Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic

WHAT is a constraint?

The two classification axes are independent questions:

Holonomic vs non-holonomic — Can the constraint be written as an equation purely in coordinates and time?
Rheonomic vs scleronomic — Does time appear explicitly?

Axis 1 — Holonomic vs Non-holonomic

WHY does "non-integrable velocity" count as non-holonomic?

Axis 2 — Rheonomic vs Scleronomic

Figure — Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic

Worked examples

Worked example 4 — Rolling disk (the classic non-holonomic)

A vertical disk rolling without slipping in the plane, position $(x,y)$ , heading $\phi$ , radius $R$ . Rolling gives $dx = R\cos\phi\, d\theta,\qquad dy=R\sin\phi\,d\theta,$ where $\theta$ is the roll angle. Why this step? These mix $x,y,\phi,\theta$ in a velocity relation. Eliminating $d\theta$ : $dx\sin\phi - dy\cos\phi = 0$ . Test integrability — it is not an exact differential and has no integrating factor (3+ coupled variables), so it cannot be integrated to $f(x,y,\phi)=0$ . → Non-holonomic, scleronomic. The disk can be rolled to any $(x,y,\phi)$ , proving configurations aren't restricted — only velocities are.

Common mistakes

Flashcards

What defines a holonomic constraint?

It can be written as an equation

f(\mathbf r_1,\dots,\mathbf r_N,t)=0

involving only coordinates and time.

How does a holonomic constraint affect degrees of freedom?

Each independent one removes one coordinate:

n=3N-k

What makes a velocity constraint non-holonomic?

It is non-integrable — no

f

(even with an integrating factor) has it as its differential.

Scleronomic vs rheonomic?

Scleronomic = no explicit

t

(frozen surface); rheonomic = explicit

t

(moving surface).

Why does rheonomic threaten energy conservation?

\mathbf r=\mathbf r(q,t)

gains a

\partial\mathbf r/\partial t

term; constraint surface does work, so mechanical energy needn't be conserved.

Classify a bead on a wire spinning at fixed

\omega

Holonomic (

y-x\tan\omega t=0

) and rheonomic.

Classify a rolling vertical disk.

Non-holonomic (non-integrable rolling) and scleronomic.

Classify a particle confined inside a box.

Non-holonomic (inequality) and scleronomic.

Exactness test for

A\,dx+B\,dy=0

Integrable when

\partial A/\partial y=\partial B/\partial x

Why can't genuine non-holonomy happen with only 2 variables?

In 2 variables an integrating factor always exists, so any such Pfaffian is integrable.

Recall Feynman: explain to a 12-year-old

Imagine a toy train. A holonomic rule is like "the train must stay on the painted line on the floor" — you can write down exactly where it's allowed to be. A non-holonomic rule is like "the wheels can't skid sideways" — that doesn't pin down where the train ends up; the train can still reach any spot on the floor, it just can't slide sideways to get there. Scleronomic = the painted line never moves. Rheonomic = someone is dragging the painted line around while the train tries to follow it. Two totally different things: one is what shape the rule is, the other is whether the rule changes over time.

Connections

Generalized Coordinates — holonomic constraints define the independent $q_i$ .
Lagrangian Mechanics — holonomic constraints let you write $L(q,\dot q,t)$ directly.
D'Alembert's Principle — handles constraint forces via virtual work.
Lagrange Multipliers — the standard tool for non-holonomic / Pfaffian constraints.
Kinetic Energy and the Hamiltonian — scleronomic ⇒ $H=$ total energy.
Virtual Displacements — rheonomic vs scleronomic differ in whether $\delta\mathbf r$ allows $\partial\mathbf r/\partial t$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, constraints ka matlab hai system pe lagi hui geometric rules. Newton ke equations mein humein har force chahiye, including wo unknown forces jaise rod ki tension ya surface ka normal reaction. Ye forces pareshan karte hain kyunki pehle se pata nahi hota. To trick ye hai: force likhne ke bajaye hum geometric condition likh dete hain, jaise "ball wire pe hi rahegi" ya "rod ki length fixed hai". Isse coordinates kam ho jaate hain — yahi Lagrangian mechanics ka asli motivation hai.

Do alag-alag classification hain, dono independent. Pehla axis: holonomic vs non-holonomic. Agar constraint ko ek poora equation $f(\mathbf r,t)=0$ ki tarah likh sakte ho (sirf positions aur time mein) to wo holonomic hai — ye ek coordinate kam kar deta hai. Agar constraint sirf velocities pe ho aur usko integrate karke equation mein nahi badal sakte (jaise rolling disk jo sideways skid nahi kar sakti), to wo non-holonomic hai. Inequality constraints (jaise gas box ke andar) bhi non-holonomic hote hain.

Doosra axis: scleronomic vs rheonomic. Yahan sirf ek hi sawaal hai — kya time $t$ explicitly equation mein aata hai? Agar nahi (surface frozen hai) to scleronomic. Agar haan (jaise wire ko fixed speed se ghuma rahe ho, $y=x\tan\omega t$ ) to rheonomic. Important baat: rheonomic hone par energy conserve nahi hoti, kyunki bahar se koi system ko drive kar raha hai.

Sabse common galti: log sochte hain har velocity-wala constraint non-holonomic hota hai — galat! Agar wo integrate ho jaye to wo holonomic hi tha. Aur doosri galti: rheonomic ko non-holonomic samajh lena. Yaad rakho dono axes alag hain — bead on spinning wire holonomic AND rheonomic hai. Har constraint ka ek label first pair se, ek label second pair se.

Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic

WHAT is a constraint?

Axis 1 — Holonomic vs Non-holonomic

WHY does "non-integrable velocity" count as non-holonomic?

Axis 2 — Rheonomic vs Scleronomic

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Analytical Mechanics

Connections