Classical Mechanics Basis

The Degeneracy Problem

Two classic pairs share SI dimensions:

Pair	Shared Dimension	Hidden Qualifier
Torque vs Energy	ML²T⁻²	Angle (per radian)
Surface tension vs Spring constant	MT⁻²	Geometric character

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Torque vs Energy

The Degeneracy

Quantity	SI Unit	SI Dimension
Energy	Joule (J)	ML²T⁻²
Torque	Newton-meter (N·m)	ML²T⁻²

Same dimension. Yet: - Energy is a scalar (state function) - Torque is a pseudovector (force × lever arm × sin θ) - Torque is energy per unit angle: τ = dW/dθ

Extended Basis

from ucon.basis import Basis, BasisComponent, Vector
from ucon.core import Dimension, Unit
from fractions import Fraction

MECHANICS = Basis("Mechanics", [
    BasisComponent("mass", "M"),       # 0
    BasisComponent("length", "L"),     # 1
    BasisComponent("time", "T"),       # 2
    BasisComponent("angle", "A"),      # 3 — the hidden qualifier
])

Dimensional Vectors

Quantity	Vector	Interpretation
Energy	M¹L²T⁻²A⁰	Work done
Torque	M¹L²T⁻²A⁻¹	Work per radian
Angle	M⁰L⁰T⁰A¹	Rotation measure
Angular velocity	M⁰L⁰T⁻¹A¹	Radians per second
Angular momentum	M¹L²T⁻¹A¹	Rotational inertia × ω

Implementation

# Dimensions
energy = Dimension(
    vector=Vector(MECHANICS, (1, 2, -2, 0)),
    name="energy"
)  # M¹L²T⁻²A⁰

torque = Dimension(
    vector=Vector(MECHANICS, (1, 2, -2, -1)),
    name="torque"
)  # M¹L²T⁻²A⁻¹

angle = Dimension(
    vector=Vector(MECHANICS, (0, 0, 0, 1)),
    name="angle"
)  # A¹

# Units
joule = Unit(name="joule", shorthand="J", dimension=energy)
newton_meter = Unit(name="newton_meter", shorthand="N·m", dimension=torque)
radian = Unit(name="radian", shorthand="rad", dimension=angle)

Dimensional Algebra

Torque × Angle = Energy:

Work = τ × θ
M¹L²T⁻²A⁰ = (M¹L²T⁻²A⁻¹) × (A¹)  ✓

Energy / Angle = Torque:

τ = dW/dθ
M¹L²T⁻²A⁻¹ = (M¹L²T⁻²A⁰) / (A¹)  ✓

Safety

joule(100) + newton_meter(50)
# raises: incompatible dimensions (A⁰ vs A⁻¹)

# Correct: convert torque to energy via angle
work = newton_meter(50) * radian(2)  # → 100 J

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Surface Tension vs Spring Constant

The Degeneracy

Quantity	SI Unit	SI Dimension	Physical Meaning
Spring constant (k)	N/m	MT⁻²	Force per displacement (1D)
Surface tension (γ)	N/m	MT⁻²	Force per length along interface (2D)

Both are N/m = kg/s², but: - Spring constant: restoring force per unit displacement - Surface tension: force per unit length of edge on a surface (or energy per area)

Alternative View

Quantity	Alternative Form	Shows
Spring constant	N/m	Force / length (1D linear)
Surface tension	J/m²	Energy / area (2D interfacial)

J/m² = (kg·m²/s²)/m² = kg/s² = N/m — same dimension.

Extended Basis

MECHANICS_EXTENDED = Basis("Mechanics-Extended", [
    BasisComponent("mass", "M"),
    BasisComponent("length", "L"),
    BasisComponent("time", "T"),
    BasisComponent("angle", "A"),
    BasisComponent("interface", "I"),    # interfacial/surface character
])

Dimensional Vectors

Quantity	Vector	Interpretation
Spring constant	M¹L⁰T⁻²A⁰I⁰	Linear restoring force
Surface tension	M¹L⁰T⁻²A⁰I¹	Interfacial energy density
Interface factor	M⁰L⁰T⁰A⁰I¹	Surface/interface qualifier

Implementation

spring_constant_dim = Dimension(
    vector=Vector(MECHANICS_EXTENDED, (1, 0, -2, 0, 0)),
    name="spring_constant"
)  # M¹T⁻²I⁰

surface_tension_dim = Dimension(
    vector=Vector(MECHANICS_EXTENDED, (1, 0, -2, 0, 1)),
    name="surface_tension"
)  # M¹T⁻²I¹

# Units
newton_per_meter = Unit(
    name="newton_per_meter",
    shorthand="N/m",
    dimension=spring_constant_dim
)

joule_per_m2 = Unit(
    name="joule_per_square_meter",
    shorthand="J/m²",
    dimension=surface_tension_dim
)

Physical Context

Spring constant: - Hooke's law: F = -kx - k has units N/m - Describes 1D elastic deformation

Surface tension: - Capillary force: F = γL (force along edge of length L) - γ has units N/m or equivalently J/m² - Describes 2D interfacial energy - Examples: water-air (~72 mN/m), mercury-air (~486 mN/m)

Why It Matters

In multiphysics simulations (e.g., microfluidics), both appear:

# Droplet on a spring (hypothetical)
spring_force = k * displacement       # N
surface_force = gamma * perimeter     # N

# Without dimensional distinction:
total = k + gamma  # Dimensionally valid in SI, physically nonsense

# With extended basis:
total = k + gamma  # raises: incompatible (I⁰ vs I¹)

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Stiffness Variants

The spring constant degeneracy extends to other stiffness measures:

Quantity	SI Dimension	Physical Context
Spring constant	MT⁻²	Translational spring
Torsional stiffness	ML²T⁻²A⁻¹	Rotational spring (torque per radian)
Bending stiffness (EI)	ML³T⁻²	Beam flexure
Surface tension	MT⁻²	Interface energy

With the extended basis, all are distinct:

# Torsional stiffness: torque per angle
torsional_stiffness = Dimension(
    vector=Vector(MECHANICS_EXTENDED, (1, 2, -2, -2, 0)),
    name="torsional_stiffness"
)  # M¹L²T⁻²A⁻² (N·m per radian)

---

Projection to SI

When you need SI compatibility:

MECHANICS_TO_SI = ConstantBoundBasisTransform(
    source=MECHANICS,
    target=SI,
    matrix=(
        (1, 0, 0, ...),  # M → M
        (0, 1, 0, ...),  # L → L
        (0, 0, 1, ...),  # T → T
        (0, 0, 0, ...),  # A → dimensionless (dropped)
    ),
    bindings=(
        ConstantBinding(
            source_component=MECHANICS["angle"],
            target_expression=Vector(SI, (0, 0, 0, ...)),  # dimensionless
            constant_symbol="rad",
            exponent=Fraction(1),
        ),
    ),
)

The binding records that angle was collapsed to dimensionless via the radian.

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Summary

Degeneracy	Hidden Dimension	Resolution
Torque vs Energy	Angle (A)	Torque is M¹L²T⁻²A⁻¹
Surface tension vs Spring constant	Interface (I)	Surface tension is M¹T⁻²I¹
Torsional vs Linear stiffness	Angle (A)	Torsional is M¹L²T⁻²A⁻²

All three are instances of the same pattern: a geometric qualifier (angle, interface) is treated as dimensionless in SI but carries physical meaning that affects how quantities combine.