Exploring Non-Linear Transformations Preserving Hyperbolic Curves

Are there non-linear transformations that map points on a hyperbola to other points on the same hyperbola? To address this question, we delve into the concepts of Hamiltonian vector fields and hyperbolas, offering a comprehensive exploration through mathematical and geometric perspectives.

Understanding Hyperbolas and Hamiltonian Vector Fields

Consider a hyperbola defined by the equation x2 - y2 p. Each point (x, y) on this hyperbola lies on one of an infinite family of hyperbolas, determined by the value of p. When p 0, the hyperbola consists of two lines, y ±x.

The Hamiltonian vector field associated with the function Hxy x2 - y2 transforms points on these hyperbolas. The equation for this vector field is:

Vector Field in Component Form

Python#160;code to express the vector field:

X_H  begin{cases} x'  -2y  y'  -2x end{cases}end{code>

Through this transformation, the function H (x, y) x2 - y2 remains constant, indicating that the hyperbolic curves are preserved under the transformation.

Reparametrization and Non-Linear Solutions

Although the flow of linear vector fields is linear, non-linear transformations can be constructed that map points on a hyperbola to other points on the same hyperbola. One such transformation is given by the reparametrization of the vector field:

Transformed Vector Field

The reparametrized vector field is defined as:

Y  frac{1}{x^2y^2   1} X_H.

The flow ψt of this vector field provides a non-linear solution to the question, demonstrating that points can be mapped to other points on the same hyperbola in a non-linear manner.

Characterizing Non-Linear Transformations

Non-linear transformations that preserve hyperbolic curves can be described by a family of functions v frac{a^2 - 1}{2a}x - frac{a^2 1}{2a}y, w frac{a^2 1}{2a}x - frac{a^2 - 1}{2a}y, where a is a non-zero real number. This family of functions ensures that the transformed points (v, w) also lie on the same hyperbola:

v2 - w2 x2 - y2

The inverse of this function is:

x frac{a^2 - 1}{2a}v - frac{a^2 1}{2a}w, y frac{a^2 1}{2a}v - frac{a^2 - 1}{2a}w

Taking a non-constant function g(p) 1 p2, we can construct a non-linear transformation that preserves the hyperbolas:

v frac{g(x^2 - y^2) - 1}{2sqrt{g(x^2 - y^2)}}x - frac{g(x^2 - y^2) 1}{2sqrt{g(x^2 - y^2)}}y, w frac{g(x^2 - y^2) 1}{2sqrt{g(x^2 - y^2)}}x - frac{g(x^2 - y^2) - 1}{2sqrt{g(x^2 - y^2)}}y

Here, g(p) ≠ 1 for some non-zero pv, ensuring that the transformation is non-linear.

Proof of Non-Linearity

To prove that the constructed transformation is non-linear, we assume that g(p) 1 for x 1, y 1 and x 1, y -1. Evaluating the transformation at these points, the hyperbola still remains, but the only solution would be the identity function, which is not possible since g(p) ≠ 1 for some non-zero pv.

The final verification is that the transformation cannot be the identity function, as the mapped points (v, w) are not equal to the original points (x, y).