How Does Parametrization Help Describe Particle Motion in Mathematics?

In summary, to describe the equation of a line in 2 dimensions, we need either a point on the line and the slope, or two points on the line. This can also be described as the trajectory of a moving point along the line, with the initial point and constant velocity being known. If the particle moves at a variable speed, the equations for the position at any time can be found through integration. Parametrization is a way to describe the position of a point along a line and is necessary in order to find the velocity and acceleration of the particle. Without parametrization, it would not be possible to find the velocity and acceleration of the particle from just the given trajectory.
  • #1
AAMAIK
47
0
TL;DR Summary
What is the purpose of parametrization
What is a parameter?
To describe the equation of a line, in 2 dimensions, we need a (point on the line + slope to measure slantiness) or two points. Another way: The trajectory of a moving point along the line. Suppose that the moving point initially is at a point of know coordinates r0=(x(t=0), y(t=0), z(t=0)) and that it moves at a constant speed on the line. From the above, the velocity vector has constant components v=(a1,a2,a3) independent of time, and the particle moves along one direction on either side of the initial point. The position of the moving point at any time , t
r=r0+Δr=r0+vΔt=>
x(t)=x(t=0)+a1*t
y(t)=y(t=0)+a2*t
z(t)=z(t=0)+a3*t
If the particle moves at variable speed
x(t)=x(t=0)+ ∫a1(t)dt
y(t)=y(t=0)+ ∫a2(t)dt
z(t)=z(t=0)+ ∫a3(t)dt
With this approach, I can describe the position of any point along the line. But I have problems with this approach, one, if given the x=2 I want to find the corresponding y and z such that the point (x=2, y,z) lies on the line. What if the trajectory of the moving particle does not go through x=2.
Secondly, this approach was based on the fact that the trajectory is along the line (given), it was made up. I don't understand parametrization, I view it as describing things that could be deduced easily from the givens of a moving point not the arbitrary curve in the plane or space it was introduced to serve to describe in the first place.
 
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  • #2
AAMAIK said:
What if the trajectory of the moving particle does not go through x=2
Then you don't find a solution

You could consider a parametrization as a kind of coordinate transform. In that sense $$x=\sin\omega t\\y = \cos\omega t$$ is also a parametrization
 
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  • #3
I wasn't trying to find a solution to the problem, I was illustrating why we need parametrization in the first place to describe trajectories in space or planes through a simple example of a line.
 
  • #4
AAMAIK said:
Summary:: What is the purpose of parametrization
What is a parameter?

But I have problems with this approach, one, if given the x=2 I want to find the corresponding y and z such that the point (x=2, y,z) lies on the line. What if the trajectory of the moving particle does not go through x=2.
If the the particle does not pass through x = 2 there should be no solution for t to the equation x(t) = 2. If There is a solution t0 to this equation, the particle passes through (2, y(t0),z(t0)).

Secondly, this approach was based on the fact that the trajectory is along the line (given), it was made up. I don't understand parametrization, I view it as describing things that could be deduced easily from the givens of a moving point not the arbitrary curve in the plane or space it was introduced to serve to describe in the first place.
That's a intersting thought. Of course you can't find the velocity or acceleration of point moving along a curve (such as a line) only from knowing the locus of points on the curve. Two particles can follow the same curve at different rates, just as two cars can travel the same road at different rates. So to have adequate "givens" to find the velocity and acceleration of a particle, you must know the rate of travel as a function of something. If we think of a curve parameterized by a variable representing time t, it is straightforward to find the rates of travel in the (x,y,z) directions by differentiation. I can imagine a situation where, we are given the locus of points (x,y,z) on the trajectory and know the particles velocity as a function of those locations. So our velocity information has the format ( vx(x,y,z), vy(x,y,z), vz(x,y,z)). How would we find the acceleration of the particle from that information? Is this a situation where acceleration can be deduced easily?
 

Related to How Does Parametrization Help Describe Particle Motion in Mathematics?

1. What is the purpose of parametrization in scientific research?

Parametrization is the process of defining and assigning values to parameters in a scientific model or experiment. This allows for the manipulation and control of variables in order to study their effects on the overall system. The purpose of parametrization is to make the model or experiment more realistic and representative of the real world, and to better understand the relationships between different variables.

2. How does parametrization improve the accuracy of scientific results?

By assigning specific values to parameters, scientists can reduce the amount of uncertainty in their models or experiments. This allows for more accurate predictions and conclusions to be drawn from the data. Parametrization also helps to identify and account for any potential biases or confounding factors that may affect the results.

3. Can parametrization be applied to all scientific disciplines?

Yes, parametrization can be used in all fields of science, from physics and chemistry to biology and environmental sciences. It is a fundamental aspect of the scientific method and is essential for conducting controlled experiments and building accurate models.

4. What are some common challenges in parametrization?

One of the main challenges in parametrization is selecting the most appropriate values for the parameters. This requires a thorough understanding of the system being studied and the potential interactions between variables. Another challenge is ensuring that the chosen values are consistent and realistic, as unrealistic or conflicting values can lead to inaccurate results.

5. How can parametrization help with the validation of scientific models?

Parametrization is crucial for model validation as it allows for the comparison of model predictions with real-world data. By adjusting and refining the parameters, scientists can improve the accuracy of their models and ensure that they are consistent with observed phenomena. This helps to build confidence in the validity of the model and its ability to accurately represent the real world.

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