How Does Hamilton’s Principle Work with Semiholonomic Constraints?

The issue is that a nonintegrable velocity constraint restricts actual motion, but it does not generally restrict variations of a path in the same way. Treating those two restrictions as interchangeable can lead to different equations of motion.

In the terminology commonly used in discussions of Goldstein, “semiholonomic” refers to applying a variational principle with specified restrictions on virtual displacements, even though the corresponding velocity constraints do not come from equations among the coordinates. Terminology can vary, so the equations and the choice of allowed variations matter more than the label.

The two kinds of constraint

A holonomic constraint can be written as an equation among coordinates and time, such as $f(q,t)=0$. Its allowed configurations lie on a surface in configuration space. The associated virtual displacements are tangent to that surface: $\frac{\partial f}{\partial q_i}\delta q_i=0$.

A common nonholonomic constraint is linear in velocities:

$$\sum_i A_{ai}(q,t)\dot q_i+b_a(q,t)=0.$$

If these differential relations cannot be integrated to coordinate constraints, the constraint is nonholonomic. A standard d’Alembert–Chetaev treatment restricts virtual displacements at a fixed time by

$$\sum_i A_{ai}(q,t)\delta q_i=0.$$

Notice that the term $b_a$ does not appear in this virtual-displacement condition. The condition concerns displacement at one instant, not the full velocity relation.

What variational principle is being used?

For ideal constraints treated with the d’Alembert–Chetaev rule, the action

$$S=\int_{t_1}^{t_2} L(q,\dot q,t)\,dt$$

is varied over virtual displacements satisfying the Chetaev conditions, with endpoint variations zero. This gives the Lagrange–d’Alembert equations

$$\frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i} =\sum_a \lambda_a A_{ai}.$$

The multipliers represent the ideal constraint forces, up to the sign convention chosen for them. The equations must be solved together with the original velocity constraints.

This is not the same as requiring every varied path to satisfy the nonholonomic velocity constraint at every time and then applying unrestricted calculus of variations. That procedure is often called the vakonomic approach. For genuinely nonholonomic constraints, it generally produces different equations. Which model applies depends on the physical setup; ideal rolling constraints are commonly modeled by d’Alembert mechanics.

Worked example: a nonholonomic constraint

Consider coordinates $(x,y,z)$ and the constraint

$$\dot y-x\dot z=0.$$

The associated one-form is $dy-x\,dz$. It is not integrable to a single coordinate constraint: its exterior derivative is $-dx\wedge dz$, and the Frobenius integrability condition fails. Thus it is a genuine nonholonomic constraint.

For the Chetaev variation, the allowed virtual displacements obey

$$\delta y-x\,\delta z=0.$$

Take a free particle with $L=\tfrac12m(\dot x^2+\dot y^2+\dot z^2)$. Here $(A_x,A_y,A_z)=(0,1,-x)$, so the d’Alembert equations are

$$m\ddot x=0,\qquad m\ddot y=\lambda,\qquad m\ddot z=-x\lambda,$$

along with $\dot y-x\dot z=0$. The multiplier supplies a force perpendicular to the permitted virtual displacements: its virtual work is $\lambda(\delta y-x\delta z)=0$.

If instead you add a multiplier directly to the Lagrangian, $L_{\mathrm{aug}}=L+\mu(\dot y-x\dot z)$, and vary all three coordinates independently, as well as the multiplier $\mu$ (whose variation imposes $\dot y-x\dot z=0$), you obtain

$$m\ddot x=-\mu\dot z,\qquad m\ddot y=-\dot\mu,\qquad m\ddot z=x\dot\mu+\mu\dot x.$$

These are the vakonomic equations, not the d’Alembert equations above. The difference is not an algebraic error: the two methods impose different variation rules.

When the methods agree

For a holonomic constraint, the admissible configurations form a submanifold, and variations of admissible paths are tangent to it. One can either eliminate dependent coordinates or use multipliers in the ordinary constrained variational principle; these approaches yield the same constrained equations under the usual regularity assumptions.

For a nonholonomic constraint, do not assume that a variation through constraint-satisfying paths is equivalent to imposing the Chetaev condition. They coincide only in special cases, including cases where the velocity constraint is actually integrable or where the particular dynamics make the resulting equations agree.

Common mistakes

• Calling every velocity constraint holonomic. A relation linear in velocities is not automatically integrable to a relation among coordinates.
• Varying the augmented Lagrangian and assuming it is d’Alembert mechanics. Independent variation of the coordinates gives vakonomic equations in general.
• Dropping the constraint equation after adding multipliers. The multiplier equations must be solved together with the original velocity constraints.
• Treating the terminology as universal. “Semiholonomic” is used differently across texts. Identify the actual constraint and state explicitly which virtual displacements are allowed.