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The equations of motion in Newtonian mechanics are always written in Cartesian coordinates. If there are constraints among the coordinates, these have to be taken into account separately. A special type of constraints, called holonomic constraints and the number of degrees of freedom is defined.
Constraints
For many systems the coordinates and velocities must satisfy constraint relations. For example for a particle moving on the surface of a sphere the
constraint relation
$$x^2+y^2+z^2=R^2$$
must be imposed separately on the solutions of EOM.
Several different types of constraints are possible
- $z=f(x,y)$ particle moves on a surface
- $f(x,y,z,\dot{x},\dot{y},\dot{z})=0.$
- For gas molecules in a cubical container the position coordinates satisfy \[ -L\leq x\leq L,\qquad -L\leq y\leq L \qquad -L\leq z\le L
Constraint relations involving only coordinates and possibly time, are called holonomic constraints. These are given by expressions of the form \begin{equation} f_j(\vec{x}_1,\vec{x}_2,\dots \vec{x}_N,t) = 0, \qquad\qquad j=1 \dots m. \end{equation} The Lagrangian formalism is applicable to systems with holonomic constraints only.
Degrees of freedom
We shall be concerned only with systems having holonomic constraints. For a system with $N$ particles, $3N$ coordinates are needed. If these coordinates satisfy $m$ relations of type $f(\vec{x},t)=0$. Only $3N-m$ coordinates will be independent and we say that the system has $3N-m$ degrees of freedom.
Generalized coordinates
For a system with \(m\) degrees of freedom, one needs \(m\) independent variables to specify the configuration of the system. A set of \(m\) functions of Cartesian coordinates is called generalized coordinates if they having the following properties:
- the generalized coordinates must be independent;
- All the Cartesian coordinates of the particles must be expressible in terms of the generalized coordinates when holonomic constraints are taken into account.
