I am not entirely sure if this question belongs to this forum, but I don't know where else to ask.

I am following a Rocket Propulsion course. There is a section concerned with the Velocity Equations during Atmospheric Re-Entry.

Velocity Equations

Where:

  • L is Lift force
  • D is Drag force
  • $\phi$ is the flight path angle
  • $\mathbf{e_r}$, $\mathbf{e_v}$, $\mathbf{e_L}$, etc. are unit vectors in the direction of the subscripted parameters

The derivation of the equations is clear for me, I just don't know where this Poisson's formula comes from. I googled it and I just find the well known differential equation or information about the statistical distribution. I know this should be a basic concept but right now it looks like I am missing something.

Thank you very much

  • 1
    Probably several people will recognize these equations instantly, but as a courtesy to us remaining readers could you perhaps mention what $\mathbf{e}$, $\mathbf{D}$, and $\mathbf{L}$ and maybe $\mathbf{\phi}$? Thanks! – uhoh Sep 13 at 18:37
  • 1
    My apologies, I will edit now the question. – Roger Pedrós Bòria Sep 13 at 18:55
  • Looks great, thank you! – uhoh Sep 13 at 23:10

It seems that this Poisson's formula is very specific from the Flight Mechanics community and I found its definition in two different books: Flight Mechanics of High-Performance Aircraft by Vinh (section 2.2) and Flight Mechanics: Theory of Flight Paths by Miele (section 4 of chapter 1).

As explained by Miele, the Poisson's formulas

show that the derivatives of the moving unit vectors are perpendicular to $\mathbf{\omega}$. The physical significance of this vector, which is called the angular velocity of the moving reference frame with respect to the fixed reference frame, is illustrated in the next section.

They can be written as

$$ \frac{d\mathbf{i}}{dt} = \mathbf{\omega} \times\mathbf{i}$$ $$ \frac{d\mathbf{j}}{dt} = \mathbf{\omega} \times\mathbf{j}$$ $$ \frac{d\mathbf{k}}{dt} = \mathbf{\omega} \times\mathbf{k}$$

where $\mathbf{\omega} = p\mathbf{i} + q\mathbf{j} + r\mathbf{k}$.

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