Linear harmonic-oscillator model

Mathematical statement

A linear harmonic oscillator has positive mass m and positive spring coefficient k, with displacement x satisfying m*x″+k*x=0. The ratio k/m is the angular-frequency-square candidate. A general closed-form solution requires separately established trigonometric-function and derivative semantics.

Representative formula (exact TeX)

H(x)=0

Assumptions

• The model requires positive rational m and k. Exercises check only model qualification, k/m, polynomial-candidate residuals at specified points and a single-state energy expression.
• A single-point energy value proves no trajectory-wide conservation; no sin/cos closed-form solution or stability proof is performed here.

Symbol scope

• H — oscillator-equation residual
• x — displacement function

Intuition, definition, and misconceptions

Positivity of m and k qualifies this model. The exact ratio k/m is called only angular frequency squared here.

For a polynomial candidate, recompute x and x″ at the specified point to obtain a residual. A single zero residual does not make it a solution on an entire interval.

The state-energy expression E=m*v²/2+k*x²/2 supplies a value at one state. Conservation separately requires proving its derivative is zero along a full solution trajectory.

Do not assert a general sin/cos solution without the supporting derivative semantics or present a single energy value as a conservation theorem.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Ordinary differential equation — The oscillator equation is a second-order ODE.
• Differentiating polynomial sums, products, and compositions — Candidate residuals require a second derivative.
• Natural-number powers and principal square roots — The angular-frequency square and energy-square terms use powers.

Verification boundary

Only model qualification and exact finite slices are checked; no closed-form solution, stability or energy-conservation theorem is claimed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add trigonometric derivatives, the general closed-form solution, phase space and energy invariants.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.