The space coordinate system that we choose is somewhat arbitrary, perhaps we may choose orthogonal (mutually perpendicular) x,y and z coordinates that align with the surface of the earth where we happen to be, or we could choose any other convenient alignment. In the same way the origin point of the coordinate system is similarly arbitrary.
This can be useful as changing the frame of reference (different observers) in an appropriate way could simplify the mathematics.
When we go on to look at dynamics we shall see that the laws of motion are independent of certain changes in the coordinate system such as:
- time shift
- space translation
- space rotation
- uniform linear velocity
We can think of these as symmetries.
This means that if we do a mechanics experiment where the frame of reference is transformed in any of these ways then we will get the same result (provided that we are consistent: if we are using some equation then all the quantities in that equation must be measured in the same coordinate system). in fact this can be stated in a stronger way: there is no way, by doing physics experiments that we can determine an absolute time, space or velocity, these things always have to be measured relative to something. This relativity principle applies both to Newtonian and Einsteinian physics although, as we will see below, the nature of the velocity transform depends on which of these we are using.
There may be other symmetries, for instance, reflection in space or time, however there are some frames of reference which will alter the laws of physics, for example, where the frame of reference is accelerating or in a constant angular velocity.
We will now look at these changes or transforms individually in more detail.
Time Shift Transform
We can change the frame of reference by shifting the timeframe as follows:
t'=t - t0
where:
- t' = time in new frame of reference (scalar value)
- t0 = time in new frame of reference when t=0 (scalar value)
- t = time in original frame of reference (scalar value)
Alternatively we could use t'=t + t0 if we defined t0 as time in original frame of reference when t'=0
Another way to express this is to represent space-time as a 4 dimensional vector in which case, for newtonian physics, we get:
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= |
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+ |
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Time is not absolute in either Newtonian or Einsteinian physics in the sense that there is no preferred origin (except perhaps the time of the big bang?) but in Einsteinian physics time is even less absolute in that time intervals change due to relative velocity.
Space Translation Transform
We can change the frame of reference by shifting the origin as follows:
x'=x - x0
where:
- x' = position in new frame of reference (vector value)
- x0 = position in new frame of reference when x=0 (vector value)
- x = position in original frame of reference (vector value)
Alternatively we could use x'=x + x 0 if we defined x 0 as time in original frame of reference when x'=0
Another way to express this is to represent space-time as a 4 dimensional vector in which case, for newtonian physics, we get:
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= |
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+ |
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Position is not absolute in either Newtonian or Einsteinian physics in the sense that there is no preferred origin but in Einsteinian physics distance is even less absolute in that distances change due to relative velocity.
Space Rotation Transform
In the same way we can rotate the space frame as follows:
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Uniform Linear Velocity Transform
The transform between moving frames depends on whether we are using Newtonian or Einsteinian physics in Newtonian physics we use the Galilean transform and in Einsteinian physics we use the Lorentz transform.
Galilean transform
Assuming the relative motion 'v' is along the x dimension then x -> x0 + vxt
or if we have components of velocity in all dimensions the transform will be:
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= |
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where:
- vx,vy,vz= relative velocity of the two reference frames in x,y and z directions.
- x,y,z= position in original frame.
- x',y',z'= position in transformed frame.
in other words point:
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is transformed to: |
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The nature of this transform is a shear (also known as skew) transform:
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—» | ![]() |
When doing this we choose to make time 'absolute' in that the time lines are left horizontal wheras the position lines are skewed althogh I guess that this is just a covention and we could have skewed the time and made the distance absolute.
more about Galilean transform.
Practical Issues
At first, it may seem a bit academic to transform these quantities into different frames-of-reference but, this is important for solving practical problems like collisions. For instance, the two objects colliding will have their inertial tensors defined in their own local coordinates, but when we work out the collision response, the impulse will have to be calculated in some common coordinate system.
What we are doing here is observing the same particle or solid object from different frames-of-reference. When the frames-of-reference are static (not moving relative to each other). Then the Newtonian laws will apply, regardless of where, or which direction, that we are looking at them from, provided that we are consistent about measuring all quantities on the same frame-of-reference.
It may be that the transform is changing with time, for example, if we are trying to solve a collision response, we might want to work in the frame-of-reference of one of the objects that is colliding. This object may be moving, so its frame-of-reference is moving with respect to the absolute coordinate system.
Do all the laws of physics apply when observing them from a moving frame-of-reference? If the frame-of-reference is moving with a constant linear velocity, then the Newtonian laws will apply just the same. A stationery object in one frame-of-reference may appear to be moving in another, but provided that we are constant about which frame-of-reference that we are working in, neither will contravene the laws. (Einsteins laws may not apply, the speed of light is the same in each frame-of-reference, relativity is not relative to the frame-of-reference - we are interested in slow speed interactions so this is not an issue for us).
However, if the frame-of-reference has angular motion (even if its constant), or if the frame-of-reference is accelerating, or if it has some irregular motion, then the Newtonian laws will not apply in this frame-of-reference.
To take an example, imagine a solid object travailing in free space, it may be spinning and have linear velocity as well. We may want to work in this objects own coordinate system, because this frame-of-reference is rotating, the Newtonian laws may not apply, for example an object which is stationary in the absolute frame-of-reference will appear to be traveling in a spiral in this objects frame of reference. An object with no forces acting on it should not be moving in a spiral, so Newtonian laws do not apply in this frame-of-reference. However, if we want to apply this inertia tensor, then we have to work in the local coordinate system of the rotating object.
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So, we have to be careful. When we are calculating torques and forces, we often want to work in its local coordinate system, but when we are calculating motion we probably want to work in absolute coordinates.
Using matrix algebra to calculate transforms to other frames-of-reference
As described here, a 4x4 matrix can be used to represent a rotation and a translation in 3 dimensions. So we can use matrix algebra to translate from one frame of reference to another.
The transform will take a 3d vector representing a point in absolute coordinates
and convert it into a 3d vector representing a point in absolute coordinates:
a
= [T]
l
It is possible to have many layers of one frames-of-reference inside another. For example, if we know the position of the moon relative to the earth [Tme] and we know the position of the earth in the frame-of-reference of the sun [Tes], then we could work out the position of a point on the moon in the frame of reference of the sun [Tms].
It turns out that transforms can be concatenated by multiplying their corresponding matrices. So [Tms] = [Tes]*[Tme].
One way to represent this is to use a scene graph. We could use a scene graph is a similar way to VRML. We could put dynamics information into the scene graph in the same way that shape information is. In this page I would like to work out the effect of transforming dynamics information in this way.
The transform allows us to plug in a coordinate in the local frame-of-reference and get the coresponding coordinate in the absolute frame-of-reference.