tautochrone problem
E923653
The tautochrone problem is a classic question in physics and calculus of variations that seeks the curve along which a bead sliding under gravity reaches the lowest point in the same time regardless of its starting position, whose solution is a cycloid.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
physics problem
ⓘ
problem in the calculus of variations ⓘ |
| alsoKnownAs | isochronous curve problem NERFINISHED ⓘ |
| application |
design of isochronous pendulum clocks
ⓘ
understanding timekeeping accuracy ⓘ |
| assumes |
frictionless motion
ⓘ
motion constrained to a fixed curve in a vertical plane ⓘ uniform gravitational field ⓘ |
| category |
classical variational problem
ⓘ
isochronous systems ⓘ |
| curveEquationForm | cycloid generated by a circle rolling along a straight line ⓘ |
| definition | problem of finding a curve along which a bead sliding under gravity reaches the lowest point in the same time regardless of starting position ⓘ |
| field |
calculus of variations
ⓘ
classical mechanics ⓘ mathematical physics ⓘ |
| historicalPeriod | 17th century ⓘ |
| inspiredDevelopment |
advances in the theory of cycloids
ⓘ
development of the calculus of variations ⓘ |
| involvesConcept |
Euler–Lagrange equation
NERFINISHED
ⓘ
conservation of energy ⓘ differential equations ⓘ isochronous motion ⓘ parametric representation of curves ⓘ time of descent under gravity ⓘ |
| keyProperty | time of descent is independent of starting point along the curve ⓘ |
| mathematicalFormulation | variational problem minimizing time of descent with isochrony constraint ⓘ |
| physicalModel | bead sliding on a wire in a vertical plane under gravity ⓘ |
| relatedTo |
brachistochrone problem
ⓘ
cycloid ⓘ pendulum theory ⓘ simple harmonic motion ⓘ |
| requires |
neglect of air resistance
ⓘ
no rolling or rotational kinetic energy of the bead ⓘ |
| solutionCurve | cycloid ⓘ |
| solutionMethod |
imposition of isochrony condition to determine curve shape
ⓘ
integration of time element along the curve ⓘ use of energy conservation to express velocity as a function of height ⓘ |
| solutionProperty | the cycloidal pendulum has a period independent of amplitude for small oscillations ⓘ |
| solutionUniqueness | the cycloid is the unique solution under standard assumptions NERFINISHED ⓘ |
| studiedBy |
Christiaan Huygens
NERFINISHED
ⓘ
Gottfried Wilhelm Leibniz NERFINISHED ⓘ Isaac Newton NERFINISHED ⓘ Johann Bernoulli NERFINISHED ⓘ |
| timeIndependenceCondition | descent time is constant for all starting points above the lowest point ⓘ |
| typicalAssumption |
point mass bead
ⓘ
rigid wire or track ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.