A constraint is holonomic if it can be written as an equation involving only the coordinates (positions) and time: [ f(q_1, q_2, ..., q_n, t) = 0 ] Where ( q_i ) are the generalized coordinates. This constraint reduces the degrees of freedom of the system.
If you are building a robot, understanding these constraints is a matter of life and death (or at least, success and crashing). non holonomic
One of the most famous puzzles in physics involves the non-holonomic nature of a falling cat. A cat starts upside down with zero net angular momentum. Even though it has nothing to "push" off of, it can twist its body in a specific sequence—expanding and contracting its limbs—to flip right-side up while maintaining that zero net momentum. This "phase shift" is a beautiful display of non-holonomic geometry in action. A constraint is holonomic if it can be
Crucially, even though the instantaneous velocity is restricted, the system can still reach any position in the configuration space (given enough time and complex maneuvers). One of the most famous puzzles in physics
Think of a unicyclist. A unicycle can go forward and backward. It can turn. But it cannot move sideways without falling over. However, a skilled unicyclist can get from point A to point B, facing any direction, eventually. The constraint doesn't prevent the destination; it only restricts the immediate path.