Momentum: Difference between revisions
Jump to navigation
Jump to search
Eric Lengyel (talk | contribs) (Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==...") |
Eric Lengyel (talk | contribs) No edit summary |
||
| Line 2: | Line 2: | ||
:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ | :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ | ||
The inertia tensor $$\mathcal I$$ is given by | |||
:$$\mathcal I = \gamma \sum {{\begin{bmatrix} | |||
-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\ | |||
0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\ | |||
0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\ | |||
0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\ | |||
0 & 0 & -mz & my & m(y^2 + z^2) & -mxy & -mzx & 0 & mctz & -mcty \\ | |||
0 & mz & 0 & -mx & -mxy & m(z^2 + x^2) & -myz & -mctz & 0 & mctx \\ | |||
0 & -my & mx & 0 & -mzx & -myz & m(x^2 + y^2) & mcty & -mctx & 0 \\ | |||
-mx & -mct & 0 & 0 & 0 & -mctz & mcty & m(c^2t^2 - x^2) & -mxy & -mzx \\ | |||
-my & 0 & -mct & 0 & mctz & 0 & -mctx & -mxy & m(c^2t^2 - y^2) & -myz \\ | |||
-mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2) | |||
\end{bmatrix}}}$$ . | |||
== See Also == | == See Also == | ||
* [[Velocity]] | * [[Velocity]] | ||
Revision as of 07:13, 11 November 2024
The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.
- $$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$
The inertia tensor $$\mathcal I$$ is given by
- $$\mathcal I = \gamma \sum {{\begin{bmatrix} -m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\ 0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\ 0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\ 0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\ 0 & 0 & -mz & my & m(y^2 + z^2) & -mxy & -mzx & 0 & mctz & -mcty \\ 0 & mz & 0 & -mx & -mxy & m(z^2 + x^2) & -myz & -mctz & 0 & mctx \\ 0 & -my & mx & 0 & -mzx & -myz & m(x^2 + y^2) & mcty & -mctx & 0 \\ -mx & -mct & 0 & 0 & 0 & -mctz & mcty & m(c^2t^2 - x^2) & -mxy & -mzx \\ -my & 0 & -mct & 0 & mctz & 0 & -mctx & -mxy & m(c^2t^2 - y^2) & -myz \\ -mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2) \end{bmatrix}}}$$ .