Degrees of Freedom, Constraints

Degrees of freedom #

The minimum number of independent variables or coordinates required to specify the position of a dynamical system, consisting of one or more particles, is called the number of degrees of freedom of the system.

Constraints #

Constraints: Limitations on the motion of a system. Motion is said to be constrained motion.

Types #

1. Holonomic
   1. Constraints expressible in the form of equation of the form $f(\vec{r_1},\vec{r_2},…,t)=0$ ^077dc7
   2. If the number of particles of a system: N and constraints: m, degrees of freedom=independent coordinates=n=3N-m
   3. Generalized coordinates: $q_1,q_2,q_3,…q_n$; and each coordinate can vary independently.
   4. “Superflous coordinates”=redundant coordinates
2. Nonholonomic
   1. Constraints not expressible in this form.
   2. Each coordinate cannot always vary independently ⇒ degrees of freedom < minimum number of coordinates required

Can be also be classified as

1. Conservative
2. Dissipative: Constraint forces do work⇒Total work is not constant.

.

1. Rheonomous: Constraint contains time as an explicit variable. Generally dissipative.
2. Scleronomous: Constraints don’t contain time as an explicit variable.