Momentum: Difference between revisions
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Eric Lengyel (talk | contribs) (Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==...") |
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:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ | :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ | ||
The inertia tensor $$\mathcal I$$ is given by | |||
:$$\mathcal I = {\Large\sum} \gamma m{\left[\begin{array}{cccc|ccc|ccc} | |||
-1 & 0 & 0 & 0 & 0 & 0 & 0 & -x & -y & -z \\ | |||
0 & 1 & 0 & 0 & 0 & z & -y & -ct & 0 & 0 \\ | |||
0 & 0 & 1 & 0 & -z & 0 & x & 0 & -ct & 0 \\ | |||
0 & 0 & 0 & 1 & y & -x & 0 & 0 & 0 & -ct \\ | |||
\hline | |||
0 & 0 & -z & y & y^2 + z^2 & -xy & -zx & 0 & ctz & -cty \\ | |||
0 & z & 0 & -x & -xy & z^2 + x^2 & -yz & -ctz & 0 & ctx \\ | |||
0 & -y & x & 0 & -zx & -yz & x^2 + y^2 & cty & -ctx & 0 \\ | |||
\hline | |||
-x & -ct & 0 & 0 & 0 & -ctz & cty & c^2t^2 - x^2 & -xy & -zx \\ | |||
-y & 0 & -ct & 0 & ctz & 0 & -ctx & -xy & c^2t^2 - y^2 & -yz \\ | |||
-z & 0 & 0 & -ct & -cty & ctx & 0 & -zx & -yz & c^2t^2 - z^2 | |||
\end{array}\right]}$$ . | |||
As in classical mechanics, the inertia tensor $$\mathcal I$$ has units of mass × length<sup>2</sup>. | |||
== See Also == | == See Also == | ||
* [[Velocity]] | * [[Velocity]] | ||
Latest revision as of 01:12, 20 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
- $$\mathcal I = {\Large\sum} \gamma m{\left[\begin{array}{cccc|ccc|ccc} -1 & 0 & 0 & 0 & 0 & 0 & 0 & -x & -y & -z \\ 0 & 1 & 0 & 0 & 0 & z & -y & -ct & 0 & 0 \\ 0 & 0 & 1 & 0 & -z & 0 & x & 0 & -ct & 0 \\ 0 & 0 & 0 & 1 & y & -x & 0 & 0 & 0 & -ct \\ \hline 0 & 0 & -z & y & y^2 + z^2 & -xy & -zx & 0 & ctz & -cty \\ 0 & z & 0 & -x & -xy & z^2 + x^2 & -yz & -ctz & 0 & ctx \\ 0 & -y & x & 0 & -zx & -yz & x^2 + y^2 & cty & -ctx & 0 \\ \hline -x & -ct & 0 & 0 & 0 & -ctz & cty & c^2t^2 - x^2 & -xy & -zx \\ -y & 0 & -ct & 0 & ctz & 0 & -ctx & -xy & c^2t^2 - y^2 & -yz \\ -z & 0 & 0 & -ct & -cty & ctx & 0 & -zx & -yz & c^2t^2 - z^2 \end{array}\right]}$$ .
As in classical mechanics, the inertia tensor $$\mathcal I$$ has units of mass × length2.