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black_hole
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Homework Statement
We are given L = 1/2mv2 - mgz.
a) Find the equations of motion.
b) Take x(0) [vector] = 0; v(0) [vector] = v0 [vector] ; v0z > 0 and find x(τ) [vector] and v(τ) [vector], such that z(τ) = 0; τ≠0.
c) Find S.
Homework Equations
Euler-Lagrange equation and definition of action.
The Attempt at a Solution
a) Using the Euler-Lagrange equation:
∂L/∂x - d/dt(∂L/∂v) = 0
-mg - d/dt(mv) = 0
-mg = d/dt(mv)
F = -dp/dt
b) If F = -dp/dt, then a = -g. So;
v = -vt + v0 and x = -1/2gt2 + v0t
Hmmm, that's where I'm stuck; I'm not sure how to implement the last two constraints. Maybe if z(τ) = 0 but τ≠0 then z = 1/2gt2 - vozt = 0 when t = (z * v0z)/g = τ ?
(I'm not explicitly given v0z...)
c) S = ∫Ldt = 1/2m∫v2dt - mg∫zdt
If my above expressions for v and z are correct then I presume I can use those tot find the integrals...?