Calculus of Variations
28 questions across 2 topics. Also examined in CNET.
Variation of a Functional, Euler-Lagrange Equation, Conditions for Extrema
- 82Calculus of variations — polar arc lengthMSET June 2025Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 83Calculus of variations — geodesics in R^3MSET June 2025Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 84Calculus of variations — speed v=x gives circles centred on the y-axisMSET June 2025Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 85Calculus of variations — Euler's equation when F=F(x,y')MSET June 2025Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 86Calculus of variations — a free end-point adds a natural boundary conditionMSET June 2025Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 87Optimisation — coercive functions grow at infinityMSET June 2025Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 83Isoperimetric problem — Euler–Lagrange with a Lagrange multiplierMSET April 2024Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 84First integral of a second-order Euler–Lagrange (Euler–Poisson) equationMSET April 2024Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 85Solving a boundary value problem via Euler–LagrangeMSET April 2024Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 83The isoperimetric theorem — the circle maximises area for fixed perimeterMSET March 2023Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 84An extremal problem reducing to 4y=x(x-1)MSET March 2023Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 86The fully expanded Euler–Lagrange equationMSET March 2023Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 82An isoperimetric-free extremal problem — when the boundary data becomes inconsistentMSET September 2021Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 83First integral of Euler–Poisson when f omits yMSET September 2021Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 84Solving a boundary value problem from Euler–LagrangeMSET September 2021Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 85The brachistochrone is a cycloidMSET September 2021Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 82The extremal family for int(y^2-y'^2)dx, satisfying one endpoint conditionMSET December 2020Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 83Computing the first variation term by termMSET December 2020Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 84Beltrami's identity for an x-free integrand gives a hyperbolic-sine familyMSET December 2020Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 82A resonant boundary condition leaves one constant completely freeMSET June 2019Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 85The geodesic on a sphereMSET June 2019Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 59Euler–Lagrange for a double integral: one divergence term per variableMSET January 2018 Paper IIIVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 60F free of y: the first integral Fy'= const gives y'propto xMSET January 2018 Paper IIIVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 59Euler–Poisson with fy=0 integrates once: fy'-frac ddxfy"= constMSET April 2017 Paper IIIVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 60y^(4)=y, and sin x meets all four end conditionsMSET April 2017 Paper IIIVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 61The Dirichlet integral leads to Laplace's equationMSET April 2017 Paper IIIVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy