Intrinsic coordinates and an intrinsic description of motion

In summary, for 2D motion, the velocity, position, and acceleration of a point object can be described using either the fixed basis vectors or the tangential and normal unit vectors. Both options can be used, as time and arc-length are related. The tangential and normal vectors are called intrinsic as they do not require a basis vector to span the space. This description may be more useful for an observer following the curve, such as someone on a moving ship.
  • #1
fog37
1,568
108
Hello,

For 2D motion, I understand that velocity, position and acceleration of a point object can be described using the fixed basis vector ##\hat {i}## and ##\hat {j}## and the rectangular coordinates ##x(t)## and ##y(t)## which which are functions of time ##t##.

Another option is to use the tangential (to the trajectory) unit vector ##\hat {e}_t## and the normal (to the trajectory) unit vector ##\hat {e}_n##. Are the components of the tangential and normal unit vectors supposed to be functions of time ##t## or functions of the scalar arc-length ##s##?

Also, why are the tangential and normal unit basis vectors called intrinsic? When is the motion description with these two basis vectors more useful than a description involving the traditional rectangular coordinates?

Thanks!
 
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  • #2
Too many questions :-)

First: time ##t##, and arc-length ##s##. They are related, aren't they? i.e. ##s=s\left(t\right)## is defined, and, as long as time-travel is not allowed, ##t=t\left(s\right)## is also defined. So there is no difference which one you use to describe the motion - your choice.

Second: Tangential and normal vector would be called intrinsic because they do not need a definition of basis vector to span the space, i.e. the tangential and normal vectors are intrinsically defined when you define a smooth curve.

Third: The definition in terms of intrinsic vectors may be useful if you consider things from the point of view of observer following the curve, i.e. when you sit on the ship that is going somewhere, and you see a plane flying past, it is easy to say whether the plane's trajectory is parallel or perpendicular to that of the ship.
 

FAQ: Intrinsic coordinates and an intrinsic description of motion

What are intrinsic coordinates?

Intrinsic coordinates refer to a set of coordinates that are defined within a specific system or object, rather than being dependent on an external reference frame. They are used to describe the position, orientation, and motion of an object in a way that is independent of any external factors.

What is an intrinsic description of motion?

An intrinsic description of motion is a way of describing the movement of an object without reference to any external frame of reference. It is based on the object's own intrinsic properties, such as its position, velocity, and acceleration, rather than its relationship to other objects or systems.

Why is an intrinsic description of motion useful?

An intrinsic description of motion allows for a more accurate and comprehensive understanding of an object's movement, as it is not affected by external factors such as the observer's perspective or the presence of other objects. It also allows for easier comparison and analysis of motion between different objects or systems.

How are intrinsic coordinates and an intrinsic description of motion related?

Intrinsic coordinates are used to define an object's intrinsic properties, which are then used to describe its motion intrinsically. In other words, intrinsic coordinates are the building blocks of an intrinsic description of motion.

What are some examples of systems that use intrinsic coordinates and an intrinsic description of motion?

Some examples include rigid bodies, fluid systems, and celestial bodies. Intrinsic coordinates and an intrinsic description of motion are also commonly used in fields such as physics, engineering, and computer graphics to model and analyze the movement of objects.

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