In the relationship Δϕ1+ Δϕ2 = Δϕ2+Δϕ1 valid if Δϕ1 and Δϕ2 refer to diffrent axes of rotayion ? Is it valid if they refer to diffrent rotations about the same axis ?

To address your question about the relationship Δϕ1 + Δϕ2 = Δϕ2 + Δϕ1, we need to delve into the concepts of rotational angles and how they interact in different contexts. This equation essentially expresses the commutative property of addition, which holds true regardless of the specific nature of the quantities involved, as long as they are additive. Let's break this down further.

Understanding Rotational Angles

In physics, particularly in the study of rotational dynamics, Δϕ1 and Δϕ2 represent changes in angular position (or rotation) about specific axes. The key point here is that these angles can refer to different axes of rotation or even different rotations about the same axis.

Different Axes of Rotation

When Δϕ1 and Δϕ2 refer to different axes of rotation, the relationship still holds true. For example, if you have one rotation about the x-axis and another about the y-axis, you can still add these angular displacements together. The result will be a vector sum of the rotations, which can be represented in a three-dimensional space. The commutative property applies here because the order in which you add these angular displacements does not affect the total rotation.

• Example: If Δϕ1 = 30° about the x-axis and Δϕ2 = 45° about the y-axis, then:
• Δϕ1 + Δϕ2 = 30° + 45° = 75° (in terms of a vector sum).

Same Axis of Rotation

Now, if Δϕ1 and Δϕ2 refer to different rotations about the same axis, the equation still holds. In this case, you can simply add the angles directly since they are both contributing to the same rotational effect. The order of addition remains irrelevant, which is a hallmark of the commutative property.

• Example: If Δϕ1 = 30° and Δϕ2 = 45°, both about the z-axis, then:
• Δϕ1 + Δϕ2 = 30° + 45° = 75° about the z-axis.

Conclusion on Validity

In summary, the relationship Δϕ1 + Δϕ2 = Δϕ2 + Δϕ1 is valid whether Δϕ1 and Δϕ2 refer to different axes of rotation or different rotations about the same axis. This reflects the fundamental nature of angular displacement as an additive quantity, allowing for flexibility in how we analyze and combine rotations in physics.