Trying to get an explicit function(ish)

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In summary: I just re-read the article on it, and it’s a bit over my head, but I’ll give it a try.In summary, the conversation discusses the process of solving for tp in a complex equation involving coordinates, constants, and trigonometric functions. The experts suggest using approximations and tools like Newton's method due to the lack of a closed solution. They also mention the possibility of using the implicit function theorem to find a local expression. The person seeking help expresses their dissatisfaction with the limitation and thanks the experts for their advice.
  • #1
Pencilvester
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How would I go about solving for ##t_p## in the following equation:$$t_p - t + vy \cos {(2 \pi \omega t_p )} - vx \sin {(2 \pi \omega t_p )} = 0$$where our input is a point in ##ℝ^3## with coordinates ##t##, ##x##, and ##y##, and where ##v## and ##\omega## are constants. I’m pretty sure it can’t be a function exactly, as I’m pretty sure most, if not all input points will each yield 2 distinct outputs. If it matters to you, ##|v| < 1##, but I don’t think that it’s relevant to this problem. And this isn’t any kind of homework problem. I’m not in school, I’m just trying to analyze what a coordinate transformation might look like for going from coordinates of an inertial observer in flat spacetime to coordinates of an observer tracing out a helix in spacetime (or a circle in space), and I’m running into this limitation in my mathematical abilities, so any help would be much appreciated.
 
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In general there is no closed solution - you can only find approximations. It is possible to combine the cosine and sine term to a single sine (or cosine) with an additional phase, that makes the problem a bit easier to look at. tp cannot be too different from t as the other terms cannot get larger than vx or vy, respectively, this gives you a region to look for solutions. Once you found a tp that is not too far away from a solution tools like Newton's method will work.
 
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mfb said:
In general there is no closed solution - you can only find approximations. It is possible to combine the cosine and sine term to a single sine (or cosine) with an additional phase, that makes the problem a bit easier to look at. tp cannot be too different from t as the other terms cannot get larger than vx or vy, respectively, this gives you a region to look for solutions. Once you found a tp that is not too far away from a solution tools like Newton's method will work.
Dang. That’s dissatisfying for me. Well at least I didn’t spend much time trying to find a general solution before posting this. Thanks!
 

1. What is an explicit function?

An explicit function is a mathematical function where the dependent variable (usually denoted as y) is expressed solely in terms of the independent variable (usually denoted as x). This means that for any given value of x, the value of y can be determined using the function.

2. Why is it important to try and get an explicit function?

Having an explicit function allows us to easily calculate the value of the dependent variable for any given value of the independent variable. This is especially useful in scientific research and data analysis.

3. What are some common methods for obtaining an explicit function?

Some common methods include solving for the dependent variable in terms of the independent variable, using algebraic manipulations and transformations, and using calculus techniques such as differentiation and integration.

4. Are there any limitations to obtaining an explicit function?

Yes, there are some cases where it may not be possible to obtain an explicit function for a given relationship. This can happen when the relationship is too complex or when there is not enough information available.

5. Can we use technology to help us get an explicit function?

Yes, there are many software programs and online tools available that can help us find explicit functions for various relationships. These tools use advanced mathematical algorithms and techniques to solve for the dependent variable in terms of the independent variable.

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