Mechanics of particles and systems Definition

Holonomic constraint

Builds on Euclidean space Smooth function

Definition

Consider a system of $N$ point particles with positions $x_1, \dots, x_N \in \R^3$ , and write $x = (x_1, \dots, x_N) \in \R^{3N}$ . Let $f$ be a real-valued smooth function on an open subset of $\R^{3N} \times \R$ . A holonomic constraint on the system is a condition $f(x, t) = 0$ that the positions must satisfy at every time $t$ . Holonomic constraints $f_1 = 0, \dots, f_k = 0$ are independent when the gradients of $f_1, \dots, f_k$ in $x$ are linearly independent wherever all the constraints hold.

Remarks

Example: a pendulum of length $\ell > 0$ with pivot at the origin of $\R^3$ satisfies $f(x) = \abs{x}^2 - \ell^2 = 0$ . Its gradient $2x$ is nonzero on the sphere $\abs{x} = \ell$ , so the constraint is independent. Non-example: a particle that must stay in the half-space $z \ge 0$ is restricted by an inequality, not by an equation, so the restriction is not a holonomic constraint. A constraint linear in the velocities is also called holonomic when it is equivalent to a constraint on positions. For instance, $\dot z = 0$ is equivalent to $z = c$ for a constant $c$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`JTN4TXDRP1`
URI	`m2://JTN4TXDRP1`
Kind	definition
Writing	W0 stub
MSC	70F20, 70G10
Defines	Holonomic constraint
Uses	Euclidean space, Smooth function
Changed	3 November 2026

Holonomic constraint

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here