We consider a non-relativistic particle of mass m in a central potential V(r), where r =x2 +

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We consider a non-relativistic particle of mass m in a central potential V(r), where r =√x2 + y2 + z2. We denote the velocity v ≡ ˙r and v2 its square. We study the problem in spherical coordinates (r, θ, φ) defined byimage text in transcribed

The square of the velocity isimage text in transcribed

1. Write the Lagrangian of the particle in spherical coordinates.
2. Calculate the conjugate momenta pr , pθ, and pφ.
3. Show that the momentum pφ is equal to the z component of the angular momentum Lz whose expression in Cartesian coordinates is Lz = xpy − ypx.
4. To what invariance law does the conservation of Lz correspond?
5. If the particle is charged and placed in a magnetic field B parallel to the z axis, is the component Lz conserved?

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Variational Principles In Physics

ISBN: 9783031216916

2nd Edition

Authors: By Jean-Louis Basdevant

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