3.1 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

1. $$z = f(x, y)$$ particle moves on a surface
2. $$f(x, y, z, \dot{x}, \dot{y}, \dot{z}) = 0$$
3. For gas molecules in a cubical container the position coordinates satisfy
$$−L \leq x \leq L, \hspace{2em} −L \leq y \leq L \hspace{2em} − L \leq z \leq L$$ Constraint relations involving only coordinates and possibly time, are called holonomic constraints. These are given by expressions of the form $$f_j ( \vec x_1, \vec x_2, . . . \vec x_N , t) = 0, \hspace{2em} j = 1 . . . m$$

3.2 Degrees of freedom

In our course 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.

3.3 Generalized coordinates

For a system with $$m$$ degrees of freedom, one needs $$m$$ generalized coordinates having the following properties:

1. the generalized coordinates must be independent;
2. All the Cartesian coordinates of the particles must be expressible in terms of the generalized coordinates when holonomic constraints are taken into account.