The equations of motion of a particle in a constant force field (such as its wei

Question

The equations of motion of a particle in a constant force field (such as its weight close to the surface of the Earth) are invariant under translations in all directions. And yet, momentum in the direction of the force is not conserved! What's the deal with Nother? Write down the Lagrangian of a particle of mass m in three dimensions feeling a constant force F rightarrow with components Fx, Fy, Fz in terms of the Cartesian coordinates of the particle x,y,z. Verify that the transformation delta xi = with a constant vector, is a symmetry. Find the conserved quantity corresponding to this symmetry. Is it the momentum? What does it represent?

Explanation / Answer

F? =md2r? dt2 v? =?F? m dt=??(r)r? m dt which is equal to : ?(r)tr? m+c (I am not sure what I am doing at this point. Is my integrated expression correct?) Assuming it is, we get: Angular Momentum L=m(r? ×v? )=r? ×(?(r)tr? +c) Now I don't know what to do with the constant term, but I do know that r? ×kr? =0

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