How to Derive Time Derivative of Spherical Unit Vectors?

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SUMMARY

The discussion focuses on deriving the time derivative of spherical unit vectors in the context of a spherical pendulum problem. The unit vectors are expressed in Cartesian coordinates as follows: r = sin(θ)cos(φ) i + sin(θ)sin(φ) j + cos(θ) k, θ = cos(θ)cos(φ) i + cos(θ)sin(φ) j - sin(θ) k, and φ = -sin(φ) i + cos(φ) j. The key conclusion is that to derive the time derivatives of these unit vectors, one must treat the angles θ and φ as functions of time, while the unit vectors themselves remain constant.

PREREQUISITES
  • Understanding of spherical coordinates and their relation to Cartesian coordinates.
  • Familiarity with vector calculus, specifically derivatives of vector functions.
  • Knowledge of trigonometric functions and their derivatives.
  • Basic concepts of pendulum motion and dynamics.
NEXT STEPS
  • Study the derivation of vector functions in calculus, focusing on time derivatives.
  • Learn about the application of spherical coordinates in physics, particularly in pendulum dynamics.
  • Explore the relationship between angular motion and time derivatives in spherical coordinates.
  • Investigate the use of Lagrangian mechanics for analyzing pendulum systems.
USEFUL FOR

Physics students, mechanical engineers, and anyone studying dynamics and kinematics of pendulum systems will benefit from this discussion.

General-Simon
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Here I am doing a spherical pendulum problem, and i was asked to represent the unit vectors of spherical coor in terms of Cartesian coor, which i have already solved:
r=sinθcosφ i + sinθsinφ j + cosθ k
θ=cosθcosφ i + cosθsinφ j - sinθ k
φ= -sinφ i + cosφ j
where φ is the angle on the X-Y plane, between x and the position, and θ is between position and z

now i want to know how to process to the next step: derive the time derivative of each unit vector, in terms of spherical unit vectors.
 
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Just derive it
I don't know what's the problem
 
netheril96 said:
Just derive it
I don't know what's the problem

ok, Thanks, I already solved it. considering the θ and φ also as the function of t, but the unit vectors do not.
 

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