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In classical mechanics a system may be defined as holonomic if all constraints can be expressible as a function of only state or only a derivative of state (but not both) i.e.:
Path Dependence and Integrability: Holonomic constraints imply that if a constraint exists in terms of position only, we can integrate this constraint to define the allowed paths or configurations explicitly. However, if the constraint depends on both position and velocity (or just velocity), it means the constraint affects not just where the system can go, but how it gets there. This coupling with velocity introduces path dependence in the allowed motions, which generally makes it impossible to reduce the constraint to a function of configuration alone.
Degrees of Freedom: For holonomic constraints, we can reduce the degrees of freedom by working directly with coordinates that satisfy the constraint, which simplifies analyses. With non-holonomic constraints, the number of independent variables needed to describe the system’s evolution can’t be reduced in this way since the path of the system (i.e., its velocity) plays a crucial role in determining its permissible configurations at each point. This also affects how we can calculate possible motions and forces in the system since non-holonomic constraints often require the use of generalized velocities or more complex expressions.
Dynamical Implications: When a constraint involves velocity directly, it may restrict certain types of motion (like no slipping or rolling without slipping) that require the use of forces and torques dynamically and lead to equations of motion that can’t be derived from potential energy alone. This makes non-holonomic systems dynamically richer and often harder to analyze than holonomic ones, as they can’t usually be solved using traditional energy conservation principles.