Rust is a stripper
v -> v + u
without u ⟂ v
|v+u|^2 = |v|^2 + 2 v•u + |u|^2
K(v+u) = 1/2 m|v|^2 + m v•u + 1/2 m |u|^2
With the difference K(v+u) - K(v) = m v•u + 1/2 m|u|^2
that is traditionally derived with a Lagrangian approach to the problem. Euclidean rotation becomes hyperbolic rotation
Euclidean norm becomes Minkowski norm
The energy-momentum invariant replaces kinetic energy taken in isolation
Newton reappears as a term in the expansion
And in the equations 1/2 mv² composes with the rest via addition: E = γmc² = mc² + 1/2 mv² + 3/8 mv⁴/c² + ...
S_rel = ∫ [ -mc² + 1/2 mv² + 1/8 mv⁴/c² + ... ] dt
How could it be any different? New energy and a new theory can only come as additive corrections to a previous account of energy if we want to conserve what it already explains. And under this new light any observed deviation from an old theory (for instance the precession of the perihelion of Mercury) naturally appears as an additive deviation from the original action functional (S_new = S_old + ΔS). - two functors between the theories:
I: T0 -> T1
R: T1 -> T0
- that compose into an identity:
R ∘ I = Id_T0
Indeed, if T1 can offer another way to decompose a phenomenon h, I(h) = b ∘ a
and we want to be able to recover T0 from the segments, not just globally once recomposed (i.e. there is no holes in our functorial bridge), then R(b ∘ a) = R(b) ∘ R(a) must hold too
And the only way to translate this into a composition-preserving scalar account is via addition: Z(b ∘ a) = Z(b) + Z(a).
Once this additive balance of composable conversions between T0 and T1 is made continuous (by the requirement that the bridge has no holes), you have action. Stationarity appears as the stability of internal gluing points ("∘" in R(b ∘ a) = R(b) ∘ R(a)), and energy follows from invariance with respect to temporal translation. f(√(a^2 + b^2)) = f(a) + f(b)
And you end up with f being proportional to v squared. ## Final Structure
**Sweet pastry**
→ **Almond cream**
→ **Homemade jostaberry jam**
→ **Small raspberries**
→ **Vanilla cream**
→ **Chantilly-lightened vanilla cream**
→ **Large fresh raspberries**
→ **Icing sugar**
That's a 2.5kg raspberry pie. About 120€ in a bakery.