Classical Mechanics · Constraints and Lagrangian mechanics
The brachistochrone first integral generates a cycloid
Problem
Take $y$ positive downward from a release point and let a frictionless bead start from rest. Derive the travel-time functional for a graph $y(x)$, use the Beltrami identity to obtain $y(1+y'^2)=2a$, and verify that $x=a(\theta-\sin\theta)$, $y=a(1-\cos\theta)$ solves it. For the endpoint $\theta=\pi$ with $a=1.00\,\mathrm m$, compute horizontal drop, vertical drop, and travel time. State the endpoint and cusp caveats.
Hint
Write time as path length divided by the energy-determined speed.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Classical Mechanics sample problem: Try the free sample problem.
More Classical Mechanics practice problems
- Angular momentum about a translating point carries a transport termNewtonian mechanics and conservation laws
- A discrete spatial symmetry does not create angular-momentum conservationNewtonian mechanics and conservation laws
- Rayleigh dissipation supplies the correct damping signConstraints and Lagrangian mechanics
- Radial speed contributes no angular momentumCentral-force motion and scattering
- The Kepler virial theorem fixes mean energiesCentral-force motion and scattering
- A coordinate-dependent mass freezes at linear orderSmall oscillations and normal modes
- A diatomic lattice opens an acoustic-optical gapSmall oscillations and normal modes
- Principal axes of a coupled planar inertia tensorRigid bodies and rotating frames
- Energy ellipse and enclosed action of an oscillatorHamiltonian mechanics and canonical structure