Calculus of Variations
Gelfand & Fomin
Elements Of The Theory
13 problems
- Find the extremal of ∫ √(1 + y′²)/x dx with fixed endpoints
- Find the extremals of ∫ (12xy + y′²) dx
- Find the extremals of ∫ (3x + √y′) dx
- Find the extremals of ∫ (x + y² + 3y′) dx
- Find the extremals of ∫ (xy′ + y′²) dx
- Find the extremals of ∫ (y² − y′²) dx
- Find the extremals of ∫ (y² + x²y′) dx
- Find the extremal of ∫ (1 + y″²) dx with fixed endpoints and slopes
- Find the extremal of ∫ (y″² − y² + x²) dx with fixed endpoints and slopes
- Find the curve of revolution with minimum surface area
- Find the extremals of ∫ (2yz − 2y² + y′² − z′²) dx
- Find the extremals of ∫ (y′² + z′² + 2yz) dx with fixed endpoints
- Find the extremals of ∫ (y′² + z′² + y′z′) dx
Further Generalizations
3 problems
The General Variation Of A Functional
7 problems
- Find the extremals of ∫ (y′² + z′² + 2yz) dx with y(0) = z(0) = 0 and the right endpoint free in the plane x = x₁
- Find the extremals of ∫₀^{π/4} (y² − y′²) dx with y(0) = 0 and the right endpoint free on x = π/4
- Derive the general variation of ∫ F(x, y, y′, y″) dx
- Derive the general variation of ∫ F(x, y, y′) dx + G(x₀, y₀, x₁, y₁)
- Prove that ∫ (ay′² + byy′ + cy²) dx with a ≠ 0 and fixed endpoints has no broken extremals
- Derive the transversality conditions for ∫ F(x, y, y′) dx + G(x₀, y₀, x₁, y₁) with endpoints on the curves y = φ(x) and y = ψ(x)
- Derive the transversality conditions for ∫ F(x, y, z, y′, z′) dx with endpoints on the surfaces x = φ(y, z) and x = ψ(y, z)
The Canonical Form of the Euler Equations
3 problems
- Write and interpret the canonical Euler equations for the harmonic oscillator action ½ ∫ (mẋ² − kx²) dt
- Use the canonical Euler equations to find the extremals of ∫ √(x² + y²) √(1 + y′²) dx
- Show that the principle of least action for a system of n particles is equivalent to Newton's equations of motion
Direct Methods
7 problems
- Find the exact extremal of ∫₀¹ (y′² − y² − 2xy) dx with y(0) = y(1) = 0
- Find the exact extremal of ∫₀² (y′² + y² + 2xy) dx with y(0) = y(2) = 0
- Implement the Ritz method in Julia as a general solver for 1D functionals ∫ₐᵇ F(x, y, y′) dx
- Use the Ritz method to find a one-term approximate minimizer of ∬ (u_x² + u_y² − 2u) dx dy on a square, with u = 0 on the boundary
- Use the Ritz method to approximate the extremal of ∫₀¹ (x³y″² + 100xy² − 20xy) dx with y(1) = y′(1) = 0
- Use the Ritz method to find one- and two-term approximate minimizers of ∫₀¹ (y′² − y² − 2xy) dx with y(0) = y(1) = 0
- Use the Ritz method to find one- and two-term approximate minimizers of ∫₀² (y′² + y² + 2xy) dx with y(0) = y(2) = 0