How to calculate the index number for a curve around a linear system's fixed point without integrals?
$begingroup$
We know $$ phi = tan^{-1} frac{dot{y}}{dot{x}},$$ yet so far I've only been able to calculate the index of curves by using the integral
$$
frac{1}{2pi} oint_C frac{dot{y}ddot{x} - dot{x}ddot{y}}{dot{x}^2 + dot{y}^2} dt.
$$
I'm only working with simple $2times 2$ linear system fixed points, i.e. centers, saddles, stable nodes, etc...
While my method works, with $x = cos t$, $y = sin t$, and $t in [0,2pi]$, I was told very quickly by my prof. that we can also directly calculate it with only the difference of $phi$ at $t = 0, 2pi$. Yet, every parameterization has cancelled out to zero when I calculate arctan: when the index is suppose to 1.
It seems like my parameterization will always cancel out since $0 equiv 2pi pmod{2pi}$. What am I missing?
ordinary-differential-equations winding-number
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
$endgroup$
add a comment |
$begingroup$
We know $$ phi = tan^{-1} frac{dot{y}}{dot{x}},$$ yet so far I've only been able to calculate the index of curves by using the integral
$$
frac{1}{2pi} oint_C frac{dot{y}ddot{x} - dot{x}ddot{y}}{dot{x}^2 + dot{y}^2} dt.
$$
I'm only working with simple $2times 2$ linear system fixed points, i.e. centers, saddles, stable nodes, etc...
While my method works, with $x = cos t$, $y = sin t$, and $t in [0,2pi]$, I was told very quickly by my prof. that we can also directly calculate it with only the difference of $phi$ at $t = 0, 2pi$. Yet, every parameterization has cancelled out to zero when I calculate arctan: when the index is suppose to 1.
It seems like my parameterization will always cancel out since $0 equiv 2pi pmod{2pi}$. What am I missing?
ordinary-differential-equations winding-number
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
$endgroup$
add a comment |
$begingroup$
We know $$ phi = tan^{-1} frac{dot{y}}{dot{x}},$$ yet so far I've only been able to calculate the index of curves by using the integral
$$
frac{1}{2pi} oint_C frac{dot{y}ddot{x} - dot{x}ddot{y}}{dot{x}^2 + dot{y}^2} dt.
$$
I'm only working with simple $2times 2$ linear system fixed points, i.e. centers, saddles, stable nodes, etc...
While my method works, with $x = cos t$, $y = sin t$, and $t in [0,2pi]$, I was told very quickly by my prof. that we can also directly calculate it with only the difference of $phi$ at $t = 0, 2pi$. Yet, every parameterization has cancelled out to zero when I calculate arctan: when the index is suppose to 1.
It seems like my parameterization will always cancel out since $0 equiv 2pi pmod{2pi}$. What am I missing?
ordinary-differential-equations winding-number
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
$endgroup$
We know $$ phi = tan^{-1} frac{dot{y}}{dot{x}},$$ yet so far I've only been able to calculate the index of curves by using the integral
$$
frac{1}{2pi} oint_C frac{dot{y}ddot{x} - dot{x}ddot{y}}{dot{x}^2 + dot{y}^2} dt.
$$
I'm only working with simple $2times 2$ linear system fixed points, i.e. centers, saddles, stable nodes, etc...
While my method works, with $x = cos t$, $y = sin t$, and $t in [0,2pi]$, I was told very quickly by my prof. that we can also directly calculate it with only the difference of $phi$ at $t = 0, 2pi$. Yet, every parameterization has cancelled out to zero when I calculate arctan: when the index is suppose to 1.
It seems like my parameterization will always cancel out since $0 equiv 2pi pmod{2pi}$. What am I missing?
ordinary-differential-equations winding-number
ordinary-differential-equations winding-number
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
edited Nov 26 '18 at 21:25
Math Student 99
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
asked Nov 26 '18 at 21:18
Math Student 99Math Student 99
112
112
add a comment |
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