5.1 Generalized coordinates and constraints

Generalized coordinates simplify complex systems by reducing variables and incorporating constraints. They're crucial for describing system configurations and formulating equations of motion in classical mechanics, whether using Cartesian, polar, or angular coordinates.

Constraints play a key role in shaping system behavior. Holonomic constraints involve only coordinates and time, while non-holonomic constraints include velocities. Lagrange multipliers help incorporate these constraints into equations of motion, allowing for easier analysis of constrained systems.

Generalized Coordinates and System Constraints

Role of generalized coordinates

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• Generalized coordinates
  • Independent variables fully describe system configuration $[q_i](https://www.fiveableKeyTerm:q_i)$ where $i = 1, 2, ..., n$
  • Number of coordinates equals degrees of freedom
• Simplify complex systems
  • Reduce variables needed to describe system
  • Automatically incorporate certain constraints
  • Facilitate easier formulation of equations of motion
• Examples
  • Cartesian coordinates $(x, y, z)$
  • Polar coordinates $(r, \theta)$
  • Angular coordinates for rotational systems (pitch, yaw, roll)

Holonomic vs non-holonomic constraints

• Holonomic constraints
  • Expressed as equations involving only coordinates and time $f(q_1, q_2, ..., q_n, t) = 0$
  • Fixed length pendulum, bead on a wire
• Non-holonomic constraints
  • Cannot be expressed solely in terms of coordinates and time
  • Involve velocities or differentials of coordinates
  • Rolling without slipping, knife-edge constraint
• Scleronomic constraints
  • Time-independent (fixed-length rod)
• Rheonomic constraints
  • Time-dependent (piston-cylinder system)

Expression of constraints

• Lagrange multipliers
  • Additional variables $\lambda_j$ incorporate constraints
  • Treat constrained systems as unconstrained
• Constraint equations
  • Express holonomic constraints $f_j(q_1, q_2, ..., q_n, t) = 0$
• Augmented Lagrangian
  • $L^* = L + \sum_j \lambda_j f_j(q_1, q_2, ..., q_n, t)$
  • $L$ represents original system Lagrangian
• Virtual work of constraint forces
  • $\delta W_c = -\sum_j \lambda_j \delta f_j$

Equations of motion derivation

• Lagrange's equations of motion
  • $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_i}}\right) - \frac{\partial L}{\partial q_i} = Q_i$
  • $Q_i$ represents generalized forces (gravity, friction)
• Equations of motion with constraints
  • $\frac{d}{dt}\left(\frac{\partial L^*}{\partial \dot{q_i}}\right) - \frac{\partial L^*}{\partial q_i} = 0$
• Derivation steps
  1. Express kinetic and potential energies using generalized coordinates
  2. Form Lagrangian $L = T - V$
  3. Add constraint terms to create augmented Lagrangian $L^*$
  4. Apply Lagrange's equations to $L^*$
• Solve resulting system
  • Equations of motion for generalized coordinates
  • Constraint equations (ensure system consistency)