Momentum - Revision history

Eric Lengyel at 01:12, 20 November 2024

2024-11-20T01:12:20Z

← Older revision		Revision as of 01:12, 20 November 2024
Line 5:		Line 5:
	The inertia tensor $$\mathcal I$$ is given by		The inertia tensor $$\mathcal I$$ is given by

	:$$\mathcal I = \~~gamma~~ \sum{\left[\begin{array}{cccc\|ccc\|ccc}		:$$\mathcal I = {\Large\sum} \gamma m{\left[\begin{array}{cccc\|ccc\|ccc}
	-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\		-1 & 0 & 0 & 0 & 0 & 0 & 0 & -x & -y & -z \\
	0 & m & 0 & 0 & 0 & mz & -my & -~~mct~~ & 0 & 0 \\		0 & 1 & 0 & 0 & 0 & z & -y & -ct & 0 & 0 \\
	0 & 0 & m & 0 & -mz & 0 & mx & 0 & -~~mct~~ & 0 \\		0 & 0 & 1 & 0 & -z & 0 & x & 0 & -ct & 0 \\
	0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -~~mct~~ \\		0 & 0 & 0 & 1 & y & -x & 0 & 0 & 0 & -ct \\
	\hline		\hline
	0 & 0 & -mz & my & m(y^2 + z^2) & -~~mxy~~ & -~~mzx~~ & 0 & ~~mctz~~ & -~~mcty~~ \\		0 & 0 & -z & y & y^2 + z^2 & -xy & -zx & 0 & ctz & -cty \\
	0 & mz & 0 & -mx & -~~mxy~~ & m(z^2 + x^2) & -~~myz~~ & -~~mctz~~ & 0 & ~~mctx~~ \\		0 & z & 0 & -x & -xy & z^2 + x^2 & -yz & -ctz & 0 & ctx \\
	0 & -my & mx & 0 & -~~mzx~~ & -~~myz~~ & m(x^2 + y^2) & ~~mcty~~ & -~~mctx~~ & 0 \\		0 & -y & x & 0 & -zx & -yz & x^2 + y^2 & cty & -ctx & 0 \\
	\hline		\hline
	-mx & -~~mct~~ & 0 & 0 & 0 & -~~mctz~~ & ~~mcty~~ & m(c^2t^2 - x^2) & -~~mxy~~ & -~~mzx~~ \\		-x & -ct & 0 & 0 & 0 & -ctz & cty & c^2t^2 - x^2 & -xy & -zx \\
	-my & 0 & -~~mct~~ & 0 & ~~mctz~~ & 0 & -~~mctx~~ & -~~mxy~~ & m(c^2t^2 - y^2) & -~~myz~~ \\		-y & 0 & -ct & 0 & ctz & 0 & -ctx & -xy & c^2t^2 - y^2 & -yz \\
	-mz & 0 & 0 & -~~mct~~ & -~~mcty~~ & ~~mctx~~ & 0 & -~~mzx~~ & -~~myz~~ & m(c^2t^2 - z^2)		-z & 0 & 0 & -ct & -cty & ctx & 0 & -zx & -yz & c^2t^2 - z^2
	\end{array}\right]}$$ .		\end{array}\right]}$$ .

			As in classical mechanics, the inertia tensor $$\mathcal I$$ has units of mass × length<sup>2</sup>.

	== See Also ==		== See Also ==

	* [[Velocity]]		* [[Velocity]]

Eric Lengyel at 23:41, 13 November 2024

2024-11-13T23:41:40Z

← Older revision		Revision as of 23:41, 13 November 2024
Line 5:		Line 5:
	The inertia tensor $$\mathcal I$$ is given by		The inertia tensor $$\mathcal I$$ is given by

	:$$\mathcal I = \gamma \sum {{\left[\begin{array}{cccc\|ccc\|ccc}		:$$\mathcal I = \gamma \sum{\left[\begin{array}{cccc\|ccc\|ccc}
	-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\		-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\
	0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\		0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\
Line 18:		Line 18:
	-my & 0 & -mct & 0 & mctz & 0 & -mctx & -mxy & m(c^2t^2 - y^2) & -myz \\		-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)		-mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2)
	\end{array}\right]}}$$ .		\end{array}\right]}$$ .

	== See Also ==		== See Also ==

	* [[Velocity]]		* [[Velocity]]

Eric Lengyel at 07:17, 11 November 2024

2024-11-11T07:17:21Z

← Older revision		Revision as of 07:17, 11 November 2024
Line 5:		Line 5:
	The inertia tensor $$\mathcal I$$ is given by		The inertia tensor $$\mathcal I$$ is given by

	:$$\mathcal I = \gamma \sum {{\begin{~~bmatrix~~}		:$$\mathcal I = \gamma \sum {{\left[\begin{array}{cccc\|ccc\|ccc}
	-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\		-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\
	0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\		0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\
	0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\		0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\
	0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\		0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\
			\hline
	0 & 0 & -mz & my & m(y^2 + z^2) & -mxy & -mzx & 0 & mctz & -mcty \\		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 & 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 \\		0 & -my & mx & 0 & -mzx & -myz & m(x^2 + y^2) & mcty & -mctx & 0 \\
			\hline
	-mx & -mct & 0 & 0 & 0 & -mctz & mcty & m(c^2t^2 - x^2) & -mxy & -mzx \\		-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 \\		-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)		-mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2)
	\end{~~bmatrix~~}}}$$ .		\end{array}\right]}}$$ .

	== See Also ==		== See Also ==

	* [[Velocity]]		* [[Velocity]]

Eric Lengyel at 07:13, 11 November 2024

2024-11-11T07:13:07Z

@@ Line 2: / Line 2: @@
 :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$
 == See Also ==
 * [[Velocity]]

Eric Lengyel: Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==..."

2024-11-10T23:49:23Z

Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==..."

New page

The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.

:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$

== See Also ==

* [[Velocity]]