dt=\mathbf F\cdot\mathbf v$. Divergence as $v\to c$ is a feature within this model, not an account of crossing that boundary. Radiation, internal excitation and background changes require an expanded energy balance.

Human review concerns whether inertia closure and static energy derived from one microscopic structure support the common normalization. That question is distinct from the correctness of the definite integral; see OP-DYN-01 (Chinese) · Chinese original.

08Sources and experimental interface · Sources

This page follows Chapter 2 §III, final manuscript §III, Eqs. (2.22)–(2.24b) and the unnumbered intermediate steps. Universal coupling of energy, inertial and gravitational mass is discussed in Experimental Relations.