Sicm

SICM Skill - Structure and Interpretation of Classical Mechanics

"The variational formulation of mechanics is coordinate-independent and leads to a systematic computational approach." — Sussman & Wisdom

Overview

SICMUtils is a Clojure/​ClojureScript implementation of scmutils—the computer algebra system from MIT's SICM course. It provides:

• Generic arithmetic extensible across numeric types
• Symbolic computation with automatic differentiation
• Functional differential geometry on manifolds
• Lagrangian/​Hamiltonian mechanics with Noether's theorem

Trit: 0 (ERGODIC - Coordinator between SICP abstractions and physical mechanics)

Architecture

┌────────────────────────────────────────────────────────────┐
│                    sicmutils.env                           │
│           (batteries-included namespace)                   │
└─────────────────────────┬──────────────────────────────────┘
                          │
    ┌─────────────────────┼─────────────────────┐
    │                     │                     │
┌───▼────────┐      ┌─────▼──────┐       ┌──────▼─────┐
│ mechanics/ │      │  calculus/ │       │ numerical/ │
│ Lagrangian │      │  manifold  │       │ integration│
│ Hamiltonian│      │ vector-fld │       │ optimize   │
│ Noether    │      │ form-field │       │ ODE solve  │
└────────────┘      └────────────┘       └────────────┘
        │                 │                    │
        └─────────────────┼────────────────────┘
                          │
    ┌─────────────────────┼─────────────────────┐
    │                     │                     │
┌───▼────────┐      ┌─────▼──────┐       ┌──────▼─────┐
│ simplify/  │      │ operator   │       │ derivative │
│ polynomial │      │ composition│       │ D, partial │
│ rules      │      │ algebra    │       │ dual nums  │
└────────────┘      └────────────┘       └────────────┘
        │                 │                    │
        └─────────────────┼────────────────────┘
                          │
    ┌─────────────────────┼─────────────────────┐
    │                     │                     │
┌───▼────────┐      ┌─────▼──────┐       ┌──────▼─────┐
│  generic   │      │   value    │       │ structure  │
│  +,-,*,/   │      │  protocol  │       │  up/​down   │
│  extensible│      │ kind,zero? │       │  vectors   │
└────────────┘      └────────────┘       └────────────┘

Core Namespaces (GF(3) Decomposition)

MINUS (-1): Verification/​Structure

ERGODIC (0): Coordination/​Transformation

PLUS (+1): Generation/​Execution

Key Concepts

1. Generic Arithmetic (Extensible Tower)

(require '[sicmutils.env :as e])

;; Works on numbers
(e/​* 3 4) ;=> 12

;; Works on symbols
(e/​* 'x 'y) ;=> (* x y)

;; Works on functions
((e/​* e/​sin e/​cos) 'x) ;=> (* (sin x) (cos x))

;; Extend to custom types via protocols

2. Automatic Differentiation (D Operator)

;; D is the derivative operator
(e/​D e/​sin)          ;=> cos
((e/​D e/​sin) 'x)     ;=> (cos x)

;; Higher derivatives
((e/​expt e/​D 2) e/​sin)  ;=> -sin

;; Partial derivatives
((e/​partial 0) (fn [[x y]] (e/​* x y)))  ; ∂/∂x(xy) = y

3. Structures (Up/​Down Vectors)

;; Up = contravariant (coordinate-like)
(e/​up 1 2 3)

;; Down = covariant (momentum-like)
(e/​down 1 2 3)

;; Lagrangian state: (t, q, q̇)
(e/​up 't (e/​up 'x 'y) (e/​up 'vx 'vy))

;; Hamiltonian state: (t, q, p)
(e/​up 't (e/​up 'x 'y) (e/​down 'px 'py))

4. Lagrangian Mechanics

(require '[sicmutils.mechanics.lagrange :as L])

;; Free particle Lagrangian: L = T - V = ½mv²
(defn L-free-particle [mass]
  (fn [[_ _ v]]
    (e/​* (e// 1 2) mass (e/​square v))))

;; Euler-Lagrange equations
(L/​Lagrange-equations (L-free-particle 'm))

;; Action functional
(L/​Lagrangian-action L q t1 t2)

5. Manifolds and Differential Geometry

(require '[sicmutils.calculus.manifold :as m])
(require '[sicmutils.calculus.coordinate :as c])

;; Define R² with rectangular coordinates
(def R2 (m/​make-manifold m/​Rn 2))
(def rect (c/​coordinate-system-at 'rectangular :origin R2))

;; Vector fields
(def d:dx (c/​vector-field rect 0))
(def d:dy (c/​vector-field rect 1))

;; Differential forms
(def dx (c/​oneform-field rect 0))
(def dy (c/​oneform-field rect 1))

;; Lie derivative, exterior derivative, etc.

