📂Classical Mechanics

What is a Constraint in Physics?

Definition

A particle, or a system of particles, is said to undergo constrained motion when it moves only within a geometrically confined area (such as given curves or surfaces), and such restrictions themselves are called constraints.

Explanation

In Korean, it is commonly referred to as 구속조건, but in English, it is just called a constraint, not a constraint condition.

Simple examples of constrained motion include circular motion and pendulum motion. The constraint condition of an object moving in a circular trajectory with a radius of $r$ on a 2-dimensional plane is $x^{2} + y^{2} = r^{2}$. The number of degrees of freedom of a system of particles is obtained by subtracting the number of (holonomic) constraints from the total number of coordinates. Representing the coordinates of a system with $n$ degrees of freedom by $n$ coordinates that are independent of the constraints is called generalized coordinates.

Holonomic

When the constraints are exclusively equations related to position and time, they are referred to as holonomic. If all constraints of a system of particles are holonomic, then the system is considered holonomic.

Suppose there are $N$ particles moving in 3 dimensions. When the system of particles has $m$ constraints, being holonomic means that the system satisfies the following equation for constraints $f_{j}$.

$$f_{j}(x_{i}, y_{i}, z_{i}, t) = 0,\quad i=1,2,\dots,N \quad j=1,2,\dots,m$$

Specifically, the constraint of a particle moving on a spherical surface with a radius of $r$ is

$$f(x,y,z) = r^{2} - x^{2} - y^{2} - z^{2} = 0$$

thus it is holonomic. Conversely, the constraint of a particle moving outside a sphere is $r^{2} - x^{2} - y^{2} - z^{2} \ge 0$, and so it’s not holonomic. In simple terms, holonomic means constraints that can reduce degrees of freedom.