example of isogonal trajectory
Determine the curves which intersect the origin-centered circles at an angle of 45∘.
The differential equation of the circles x2+y2=R2 is 2xdx+2ydy=0, i.e.
xy+dydx= 0. |
Thus, by the model (2) of the parent entry (http://planetmath.org/IsogonalTrajectory), the differential equation of the isogonal trajectory reads
xy+y′-tanπ41+y′tanπ4= 0, | (1) |
which can be rewritten as
y′=y-xy+x=yx-1yx+1. |
Here, one may take yx:=t as a new variable (see ODE types reductible to the variables separable case), when
y=xt,y′=dydx=t+xdtdx, |
and in the resulting equation
t+xdtdx=t-1t+1 |
one can separate the variables (http://planetmath.org/SeparationOfVariables):
1+t1+t2dt=-dxx |
Multiplying here by 2 and integrating then give
2arctant+ln(1+t2)=-2lnx+lnC2≡-lnx2C2, |
or equivalently
lnx2+x2t2C2=-2arctant. |
This is
ln√x2+y2C=-arctanyx, |
i.e.
√x2+y2=Ce-arctanyx. |
Expressing this in the polar coordinates r,φ gives the family of the integral curves of the equation (1) in the form
r=Ce-φ. |
Consequently, the family of the isogonal trajectories consists of logarithmic spirals.
Title | example of isogonal trajectory |
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Canonical name | ExampleOfIsogonalTrajectory |
Date of creation | 2013-03-22 18:59:23 |
Last modified on | 2013-03-22 18:59:23 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 10 |
Author | pahio (2872) |
Entry type | Example |
Classification | msc 51N20 |
Classification | msc 34A26 |
Classification | msc 34A09 |
Synonym | isogonal trajectories of concentric circles |
Related topic | IsogonalTrajectory |