Question

Prove that among the Lissajous curves with \(\omega_{2}=n \omega_{1}\) is the graph of a polynomial of degree \(n\). This polynomial is called the Chebyshev polynomial. $$ T_{n}(x)=\cos (n \arccos x) $$

Short answer

Answer: The Chebyshev polynomial of degree n for Lissajous curves with a relation 𝜔2=n𝜔1 is given by T_n(x) = cos(n * arccos(x)).

Step-by-step solution

Understanding Lissajous curves and given relation

Lissajous curves are defined by the following parametric equations: $$ x(\theta)=\cos(\omega_{1}\theta), \quad y(\theta)=\cos(\omega_{2}\theta). $$ From the given exercise, we have \(\omega_{2}=n\omega_{1}\), where \(n\) is an integer.

Substituting the value of \(\omega_{2}\) into the Lissajous curves equation

Let's substitute \(\omega_{2}=n\omega_{1}\) into the Lissajous curves equation: $$ x(\theta)=\cos(\omega_{1}\theta), \quad y(\theta)=\cos(n\omega_{1}\theta). $$

Eliminating the parameter \(\theta\)

To eliminate the parameter \(\theta\), let's use the inverse trigonometric function \(\arccos\) (cosine inverse). For, \(x(\theta)=\cos(\omega_{1}\theta)\), we have: $$ \omega_{1}\theta = \arccos(x(\theta)). $$ We have to eliminate \(\theta\) from \(y(\theta)=\cos(n\omega_{1}\theta)\). For this, substitute \(\omega_{1}\theta\) from the above equation to get, $$ y(\theta)=\cos(n \arccos x(\theta)). $$

Concluding the proof

Now, we have eliminated the parameter \(\theta\) and have an equation in terms of x and y: $$ y(\theta)=\cos(n \arccos x(\theta)). $$ As the equation defines a polynomial of degree \(n\), it is called the Chebyshev polynomial. The proof is complete.

Key concepts

Lissajous Curves

Lissajous curves are complex and beautiful patterns that occur when two motions at right angles to each other are combined. They're named after Jules Antoine Lissajous, who studied these figures in 1857.

Think of them as the figures you can create by moving a laser pointer in one hand up and down while moving another pointer left to right with the other hand. If the frequency of these motions is the same, you get a circle or an ellipse, but if the frequencies are different, you might get something that looks rather like a complicated knot.

Mathematically, Lissajous curves can be described using parametric equations, which express the coordinates of the points making up the curve as functions of a variable called a parameter. For Lissajous curves, the parameter is often time and the equations involve trigonometric functions like sine and cosine. An example of these equations is: \( x(t) = A \times \text{sin}(a \times t+\text{phase})\, y(t) = B \times \text{sin}(b \times t) \).

When the frequency ratio (b/a or a/b) is a rational number, the Lissajous curve closes on itself after a finite number of oscillations and looks particularly neat. In the context of our exercise, the relationship \( \omega_{2} = n \omega_{1} \) constrains the Lissajous curve to a shape that can be defined by a polynomial, specifically the Chebyshev polynomial of the first kind, \( T_{n}(x) \).

Parametric Equations

Parametric equations are a set of equations in which the system's state is described using one or more independent variables called parameters. They are essential for describing motion and geometry in a more dynamic way compared to traditional 'y as a function of x' descriptions.

For instance, consider the simple case of a circle. We can describe it parametrically with \( x(t) = r \times \text{cos}(t) \) and \( y(t) = r \times \text{sin}(t) \) where \( r \) is the radius and \( t \) could represent time or an angle in radians. It's this parametric approach which allows for the beautiful complexity of Lissajous curves.

With parametric equations, relationships between two quantities can be visualized more flexibly and movements through space can be portrayed more vividly. They enable us to express curves in the plane and in three-dimensional space that might be impossible or cumbersome to describe otherwise. The concept proves pivotal in advanced fields of mathematics, physics, engineering, and computer graphics.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backwards from a trigonometric ratio to an angle measure. They serve as the 'undo button' for the trigonometric functions like sine, cosine, and tangent.

When we have \( y = \text{cos}(x) \) and we want to find \( x \) based on a known \( y \) value, we turn to \( x = \text{arccos}(y) \) to retrieve the original angle in radians. This inversion process is crucial when we are dealing with parametric equations and need to eliminate the parameter to shift from a pair of equations into a single relationship between \( x \) and \( y \) directly.

In our main exercise context, using the inverse cosine function, \( \text{arccos} \) was the key step in transforming a Lissajous curve defined parametrically into the Chebyshev polynomial, \( T_{n}(x) = \text{cos}(n \times \text{arccos}(x)) \). Such transformations are useful not only in mathematical theory but have practical applications in various branches of science and engineering, such as signal processing and robotics.