Angular Momentum

Table of Contents

Time derivatives in a rotating frame

Let angular_momentum_406b440e753d69236dae0ac54e25bfaa615eb3c7.png, angular_momentum_b3cfb4d1b971f84df01956dc097853fd6a3bf93b.png and angular_momentum_47ea5171e1e444a1ea465c2b9c94949d6325c89a.png be unit basis vectors in the rotating frame, with a rotational speed of angular_momentum_cdfd3a815ed27b71796fe6289ce253c6c19a2803.png about the axis angular_momentum_85a1e8e3a60016f8a8c3f1f030b64485b85a5dde.png. Letting angular_momentum_dccddd4b3f21520c5264bb6d6d756f95eb0ab371.png be the unit vector in this frame, we have:

angular_momentum_a51c4cbb407668b70b239c472d505cd2cec2a68a.png

Then if we have a function angular_momentum_97c3cb721cd6d76849d88aff3353c3110783a9ae.png, taking the derivative in the rotational frame :

angular_momentum_9314da8b4fe79faff0c3a2a1f670d94311f0722c.png

Which describes the change of angular_momentum_97c3cb721cd6d76849d88aff3353c3110783a9ae.png wrt. angular_momentum_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png using the rotational frame.

Euler's Equations (rigid body dynamics)

From Time derivatives in a rotating frame we have:

angular_momentum_6492fcd4a60465a5270092469de4947697fa4f37.png

where angular_momentum_348fdd4928a3267bb8bf3b3788e8cb9c7759d940.png is the moment of inertia tensor calculated in the intertial frame .

It's often more useful to change to coordinates to the rotating frame .

With rotating frame we mean that the coordinate axes are rotating about a fixed axis.

Why is this useful? In this frame the moment of inertia tensor is constant (and diagonal)!

Moment of inertia

Covariance and contravariance

  • Describe how the quantitive description of a certain geometric or physical entities change with a change of basis

Covariance

  • Components of a covector change in the same way as changes in scale of the reference axes → the covector and the reference axes are covariant

Contravariance

  • Components of a contr