6. Simplification

(require '[sicmutils.simplify :as s])

;; Simplify expressions
(s/​simplify (e/​+ (e/​square (e/​sin 'x)) 
                  (e/​square (e/​cos 'x))))
;=> 1

;; Render to LaTeX
(e/​->TeX (e/​simplify expr))

;; Render to infix
(e/​->infix (e/​simplify expr))

SICP → SICM Bridge

The progression from SICP to SICM:

Shared Patterns

;; SICP: deriv procedure
(define (deriv exp var) ...)

;; SICM: D operator on symbolic expressions
((D (literal-function 'f)) 'x)

Incremental Update Schema

For semantic world closure with incremental updates:

(def sicm-acset-schema
  {:objects [:Namespace :Symbol :Expression :Operator]
   :morphisms {:depends [:Namespace :Namespace]
               :defines [:Namespace :Symbol]
               :uses [:Expression :Symbol]
               :applies [:Operator :Expression]}
   :attributes {:trit [:Namespace :GF3]
                :kind [:Symbol :Keyword]
                :arity [:Operator :Int]}})

;; Incremental update: only recompute changed subgraph
(defn incremental-update [delta-expr sicm-graph]
  (let [affected (transitive-deps delta-expr sicm-graph)
        unchanged (set/​difference (all-nodes sicm-graph) affected)]
    (-> sicm-graph
        (invalidate-nodes affected)
        (recompute-nodes affected)
        (verify-gf3-conservation))))

World Closure

SICM completes the semantic world by bridging:

SICP (abstraction)  ←→  SICM (mechanics)  ←→  FDG (geometry)
     λ-calculus              variational         manifolds
     procedures              Lagrangians         vector fields
     data abstraction        state tuples        differential forms
     interpreters            simplifiers         coordinate systems

---

End-of-Skill Interface

Commands

# Add dependency
clj -Sdeps '{:deps {sicmutils/​sicmutils {:mvn/​version "0.23.0"}}}'

# REPL with SICMUtils
clj -M -e "(require '[sicmutils.env :refer :all])"

# ClojureScript (browser/​Node)
shadow-cljs watch sicm-app

# With Babashka (limited - no symbolic)
bb -e "(require '[sicmutils.value :as v]) (v/​zero? 0)"

Integration with Gay.jl / GF(3)

Namespace → Trit Mapping

;; Use Gay.jl colors to track namespace triads
(def namespace-trits
  {"sicmutils.value"       -1  ; foundation
   "sicmutils.generic"     -1  ; arithmetic
   "sicmutils.structure"   -1  ; data
   "sicmutils.operator"     0  ; coordination
   "sicmutils.expression"   0  ; symbolic
   "sicmutils.derivative"   0  ; calculus bridge
   "sicmutils.lagrange"    +1  ; mechanics
   "sicmutils.hamilton"    +1  ; dynamics
   "sicmutils.numerical"   +1  ; execution
   })

;; GF(3) conserved per layer:
;; 3×(-1) + 3×(0) + 3×(+1) = 0 ✓

Color Coding for Expressions

;; Map expression types to Gay.jl palette
(defn expr-color [expr seed]
  (let [kind (value/​kind expr)
        idx (case kind
              ::number 1
              ::symbol 2
              ::structure 3
              ::operator 4
              ::function 5
              6)]
    (gay/​color-at seed idx)))

References

• SICM Book (Sussman & Wisdom)
• SICMUtils GitHub
• Emmy (successor)
• FDG Book (Functional Differential Geometry)
• cljdoc Reference

---

Skill: sicm
Trit: 0 (ERGODIC - bridges SICP abstraction to physical mechanics)
GF(3) Triad: sicp (-1) ⊗ sicm (0) ⊗ fdg (+1) = 0
Status: Ready for incremental semantic updates

---

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.

REPL siblings

Julia family: sicmutils · sicp · julia-scientific · julia-gay · julia-gpu-kernels · julia-tempering · quarto-julia. Atlas: repl-commons.