brachistochrone problem
E270049
The brachistochrone problem is a famous challenge in the calculus of variations that asks for the curve along which a particle will descend between two points in the shortest time under gravity, whose solution is a cycloid.
All labels observed (1)
| Label | Occurrences |
|---|---|
| brachistochrone problem canonical | 2 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in the calculus of variations ⓘ |
| application |
design of roller coaster profiles
ⓘ
optimal control theory ⓘ time-optimal motion planning ⓘ |
| asksFor | curve of fastest descent under gravity between two points ⓘ |
| assumes |
motion constrained to a vertical plane
ⓘ
no friction ⓘ particle starts from rest ⓘ particle treated as point mass ⓘ uniform gravitational field ⓘ |
| constraintType | holonomic constraint to a curve in a plane ⓘ |
| field |
calculus of variations
ⓘ
classical mechanics ⓘ mathematical physics ⓘ |
| generalization |
brachistochrone in non-uniform gravitational fields
ⓘ
relativistic brachistochrone problem ⓘ |
| historicalContext | early problem in the development of the calculus of variations ⓘ |
| mathematicalFormulation | minimization of a time functional over admissible curves ⓘ |
| nameEtymology | from Greek "brachistos" meaning shortest and "chronos" meaning time ⓘ |
| objectiveFunction | time of travel along the curve under gravity ⓘ |
| posedBy | Johann Bernoulli ⓘ |
| publishedIn | Acta Eruditorum ⓘ |
| receivedSolutionsFrom |
Gottfried Wilhelm Leibniz
ⓘ
Guillaume de l’Hôpital ⓘ
surface form:
Guillaume de l'Hôpital
Isaac Newton ⓘ Jakob Bernoulli ⓘ
surface form:
Jacob Bernoulli
Tschirnhaus ⓘ |
| relatedConcept |
Euler–Lagrange equation
ⓘ
Fermat’s principle of least time ⓘ
surface form:
Fermat's principle
geodesic ⓘ principle of least action ⓘ tautochrone problem ⓘ |
| solutionCurve | cycloid ⓘ |
| solutionFamily | one-parameter family of cycloidal arcs through given endpoints ⓘ |
| solutionProperty |
gives minimum time of descent
ⓘ
not a circular arc ⓘ not a straight line ⓘ |
| standardExampleIn |
advanced mechanics textbooks
ⓘ
introductory courses on calculus of variations ⓘ |
| teachesConcept |
difference between shortest path and quickest path
ⓘ
variational extremals may be non-intuitive curves ⓘ |
| typicalAssumption | fixed endpoints with lower point vertically below upper point or horizontally displaced ⓘ |
| usesMethod |
Euler–Lagrange equation
ⓘ
surface form:
Euler–Lagrange differential equation
Snell's law analogy ⓘ |
| usesPrinciple | conservation of mechanical energy ⓘ |
| yearPosed | 1696 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.