By P. W Bridgman
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Extra resources for A sophisticate's primer of relativity
171): ∂H 1 = pβ pγ g γβ,α + U,α ∂x α 2 ∂H = pα . 174) Then we have:8 1 [H, J ] = − p γ p β gγβ,α + U,α 2 = −p p α 8 β f α − g γ δ pδ pα f α,γ 1 γ gαβ,γ f γ + gαγ f ,β + U,α f α = 0. 176) We use the result gµα g αβ,γ gβν = −gµν,γ , obtainable via the product rule for the ordinary derivative. 175) 44 Newtonian gravity Notice the two separate pieces to the Poisson bracket: we must have both the term multiplying p α p β equal to zero, and U,α f α = 0 (which says that the coordinate transformation must be orthogonal to the force).
3 Lagrangian for U (r) 15 for example) we see are manifestations of the curvature of the space-time (expressed via the metric). In that context, it must be the ανγ term that approximates the forcing we would normally associate with a Newtonian gravitational source. 60) L = m x˙ 2 + y˙ 2 + z˙ 2 − U x 2 + y 2 + z2 2 and transform the coordinates (and velocities) directly (no metric allowed) to find the Lagrangian associated with a central potential in cylindrical coordinates, with x 1 = s, x 2 = φ, x 3 = z.
103) where the pair f (x) and g(p) are Legendre transforms of each other. There is a nice symmetry here, the same transformation takes us back and forth. Example Let’s look at an example to see how the transformation works in practice. Consider the function f (x) = α x m for integer m. We can define p from the derivative of f (x) as prescribed above: df p = α m x m−1 = p −→ x(p) = dx αm 1 m−1 . 4 The Legendre transform constructs a function g(p) from f (x) by swapping the role of the local slope of the curve for its argument.
A sophisticate's primer of relativity by P. W Bridgman