Nonholonomic mechanics · Free motion

Chaplygin Sleigh

Release a three-dimensional knife-edge sleigh. Move its center of mass, reverse the launch or put the mass over the blade. Compare the curved constrained path with a freely sliding center of mass.

Blade forward speed $0$

Yaw rate $0$

Mechanical energy $0$

Maximum sampled relative energy drift $0$

Signed lateral reaction $0$

Blade lateral velocity $0$

Geometry & initial state

Change the constraint. Follow the consequence.

Signed blade-to-mass offset

a

0.50\,\mathrm m

Radius of gyration about mass center

k

0.40\,\mathrm m

Initial blade forward speed

u_0

-1.00\,\mathrm{m/s}

Initial yaw rate

\omega_0

1.50\,\mathrm{rad/s}

Integration: standard or fine

b_{\rm fine}

0.00\,

Initial heading

\theta_0

20.00\,{}^\circ

The blade moves along the body axis but cannot move sideways. The gold sphere is the center of mass, not an added point mass. Total mass is one kilogram; the inertia is specified by the radius of gyration. The chassis and supports are schematic.

Cyan follows the mass center; gold follows the knife contact location. Dashed gray is the mass-center path after removing the blade at release, using the same initial mass-center velocity. The reference is a different experiment.

Fourth-order Runge–Kutta with cubic Hermite interpolation over eight seconds. Standard uses 4096 steps; fine uses 8192. The displayed drift checks stored step states, not a rigorous error bound. Energy conservation does not prove trajectory accuracy.

Ideal planar rigid body with one lateral knife-edge constraint. No drag, blade friction loss, lift or tipping. Mass and inertia are specified independently of the schematic chassis.

Start with the essentials

Focus question: How can rotation fade away without losing mechanical energy?
One-sentence intuition: The sideways constraint transfers energy between translation and rotation. For an offset mass center, the stable direction has the blade trailing the mass.

Core mathematical model

Knife velocity and rigid-body dynamics

\dot x=u\cos\theta,\quad\dot y=u\sin\theta,\quad\dot\theta=\omega,\qquad\dot u=a\omega^2,\quad\dot\omega=-\frac{a u\omega}{k^2+a^2}

Coordinates refer to the knife contact location, not the mass center. Positive offset points forward along the blade; mass-center inertia equals mass times squared radius of gyration.

Energy and reaction

E=\frac m2\left[u^2+(k^2+a^2)\omega^2\right],\quad N=\frac{m k^2 u\omega}{k^2+a^2},\quad P_N=Nv_{P,\perp}=0

The mass center has a lateral velocity even though the blade does not. The reaction may be nonzero while doing no work.

Common difficulties

Contact is a velocity rule

Typical misconceptionReturning a control or reducing yaw must undo motion or remove energy.

Better mental modelCheck the fixed energy total while its two terms exchange. The constraint does no work.

Run the experiment

01
Experiment 1
Release backward, then watch the forward speed reverse while energy stays constant.
What to observe: Compare the synchronized path and numerical readouts.
02
Experiment 2
Choose Mass over blade, then reverse the offset. Compare the circular and straightening paths.
What to observe: Compare the synchronized path and numerical readouts.
03
Experiment 3
Compare standard and fine integration at the largest yaw rate.
What to observe: Compare the synchronized path and numerical readouts.

Primary theory: Simoes & Colombo · rolling disk · Chang & Perlmutter · Chaplygin sleigh