na_lec_6a

na_lec_6a - The Euler-Lagrange Equation in Expanded Form...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The Euler-Lagrange Equation in Expanded Form ∂F d ⎛ ∂F ⎞ ⎟=0 −⎜ ∂y dx ⎜ ∂y ′ ⎟ ⎝ ⎠ (1) ∂F ∂ 2 F dx ∂ 2 F dy ∂ 2 F dy ′ − − − =0 ∂y ∂x∂y ′ dx ∂y∂y ′ dx ∂y ′∂y ′ dx (2) ∂F ∂ 2 F ∂ 2 F dy ∂ 2 F d 2 y − − − =0 ∂y ∂x∂y ′ ∂y∂y ′ dx ∂y ′ 2 dx 2 (3) Special Case I: F = F ( y, y ') If F = F ( y , y ') . (4) Then the Euler-Lagrange Equation reduces to F − y′ ∂F =C ∂y ′ (5) where C is an arbitrary constant. Proof: d ∂F ∂F dy′ F ( y , y′) = y′ + dx ∂y ∂y′ dx (6) d ⎛ ∂F ⎞ dy ′ ∂F d ⎛ ∂F ⎞ ⎜ y' ⎜ ∂y ' ⎟ = dx ∂y ′ + y ′ dx ⎜ ∂y ' ⎟ ⎟ ⎜ ⎟ dx ⎝ ⎠ ⎝ ⎠ (7) d⎛ ∂F ⎞ ∂F d ⎛ ∂F ⎞ ⎜ F − y′ ⎟= ⎜ ⎟ ∂y y ′ − y ′ dx ⎜ ∂y ' ⎟ ⎜ ⎟ dx ⎝ ∂y ′ ⎠ ⎝ ⎠ ⎛ ∂F d ⎛ ∂F ⎞ ⎞ = y ′⎜ ⎜ ⎟⎟ ⎜ ∂y − dx ⎜ ∂y ' ⎟ ⎟ = 0 ⎝ ⎠⎠ ⎝ which implies (5). (8) Special Case II: F = F ( x, y ') If F = F ( x, y ') . (4) Then the Euler-Lagrange Equation reduces to ∂F =C ∂y ′ where C is an arbitrary constant. This follows trivially from (1). (5) ...
View Full Document

Page1 / 2

na_lec_6a - The Euler-Lagrange Equation in Expanded Form...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online