I am confused by two different definitions of the angular momentum of a particle P about a moving point Q, with point O as the origin of the inertial frame. I learned the first definition from my sophomore dynamics course, and now I am taking analytical dynamics and the textbook gave a different definition.

The first definition looks like this:

$$\vec{h}_{P/Q} = \vec{r}_{P/Q} \times m\vec{v}_{P/O}$$

Where h_P/Q is the angular momentum of particle P w.r.t point Q; r_P/Q is the relative position of P w.r.t Q, v_P/O is the inertial velocity of particle P w.r.t a fixed point O.

The second definition looks like this:

$$\vec{h}_{P/Q} = \vec{r}_{P/Q} \times m\vec{v}_{P/Q}$$

Basically the velocities are taken w.r.t different points.

Which one is the correct one?

• Your equations are formatted in a way that's hard to understand or recognize. Can you add some explanation in words? May 25 at 20:29
• What is "velocity with respect to a fixed point"? Velocity is independent of a reference point (assuming we do not change frame) May 25 at 20:39
• @DanDan0101 velocity w.r.t a fixed point is the velocity relative to point O, which is the origin point of the inertial frame. In the second equation, velocity relative to the moving point Q is used, but it is still the velocity as seen from the inertial frame. May 25 at 20:42
• I edited your question to write the equations using MathJax/Latex that is used on this site. It is pretty easy to learn. Feel free to make a change if it does not reflect your intent. May 25 at 20:44
• @RC_23 Thank you for helping me with the formatting! May 25 at 20:46

If the momentum is calculated with respect to the origin of the inertial frame: $$\mathbf L = \mathbf r_{OP} \times \mathbf v_{OP} = (\mathbf r_{OQ} + \mathbf r_{QP}) \times \mathbf v_{OP}$$