A Self-Consistent Derivation Chain of Conservation Laws

Geometric Progressive Conservation Law（GPCL）

(Strictly valid under no external force, no external torque, one-dimensional / free curvilinear motion, consistent with the geometric view that a straight line is part of an arc.)

0. Geometric Axiom (Core Idea)

Any straight-line motion may be regarded as circular motion with radius R\to\infty.

Relation between linear speed and radius of curvature:

v=R\omega

1. Conservation of Energy ⇒ Conservation of Kinetic Energy

For a closed system with no work done by external forces: total energy is conserved

\frac{dE}{dt}=0

In free motion with no potential energy variation, E=T, so we directly obtain:

\frac{dT}{dt}=0

Kinetic energy is conserved.

2. Conservation of Kinetic Energy ⇒ Conservation of Momentum

Kinetic energy:

T=\frac12mv^2

If T=\text{const} and mass m is constant,

\Rightarrow v=\text{const}

Momentum:

p=mv

\Rightarrow \frac{dp}{dt}=0

Momentum is conserved.

3. Conservation of Momentum + Geometry ⇒ Conservation of Angular Momentum

Definition of angular momentum (relative to the center of curvature):

L=Rp

For a given trajectory, the radius of curvature R is constant. Since p=\text{const} has been established,

\Rightarrow \frac{dL}{dt}=0

Angular momentum is conserved.

The derivation chain is fully self-consistent within this framework:

\boxed{

\text{Conservation of Energy} \Rightarrow

\text{Conservation of Kinetic Energy} \Rightarrow

\text{Conservation of Momentum}

\xrightarrow{\text{Straight Line = Arc / }~L=Rp}

\text{Conservation of Angular Momentum}

}

Using a one-way logical chain, the three conservation laws are linked together.

1. Conservation of Energy

2. ⇒ Conservation of Kinetic Energy (free particle)

3. ⇒ Conservation of Momentum

4. ⇒ Further geometric unification via a straight line as a large-radius arc

5. ⇒ Conservation of Angular Momentum

This implies:

Three originally independent conservation laws become hierarchical manifestations of a single origin.

Structurally, this constitutes unification.

Schematic symbolic chain:

E\to T\to p \xrightarrow{~L=Rp~} L

Within Lagrangian Mechanics

Any curvilinear motion can be divided into infinitely many infinitesimal straight segments.

A straight line is part of an arc; in the infinitesimal limit, arc ≈ straight line.

Within each infinitesimal segment ds:

1. Motion is approximately uniform straight-line motion

2. Therefore momentum is conserved: \vec p=\text{const}

3. Taking the moment about the origin:

d\vec L = \vec r \times d\vec p

4. Since d\vec p=0 within the infinitesimal segment,

d\vec L=0

5. The entire curve is composed of infinitely many such segments ⇒ angular momentum is conserved.

Derivation:

Conservation of Momentum ⇒ Conservation of Angular Momentum

GPCL：Geometric Progressive Conservation Law

Final Complete Chain

\boxed{

\text{Conservation of Energy}

\Rightarrow

\text{Conservation of Kinetic Energy}

\Rightarrow

\text{Conservation of Momentum}

\Rightarrow

\text{Conservation of Angular Momentum}

}