Defining a generalized coordinate system

In summary, the conversation is discussing setting up a coordinate system to represent points in space relative to a parametric curve. This coordinate system is defined by the shortest distance from a point in space to the curve, the distance along the curve, and an angle relative to the curve. The conversation also mentions the Frenet-Serret system and the constraints for defining the vector in this coordinate system.
  • #1
dipole
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(Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...)

Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is the shortest distance from a point to the curve, and one is an angle defined relative to some direction.

I have some curve in space defined by:

[itex] x = x(t) [/itex]
[itex] y = y(t) [/itex]
[itex] z = z(t) [/itex]

and I want to define a coordinate system [itex] (s, \rho, \omega) [/itex] relative to this curve.

[itex] \rho [/itex] is defined by the shortest distance from a point [itex] p' = (x',y',z') [/itex] (I will use primes to denote points not on the curve) in space to the curve (assume this is unique).

[itex] s [/itex] is defined by the distance along the curve, starting from some initial point [itex] \left (x(t_0), y(t_0), z(t_0) \right) [/itex] to the point [itex] \left (x(t), y(t), z(t) \right) [/itex] such that

[tex] d \left (x', y', z', x(t), y(t), z(t) \right ) = \rho [/tex]

where [itex] d [/itex] is the euclidean distance.

In other words, say I have some point in space [itex] p' = (x',y',z') [/itex], then [itex] \rho [/itex] is the length of the smallest line segment between [itex] p' [/itex] and some point on the curve [itex] p(t) = (x(t), y(t), z(t)) [/itex]. I want to represent the vector from [itex] p(t) [/itex] to [itex] p' [/itex] by an orthogonal coordinate system that is attached to the curve.

Let [itex] \hat{s}(t) [/itex] be the unit tangent vector to the curve at the point [itex] p(t) [/itex]. This defines a plane where the point [itex] p' [/itex] lies on a circle of radius [itex] \rho [/itex] in the plane. The only other thing I need to uniquely define the point [itex] p' [/itex] in this plane is some reference direction to measure the angle at which the point [itex] p' [/itex] lies on the circle. Call this direction [itex] \hat{\omega}(t) [/itex].

Here's the kicker though - I want [itex] \hat{\omega}(t) [/itex] to be defined in such a way that when the curve is a straight line, and in the z-direction, that the coordinate system becomes plain old cylindiral coordinates, and [itex] \hat{\omega}(t) = \hat{x}[/itex] . However, when the curve is not a straight line, then [itex] \hat{\omega} [/itex] should rotate with the curve in such a way that it is always orthogonal to [itex] \hat{s}(t) [/itex], and locally if [itex] \hat{s}(t) \simeq \hat{z} [/itex] then [itex] \hat{\omega}(t) \simeq \hat{x} [/itex].

My question is how to define [itex] \hat{\omega}(t) [/itex] in such a way that satisfies these constraints.

Does this make sense? I hope at least the idea of what I want to do is clear. I further hope this problem is not ill-posed. Any help would be appreciated and please let me know if something is unclear.
 
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  • #3
Hi, yes I remembered that as I was writing this, but I'm still not sure how to enforce that the Frenet-Serret frame becomes normal cylindrical coordinates in the limit that the curvature goes to zero (assuming that any straight line will be in the z-direction).
 

What is a generalized coordinate system?

A generalized coordinate system is a mathematical framework used to describe the motion or position of a physical system. It involves choosing a set of variables, known as generalized coordinates, to represent the degrees of freedom of the system.

Why is it important to define a generalized coordinate system?

Defining a generalized coordinate system allows us to simplify and analyze complex systems by reducing the number of variables needed to describe them. It also allows us to apply mathematical principles and equations to describe the behavior of the system.

How do you choose the generalized coordinates for a system?

The choice of generalized coordinates depends on the specific system being studied. They should be independent variables that uniquely describe the position or configuration of the system. Common examples include position, angle, and velocity.

Can a system have more than one generalized coordinate system?

Yes, a system can have multiple generalized coordinate systems. This is because different sets of variables can be used to describe the same physical system. However, the choice of coordinates may affect the complexity of the equations used to describe the system.

What are the limitations of using a generalized coordinate system?

One limitation of using a generalized coordinate system is that it may not always accurately describe the behavior of a system. In some cases, more complex variables may be needed to fully capture the dynamics of the system. Additionally, the choice of coordinates may introduce certain constraints on the system that limit its behavior.

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