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Lagrangian & Hamiltonian Formulations question bank (Classical Mechanics & Electrodynamics)

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Q1.

[Classical Mechanics & Electrodynamics Problem #1] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

The matrix cannot be diagonalized under any orthogonal transformation

(B)

All eigenvalues are pure imaginary

(C)

Eigenvalues must always be positive integers

(D)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

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Q2.

[Classical Mechanics & Electrodynamics Problem #3] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(B)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(C)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(D)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

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