Mechanical Similarity¶

A homogeneous potential lets a classical trajectory generate scaled solutions when space and time are dilated by matched powers.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13427
Domain group: Natural Sciences
Origin domain: Physics
Subdomain: Classical Mechanics → Physics

Core Idea¶

Mechanical similarity is a conditional scaling principle for classical motion. If a fixed-parameter autonomous system has ordinary quadratic kinetic energy and a potential satisfying \(U(\lambda q)=\lambda^kU(q)\), a solution \(q(t)\) generates another by scaling position by \(\lambda\) and time by \(\lambda^{1-k/2}\). The trajectories have similar shapes and obey the same equations; corresponding velocities scale as \(\lambda^{k/2}\) and energies as \(\lambda^k\).[^{ref-fd3f97636835][}ref-63eb17c692d9]

Scope of Application¶

For an ideal oscillator with exactly quadratic potential (\(k=2\)), changing amplitude does not change the period at fixed mass and stiffness. For a fixed-coupling inverse-distance gravitational orbit (\(k=-1\)), orbital time scales as size to the \(3/2\) power, yielding the familiar period-squared/size-cubed relation. These are unlike applications of the same homogeneous-potential test, not claims about damped or driven motion or a central mass that changes with scale.[^{ref-fd3f97636835][}ref-63eb17c692d9]

Clarity¶

The principle means more than geometrically similar paths: the rescaled path with its matched time coordinate is another solution. It means less than “all quantities are invariant”: speed and energy generally change. It is not generic dimensional analysis or every kind of dynamical similitude; fluid-model testing can additionally require dimensionless force-ratio matching such as Reynolds-number equality.[^{ref-63eb17c692d9][}ref-2d1f2b29b276]

Manages Complexity¶

Rather than solve each scaled trajectory from scratch, check the potential's homogeneity degree. Scaled acceleration has factor \(\lambda/\tau^2\) and scaled potential force has factor \(\lambda^{k-1}\); equating them gives \(\tau=\lambda^{1-k/2}\). The resulting exponents predict corresponding times and energies while making the assumptions easy to audit. Mixed-degree potentials or extra fixed timescales require a fresh check.[^{ref-fd3f97636835][}ref-63eb17c692d9]

Abstract Reasoning¶

First hold the mechanical parameters fixed and verify a single \(k\) applies over the modeled range. Then scale every relevant coordinate together and test whether the matched time dilation preserves the equations. Finally derive ratios for periodic or transit times, velocities and energies. An exact quadratic oscillator supports exact amplitude-independent period; a small-angle approximation to a pendulum supports only a regime-limited approximation.[^ref-63eb17c692d9]

Knowledge Transfer¶

The transferable reasoning is “homogeneity degree → compatible space/time map → new solution and observable ratios.” The oscillator and Kepler problem instantiate it with different forces and exponents. Live prime Scale Invariance captures the general preserved-under-rescaling skeleton, while the proposed strict child edge records that mechanical similarity adds a classical kinetic/potential mechanism. A mere observed power law or visual resemblance is insufficient.[^{ref-fd3f97636835][}ref-63eb17c692d9]

[^ref-fd3f97636835]: L. D. Landau and E. M. Lifshitz, Mechanics, 3^rd ed. (1976), §10, equations (10.1)–(10.3), textbook scan. Indexed text was cross-checked with Fowler because direct retrieval was temporarily unavailable. [^ref-63eb17c692d9]: Michael Fowler, University of Virginia, “Mechanical Similarity and the Virial Theorem”, pp. 1–3 and author's accessible §5.2 text. [^ref-2d1f2b29b276]: NASA Glenn Research Center, “Similarity Parameters”.

Relationships to Other Abstractions¶

Current abstraction Mechanical Similarity Domain-specific

Parents (1) — more general patterns this builds on

Mechanical Similarity is a kind of Scale Invariance Prime

The equations and solution family retain their form under coordinated space–time dilation.

Hierarchy paths (2) — routes to 2 parentless roots

Mechanical Similarity → Scale Invariance → Invariance

Show alternative path (1)

Neighborhood in Abstraction Space¶

Mechanical Similarity sits in a sparse region of the domain-specific corpus (64^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Dynamical Systems & Differential Structures (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08