Physics - Energy and Momentum using Multivectors

Prerequisites

If you are not familiar with this subject you may like to look at the following pages first:

• Rigid Body Energy and Momentum
• Multivectors

using Multivectors

If we specify a velocity using a 3D multivector (explained here) and choose to define this as follows:

Scalar (e) = 1
Vector (ex,ey,ez) = vx,vy,vz
BiVector (eyz,ezx,exy) = 0
TriVector (exyz) = 1

where:		other definitions
symbol	description	type	units
ex,ey,ez	basis vectors	vector
exyz		trivector
vx,vy,vz	components of velocity in x,y and z dimensions	vector

So the velocity multivector is:

v = 1 + ex * vx + ey * vy + ez * vz + exyz

We can new calculate the square of the velocity using multi-vector multiplication which gives:

v² = x² + y² + z² + ex * 2 * vx + ey * 2 * vy + ez * 2 * vz + eyz * 2 * vx + ezx * 2 * vy + exy * 2 * vz + exyz * 2

We can new calculate m * v²/2 which gives the following:

m * v²/2 = (x² + y² + z²)*m/2+ ex * m * vx + ey * m * vy + ez * m * vz + eyz * m * vx + ezx * m * vy + exy * m * vz + exyz * m

This has the following components:

Scalar (e) = (x² + y² + z²)*m/2 = kinetic energy
Vector (ex,ey,ez) = m * vx ,m * vy, m * vz = linear momentum
BiVector (eyz,ezx,exy) = m * vx ,m * vy, m * vz = linear momentum
TriVector (exyz) = m = mass

So it is made up of constants for linear motion, for example in a perfectly elastic collision between two objects, if we calculate this quantity for both objects before the collision and after the collision, the total of both objects before the collision should be the same as the total after the collision.

Adding rotational energy

The above only includes linear motion, it would be good if we could find a way to include rotational energy and momentum. So here we try adding a rotational velocity (w) to the velocity multivector as follows:

Scalar (e) = 1
Vector (ex,ey,ez) = vx,vy,vz
BiVector (eyz,ezx,exy) = wx,wy,wz
TriVector (exyz) = 1

where:		other definitions
symbol	description	type	units
w	angular velocity	bivector	s-1
angle	angle in radians	scalar	none
t	time	scalar	s
d ... /dt	rate of change

where:

• ex,ey,ez = basis vectors
• exyz = represents tri-vector
• vx,vy,vz = components of velocity in x,y and z dimensions
• wx,wy,wz = components of angular velocity in x,y and z dimensions

So the velocity multivector is:

v = 1 + ex * vx + ey * vy + ez * vz +eyz * wx + ezx * wy + exy * wz + exyz

We can new calculate the square of this velocity which gives:

EuclideanSpace

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