Visual walkthrough — Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic
Step 1 — What "position" even means for a coin on a table
WHAT: We name the coin's state with .
WHY: Before we can ask "what can change?" we must first say "what is there to change." Four independent knobs = a configuration space of four dimensions.
PICTURE: Look at the amber dot — the contact point at . The cyan arrow is the heading . The little wedge on the rim tracks , the roll.
Step 2 — The one rule: no slipping
For a coin of radius , an angle of rim is an arc of length . So the contact point advances a distance in whatever direction the coin is pointing.
WHAT: We turn "no slipping" into a precise length: distance moved .
WHY: We need the physics ("no skid") written as numbers relating tiny changes, because tiny changes are what constraints control.
PICTURE: The rim wedge unrolls onto the table as the amber segment of length .
Step 3 — Splitting that motion into East and North
WHAT: We resolve the step into and .
WHY: We chose (and not, say, ) because we want the actual East and North lengths of a known-length step at a known angle — and cosine/sine are precisely the ratios that return those lengths.
PICTURE: The amber hypotenuse is the step; the cyan legs are its East and North shares.
Step 4 — Erasing the roll to see the pure direction rule
Divide the two equations (or cross-multiply):
WHAT: We removed and got a single relation among .
WHY: This is the sharp form of the constraint: the contact point may only slide along the heading, never sideways. Sideways motion is forbidden.
PICTURE: Green arrow = allowed (along ). Red arrow = the forbidden sideways slide the equation kills.
Step 5 — The integrability test: can this become a position rule?
If were for some , then and , and because mixed partials agree, we would need the exactness condition:
WHAT: We test whether pass exactness — but note depend on , a third variable, so we must also allow the direction. In three variables the honest test is the Frobenius condition , where .
WHY: We need a yes/no machine. Exactness (2 variables) or Frobenius (3 variables) is that machine: it detects whether an integrating factor can ever exist.
PICTURE: Left panel — an exact field whose arrows are all "downhill" of one landscape (integrable). Right panel — our rolling field, whose arrows swirl as turns, so no single landscape has them as its slope.
Compute the curl of (treating as the three axes):
Then
Result: The test returns , never zero. No integrating factor exists. The rolling rule is not the differential of any . → Non-holonomic.
Step 6 — Proof by wandering: reach any configuration anyway
WHAT: We show the reachable configurations are all of them, despite the velocity rule.
WHY: This is the physical meaning of "non-integrable": if a position equation existed, you'd be stuck on one surface forever. You are not stuck ⇒ no such surface ⇒ confirms non-holonomic, from the outside.
PICTURE: A "parallel-park" loop: roll forward, steer, roll, steer back — the coin ends up shifted sideways from where it started, a move the instantaneous rule "forbids." The net sideways shift is the signature of non-holonomy.
Step 7 — The time axis: is it scleronomic?
WHAT: We confirm no explicit time.
WHY: The two classification axes are independent (see the parent's second mistake). Having settled non-holonomic on axis 1, axis 2 is a separate yes/no.
PICTURE: A frozen table — the constraint surface does not breathe with time. Contrast with the parent's spinning wire (), which would carry a .
Verdict: the rolling coin is non-holonomic and scleronomic.
The one-picture summary
Everything on one page: the coin rolls a step along heading (top-left); that step splits into , (top-right); eliminating forbids sideways motion, (bottom-left); the swirl test gives , so no exists, yet parallel-parking reaches any (bottom-right).
Recall Feynman: the whole walkthrough in plain words
Picture a coin standing on its edge like a tiny wheel. Four things describe it: where it is (two numbers), which way it points, and how far it has rolled. The one rule is "no skidding": every bit of roll pushes it forward exactly one rim-arc, straight ahead — never sideways. Write that down and the "how far it rolled" cancels, leaving a pure direction rule: you may only move along where you point. Now ask the key question: is that a disguised map of "allowed spots"? We run a swirl test — as the coin turns, its allowed-direction arrows spin around, and the test spits out , never zero, meaning there is no hidden landscape whose slope those arrows are. So it's non-holonomic. And to prove you're not trapped: roll, turn, roll, turn back — you end up shifted sideways, a move the instant-by-instant rule "banned." The rule limited your speed directions, never your destinations. Finally, no clock appears anywhere in the equation, so it's scleronomic. Non-holonomic and scleronomic — that's the coin.
Recall Where this plugs into the rest of the vault
Because the coin's constraint is non-integrable, you cannot just delete a coordinate. You keep all of Generalized Coordinates and instead handle the rolling rule with Lagrange Multipliers inside Lagrangian Mechanics. The velocity-form constraint slots directly into D'Alembert's Principle through Virtual Displacements, and the scleronomic verdict is exactly what tells you (via Kinetic Energy and the Hamiltonian) whether energy is conserved.