Question

Is it possible to rectify the direction field of the equation \(\dot{x}=x^{2}\) on the entire plane by a timepreserving diffeomorphism?

Short answer

Answer: No, it is not possible to rectify the direction field of the equation \(\dot{x} = x^2\) on the entire plane using a time-preserving diffeomorphism.

Step-by-step solution

Recall the definition of a time-preserving diffeomorphism

A time-preserving diffeomorphism is a smooth bijective map between two manifolds which preserves the time parameter in the system of ODEs. In this case, we are looking for a coordinate transformation \(\phi(t, x) = (t, y(t, x))\), such that the direction field becomes horizontal in the new coordinate system.

Find the transformation equation for the new coordinate system

In order to find the transformation equation, we need to express the given equation in the new coordinate system \((t, y)\). Since the time \(t\) is preserved, we can write: $$\frac{dy}{dt} = \dot{y}$$ Now we need to find the relationship between \(\dot{x}\) and \(\dot{y}\) using the given equation and the coordinate transformation: $$\dot{x} = x^2 \Rightarrow \frac{d}{dt}(y^{-1}(t, x)) = y_x^{-2} \dot{y} $$ Here, \(y^{-1}(t, x)\) is the inverse function of the coordinate transformation \(y(t, x)\), and \(y_x\) is the partial derivative of \(y(t, x)\) with respect to \(x\).

Check if it is possible to make the direction field horizontal

We want the direction field to be horizontal in the new coordinate system, meaning that \(\dot{y} = 0\). So the transformed equation should look like: $$y_x^{-2} \cdot 0 = x^2$$ However, this equation implies that \(x^2 = 0\) for all \((t, x)\), which is only true for \(x = 0\). Therefore, there is no transformation that can make the entire direction field horizontal.

Conclusion

It is not possible to rectify the direction field of the equation \(\dot{x} = x^2\) on the entire plane by a time-preserving diffeomorphism.

Key concepts

Time-Preserving Diffeomorphism

Understanding time-preserving diffeomorphisms can unlock the complexities involved in mathematical transformations, particularly when dealing with ordinary differential equations (ODEs). A time-preserving diffeomorphism is a particular type of smooth, bijective function. It stands out because it maintains the 'time' dimension unchanged during the transformation between two coordinate systems or manifolds.

Imagine you are watching a dance where each dancer's path on stage can be described by an ODE, and a time-preserving diffeomorphism is like a choreographer who repositions the dancers but ensures that their timing within the music does not change. In the context of ODEs, this concept ensures that even though the spatial coordinates may undergo complex changes, the time variable, which is crucial in the analysis of dynamic systems, is left untouched. This property is essential for analyzing the behavior of systems over time without altering the temporal aspect of their evolution.

However, as seen in the example exercise, not all direction fields can be simplified using such a transformation. The constraint of preserving time can inherently limit the modifications possible to the system's direction field, implying that certain ODEs cannot be transformed to a simpler form while keeping the element of time constant.

Ordinary Differential Equations (ODEs)

Ordinary differential equations are the bread and butter of modeling the rate of change of physical phenomena. An ODE is an equation involving derivatives of a function, usually with respect to one variable, often time. It outlines the relationship between this variable and its rate of change.

Consider an ODE to be like a recipe that describes how to bake a dynamic cake — the kind that changes over time. It tells you the speed at which the cake rises (rate of change) based on the moment-by-moment status of its rise (the function). Such equations are pivotal in fields like physics, engineering, and biology, helping scholars predict patterns of movement, growth, and decay.

In a more mathematical sense, the ODE given in the exercise, \(\dot{x}=x^{2}\), can't be 'rectified,' or simplified across the entire plane because the quadratic term in \(x\) implies a parabolic growth that inherently alters with \(x\), making the field resistant to simplification throughout the entire plane while preserving time.

Coordinate Transformation

Coordinate transformation is akin to changing perspectives to view a scenario in a new way. At its core, it's about converting the points in one coordinate system to another, which can make complex problems simpler to understand or solve.

Imagine you are trying to describe the location of a treasure on a pirate map. In one coordinate system, you might use steps and turns (polar coordinates) to describe the path, while in another, you might use a grid system (Cartesian coordinates). By switching between these views, the complexity of navigating to the treasure can greatly vary.

In mathematical terms, a coordinate transformation in the context of ODEs like \(\dot{y}\) to \(\dot{x}=x^2\) attempts to find a new function \(y(t, x)\) that can simplify the original direction field under a new light. The exercise illustrates that this is not always possible, as trying to transform the direction field to make it 'horizontal' across the entire plane doesn't align with the behavior of the given function, hence making the original problem resistant to such simplification.