University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

394

Pages:

6
Author:

Jabin F.

# Properties of Levi-Civita Tensor εijk Its value is zero when two indices are equal. ε₁₁₁ = ε₁₁₂ = 0 Its sign changes on interchanging adjacent indices. ε₁₂₃ = -ε₂₁₃ = -ε₁₃₂ = ε₃₂₁ = ε₃₁₂ Its value doesn't change on changing the indices in cyclic order. ε₁₂₃ = ε₂₃₁ = ε₃₁₂ Its value is one when all indices are different and in cyclic order. ε₁₂₃ = ε₂₃₁ = ε₃₁₂ = 1 Its value is -1 when all indices are different but not in cyclic order. ε₁₃₂ = ε₃₂₁ = -1 When we sum over i, j, k, then there are 6 terms in which indices are different. εᵢⱼₖεᵢₗₘ = δⱼₗδₖₘ - δⱼₘδₖₗ Q2. Check: Solution: For i = 1, j = 1: (a^1 x b^1) - (b^1 x a^1) = a₁b₁ - b₁a₁ = 0 Similarly, for i = 2, j = 3: (a^2 x b^3) - (b^3 x a^2) = a₂b₃ - b₃a₂ = 0 P-100 The Poisson bracket {f₁, f₂} has the value: Q-10 Solution: {f₁, f₂} = {x² + y², xy} = (∂(x² + y²)/∂x)(∂xy/∂y) - (∂(x² + y²)/∂y)(∂xy/∂x) = (2x)(x) - (2y)(y) = 2x² - 2y² = 1/2i[f₁, f₂] = 1/2i(2x² - 2y²) = ix² - iy²