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# p09fl38 - Lecture 38 Relativistic mechanics(2 Dec 09...

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Lecture 38: Relativistic mechanics (2 Dec 09) Project: 2 homework-like problems from Fetter-Walecka or Goldstein on “under-represented topics” in the homework, for instance Hamilton-Jacobi theory, continuum mechanics, hydrodynamics, or relativity. A. Review: 4-vector formalism 1. The Lorentz transformation along ˆ x is ct * = γ ( ct - βx ) x * = γ ( x - βct ) y * = y ; z * = z ; β = v/c ; γ = 1 / q 1 - β 2 2. Consider this as a special case of transformations on 4-vector x 0 (= ct ) , x, y, z preserving the length r 2 - x 2 0 . ( x μ , x μ ) X μ g μ x 2 μ ; g 0 = - 1 , g 1 = g 2 = g 3 = 1 3. In matrix notation (this transform is the defining property of a 4- vector): x * μ = X ν a μν x ν X μ g μ a μν a μλ = g ν δ νλ and the inverse matrix is a - 1 νμ = g ν g μ a μν x ν = X μ g ν g μ a μν x * μ 4. Form a 4-vector from the gradient (and partial with respect to t ): 2 μ = g μ ∂x μ ; g μ ∂x * μ = X ν a μν g ν ∂x ν and the wave operator is a 4-scalar: ( 2 μ , 2 μ ) = X μ g μ 2 μ 2 μ = 2 - 1 c 2 2 ∂t 2 1

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B. Mechanics 1. r
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p09fl38 - Lecture 38 Relativistic mechanics(2 Dec 09...

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