Axial angular momentum calculation

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SUMMARY

The discussion focuses on the calculation of axial angular momentum, denoted as ##L_a##, which is defined as the component of the angular momentum vector ##\vec L## along a specified axis ##a## with respect to a reference point ##P##. The moment of inertia tensor ##\mathbf I## is introduced as a crucial element for calculating ##L_a## when the system rotates as a rigid body with angular velocity ##\omega##. It is established that the inertia tensor can vary based on the body's orientation relative to the coordinate axes, particularly when the body is not symmetric. The conversation also touches on the mathematical representation of angular momentum and the implications of using different coordinate systems.

PREREQUISITES
  • Understanding of angular momentum and its vector properties
  • Familiarity with the moment of inertia tensor and its role in rigid body dynamics
  • Knowledge of coordinate systems and transformations in physics
  • Basic concepts of covectors and pseudovectors in mechanics
NEXT STEPS
  • Study the properties of the moment of inertia tensor in various coordinate systems
  • Learn about the Parallel Axis Theorem and its applications in rigid body dynamics
  • Explore the mathematical formulation of angular momentum using wedge products
  • Investigate the implications of using body-fixed coordinate systems in dynamics
USEFUL FOR

Physicists, mechanical engineers, and students studying rigid body dynamics, particularly those interested in angular momentum calculations and the behavior of inertia tensors in different configurations.

  • #31
cianfa72 said:
In the context of Newtonian physics I believe there is no gain from the bi-vector language/definition.
This is probably true for textbook problems, where simple systems are meant to be solved by hand. But this assessment might change in the context of automated computation of more complex systems, which have a lot of symmetry, like some robots or animated characters. Here, you often have to mirror the mechanics from one side to the other. And not having to keep track which vectors are pseudo-vectors, and thus have to be negated after mirroring, might simplify the code.

Think about how quaternions where replaced in physics by vector analysis in the late 19th century, but then had a comeback in the late 20th century in computer graphics and robotics:
https://en.wikipedia.org/wiki/Quaternion#History
 
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