Constraint Forces, Virtual Work, and Generalized Coordinates

Manual: General · Subject: Physics

Solve constrained systems using generalized coordinates, holonomic and nonholonomic constraints, and the principle of virtual work.

Constraints in mechanics

Holonomic and nonholonomic constraints

A holonomic constraint can be written as $f(q_1,\dots,q_n,t)=0$ and reduces the dimension of configuration space. Nonholonomic constraints, by contrast, involve differential relations such as $\sum_i a_i(q,t)\,dq_i=0$ that may not integrate to a global position constraint.

Virtual work

For ideal constraints, the net virtual work of constraint forces vanishes for all allowed virtual displacements. This allows one to eliminate unknown reaction forces and focus on generalized forces.

Which statement best describes an ideal constraint?

It always dissipates energy Its reaction forces do no virtual work It must be holonomic It eliminates inertia

What is the defining property of a holonomic constraint?

Generalized coordinates

Choosing generalized coordinates $q_i$ adapted to the constraints reduces the dynamics to the minimal number of degrees of freedom. The kinetic energy may be expressed as a quadratic form in generalized velocities, often with coordinate-dependent coefficients.

Practical workflow for constrained systems

1

Choose coordinates that satisfy the constraints as directly as possible.
2

Express kinetic and potential energies in the reduced coordinate set.
3

Identify generalized forces, including nonconservative terms if present.
4

Derive equations of motion or apply Lagrange multipliers if reaction forces are needed.

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