# Detemining the constants in a helix equation (Calculus of Variations question)

• hookie101
In summary: Therefore, the rewritten equation becomes y=sinØ - (1/2)sin2Ø + 1.In summary, the question is about finding the equation of a frictionless wire between a point and a line in the calculus of variations. The integration and Euler equation steps are provided, but the values of the constants are unknown. The equation can be rewritten in terms of y and Ø, with A and B being determined by setting specific values for x and y.
hookie101
This question is about variable end points in calculus of variations. I understand the basic principle of how you would find the various equations, but I embarrasingly keep getting stuck on when determining the constants.

Question: Find the equation of a frictionless wire between the point (0,1) and the line x=a such that a particle slides down under gravity in least time. (Question also wants the values of the constants that would come from the Euler equation and the integration part of the problem. This is the part I'm stuck at).

The equation I arrived to was

x=(1/A^2)[Ø - (1/2)sin2Ø] + B

where Ay^1/2 = sinØ

I'm not entirely sure how I would determine A and B by only knowing the point (0,1). Considering that this is an equation of a helix, my first assumption was that when x=0, Ø=0, so leaving me with B=0. However this still leaves me with A to deal with. If anyone knows how to find the constants, I would appreciate it. It would help me even more to show me how you would find the constants for any general equation you'll eventually get from the Calculus of Variations method, since I had this sort of problem numerous times.

Additional part: Is it possible to rewrite the above equation in terms of y and Ø too?

For the additional part, the equation can be rewritten in terms of y and Ø as y=A^2[sinØ - (1/2)sin2Ø] + Bwhere Ay^1/2 = sinØ. In this case, A and B are still unknown constants and can be determined by setting y=1 when x=0, so that B=1. Additionally, when x=0, Ø=0 and thus Ay^1/2 = 0, so A=0.

## 1. What is the purpose of determining the constants in a helix equation?

Determining the constants in a helix equation is important because it allows us to fully understand and describe the shape and properties of a helix. It also helps us to accurately predict and calculate various aspects of the helix, such as its curvature and torsion.

## 2. How are the constants in a helix equation typically determined?

The constants in a helix equation are typically determined through the process of calculus of variations. This involves finding the function that minimizes a certain functional, or mathematical expression, that represents the desired properties of the helix.

## 3. What are the most commonly used constants in a helix equation?

The most commonly used constants in a helix equation are the radius, pitch, and height of the helix. These constants determine the size and shape of the helix and are essential in its description and calculation.

## 4. Can the constants in a helix equation vary?

Yes, the constants in a helix equation can vary depending on the specific helix being studied. Different helices can have different sizes, shapes, and properties, resulting in different values for the constants in their respective equations.

## 5. How does determining the constants in a helix equation relate to real-world applications?

Determining the constants in a helix equation has many real-world applications, particularly in fields such as engineering, physics, and biology. For example, helical structures are commonly found in DNA, and understanding their constants can help us better understand and manipulate genetic material. Helices are also used in various mechanical and structural designs, where knowing their constants is crucial for ensuring stability and efficiency.

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