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.