**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