Introduction
Orthogonal trajectory is a set of curves that converge at right angles with another set of curves. Electrostatics and hydrodynamics are two examples of mutually orthogonal curve families. In electrostatics, the lines of force and lines of constant potential are orthogonal, and in hydrodynamics, the streamlines and lines of constant velocity are orthogonal.In this article we will learn about orthogonal trajectories and orthogonal family of curves. We will understand orthogonal trajectories differential equations.
Define Orthogonal Trajectories
An orthogonal trajectory of the family is a curve that cuts any member of a given family of curves at right angles. It is not appropriate for the curve to pass through each family member. When they intersect, however, the angle formed by their tangents at each point of intersection is 90o. Orthogonal trajectories differential equations are used to find curves that intersect a given family of curves at right angles.
Orthogonal Trajectories Calculator
Orthogonal trajectories calculator is shown below
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Find Orthogonal Trajectories
With the help of an example, we will be able to comprehend this.
Determine the orthogonal trajectories of the straight line family y = Cx, where C is a parameter.
Sol: First, we create the differential equation for the y = Cx family of straight lines. We get the following result by differentiating the last equation with respect to x.
\[y'=C=const\]
We have to eliminate the constant C from the equation system:
\[y=Cx\]
\[y'=C\]
\[y'=\frac{y}{x}\]
The differential equation for the initial set of straight lines is obtained.
Replace \[y'\]with \[\frac{-1}{y}\]. This yields the orthogonal trajectories' differential equation:
\[-\frac{1}{y'}=\frac{y}{x}\]
\[\Rightarrow y'=-\frac{x}{y}\]
To find the algebraic equation of the family of orthogonal trajectories, we solve the last differential equation:
\[y'=-\frac{x}{y}\]
\[\Rightarrow\frac{dy}{dx}=-\frac{x}{y}\]
\[\Rightarrow ydy=-xdx\]
\[\Rightarrow \int ydy=-\int xdx\]
\[\Rightarrow\frac{y^{2}}{2}=-\frac{x^{2}}{2}+C\]
\[\Rightarrow\frac{x^{2}}{2}+\frac{y^{2}}{2}=C\]
\[x^{2}+y^{2}=2C\]
We can see that the orthogonal trajectories for the family of straight lines are concentric circles by replacing 2C with R.
\[x^{2}+y^{2}=R^{2}\]
Orthogonal Trajectories of Family of Circles
The curves that are perpendicular to the family everywhere are called orthogonal trajectories. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in the original family. We'll see how to use calculus to find orthogonal trajectories in the example below, but we'll give away the answer for now so we can draw the family of orthogonal trajectories and see that they're perpendicular to the original family.
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The curves in blue are from the original family y = kx The curves in green are
\[x^{2}+y^{2}=1\]
\[x^{2}+y^{2}=2\]
\[x^{2}+y^{2}=3\]
\[x^{2}+y^{2}=4\]
which are four of their orthogonal trajectories, part of their whole family of orthogonal trajectories given by \[x^{2}+y^{2}=C\]. Where a green and blue curve converge, note how each green circle is perpendicular to every blue line.
Orthogonal Trajectories of Parabola
The orthogonal direction of the \[y^{2} = 4ax\] parabola family is as follows:
Given the equation of the family of parabolas is \[y^{2} = 4ax\]
The parameter in the ordinary differential equation is a, which is also an arbitrary constant.
Now, on both sides, differentiating the equation with respect to x yields,
\[\frac{dy}{dx} = \frac{2a}{y}\]
\[a = \frac{y}{2} (\frac{dy}{dx})\]
substituting in the equation of the curve family will give,
\[y^{2} = 2xy(\frac{dy}{dx})\] which is a differential equation of the family of parabolas.
Now, to find the equation of the orthogonal trajectories we need to replace \[\frac{dy}{dx}\] by \[-\frac{dy}{dx}\]
and we need to solve it back
\[y^{2} = 2xy(-\frac{dy}{dx})\]
Regrouping the terms and integrating gives,
\[\int ydy=\int(-2x)dx\]
\[\Rightarrow\frac{y^{2}}{2}=-x^{2}+c\]
Where c is integration constant
regrouping the terms gives,
\[2x^{2}+y^{2}=C^{2}\]
Where C is a constant
Applications of Orthogonal Trajectories
In mathematics, orthogonal trajectories are used as curved coordinate systems (i.e. elliptical coordinates), and they also appear in physics as electric fields and their equipotential curves.
An isogonal trajectory is one in which the trajectory intersects the given curves at an arbitrary (but fixed) angle.
FAQs on Orthogonal Trajectory
1. How do you know if two curves are orthogonal?
If two intersecting curves cross at right angles, they are orthogonal. Two lines are obviously orthogonal if and only if they are perpendicular to each other. is the perpendicular to OP line that passes through P. The diameter of the line through P perpendicular to the tangent at P, on the other hand, is C.
2. What are the families of curves' orthogonal trajectories?
The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in the original family.
3. What is the condition of orthogonality of two curves?
If the slopes at the point of intersection are perpendicular, the curve is assumed to be perpendicular.
