Degrees of freedom and constraints

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SUMMARY

In mechanical systems with P holonomic constraints and N particles, the degrees of freedom are calculated as 3N-P, requiring 3N-P generalized coordinates for independent variation. In contrast, for N particles with P non-holonomic constraints, 3N coordinates are necessary to fully describe the system, but these coordinates do not vary independently due to the constraints imposed. This distinction highlights the complexity of non-holonomic systems, which remain a significant area of ongoing research, as noted in references such as "Flannery nonholonomic."

PREREQUISITES
  • Understanding of degrees of freedom in mechanical systems
  • Knowledge of holonomic and non-holonomic constraints
  • Familiarity with generalized coordinates
  • Basic principles of classical mechanics
NEXT STEPS
  • Research "Flannery nonholonomic" for advanced insights into non-holonomic constraints
  • Explore the implications of non-holonomic constraints in robotics
  • Study the mathematical formulation of generalized coordinates in constrained systems
  • Investigate current research trends in the handling of non-holonomic constraints
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Students and professionals in physics, mechanical engineering, and robotics who seek to deepen their understanding of constraints in mechanical systems and their implications on degrees of freedom.

Ahmed1029
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In case of P holonomic constraints and N particles, I have 3N-P degrees of freedom and I have to look for 3N-P generalized coordinates if I want them to vary independently, but what about non-holonomic constraints? I know if I have N particles and P non-holonomic constraints, I still need 3N coordinates to completely describe the mechanical system, but do those 3N coordinates vary independently?(assume I'm talking about any set of coordinates, not just cartesian)
 
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In systems with non-holonomic constraints, we always need more coordinates than degrees of freedom. Of course, those coordinates do not vary independently, they have to obey the non-holonomic constraint.
 
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FYI: the handling of non-holonomic constraints is still an active research area. Google for "Flannery nonholonomic" to see some relevant references.
 
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