Multipole Expansion Homework: Invariance w/ Orthogonal Rotation

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SUMMARY

The discussion focuses on demonstrating the invariance of the multipole moment expansion under orthogonal rotations. The participant seeks to explicitly show how coordinates transform during such rotations, particularly in the xy-plane, to validate the invariance of multipole terms. The reference to Exercise 2.2 indicates a need for a similar approach to coordinate transformation, emphasizing the importance of understanding the mathematical framework behind multipole expansions. This inquiry highlights the necessity of grasping both the physical implications and the mathematical techniques involved in multipole moment analysis.

PREREQUISITES
  • Understanding of multipole moments in physics
  • Familiarity with orthogonal transformations
  • Knowledge of coordinate systems and transformations
  • Basic proficiency in mathematical physics
NEXT STEPS
  • Study the mathematical derivation of multipole moments
  • Learn about orthogonal rotation matrices and their properties
  • Explore the implications of coordinate transformations in physics
  • Review Exercise 2.2 from the provided link for related concepts
USEFUL FOR

Students and researchers in physics, particularly those studying classical mechanics and electromagnetism, will benefit from this discussion, especially those focusing on multipole expansions and coordinate transformations.

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Homework Statement



Given the multipole moment of the mass distribution how would I go about determining that the multipole moment expansion is invariant. I

Homework Equations



http://cohengroup.ccmr.cornell.edu/courses/phys3327/HW2/hw2.pdf

The Attempt at a Solution



I need to explicitly show how the coordinates transform over an orthogonal rotation. I'm not sure how to do this part explicitly and for an expansion of any multipole term. The link to EX 2.2 is similar to what I'm asking but instead of a translation I need to show for a rotation of the basis.
 
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I think this is done by showing how the coordinates transform under a rotation in the xy-plane. I'm not sure how to go about doing this for an arbitrary multipole moment.
 

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