Calculate path length of solution to differential equation












2














Suppose I have a parameterized path $gamma:[0,infty)rightarrowmathbb C$ with initial condition $gamma(0)=1$ and which satisfies the following:
$$gamma'(t)=uBig(igamma(t)-gamma(t)Big)=expBig(frac{3pi}{4}iBig)frac{gamma(t)}{|gamma(t)|}$$
where $u:mathbb C^*rightarrowmathbb D$ denotes the unit vector function. In other words, $gamma(t)$ is always moving at constant unit speed towards its rotation in the plane by $pi/2$. I found computationally that
$$lim_{trightarrowinfty}gamma(t)=0$$



Questions:




  1. Is there a formula for $gamma(t)$?


  2. Does $gamma$ reach the origin in finite time? In other words, does there exist $T<infty$ such that $lim_{trightarrow T}gamma(t)=0$?



Edit: Originally we had $gamma'(t)=s(i-1)frac{gamma(y)}{|gamma(t)|}$ but without loss of generality set $s=1/sqrt{2}$ so that $gamma'(t)$ has unit magnitude. Also I realize that the path length $int_0^infty|gamma'(t)|dt$ must be infinite since $|gamma'(t)|$ is constant.










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    2














    Suppose I have a parameterized path $gamma:[0,infty)rightarrowmathbb C$ with initial condition $gamma(0)=1$ and which satisfies the following:
    $$gamma'(t)=uBig(igamma(t)-gamma(t)Big)=expBig(frac{3pi}{4}iBig)frac{gamma(t)}{|gamma(t)|}$$
    where $u:mathbb C^*rightarrowmathbb D$ denotes the unit vector function. In other words, $gamma(t)$ is always moving at constant unit speed towards its rotation in the plane by $pi/2$. I found computationally that
    $$lim_{trightarrowinfty}gamma(t)=0$$



    Questions:




    1. Is there a formula for $gamma(t)$?


    2. Does $gamma$ reach the origin in finite time? In other words, does there exist $T<infty$ such that $lim_{trightarrow T}gamma(t)=0$?



    Edit: Originally we had $gamma'(t)=s(i-1)frac{gamma(y)}{|gamma(t)|}$ but without loss of generality set $s=1/sqrt{2}$ so that $gamma'(t)$ has unit magnitude. Also I realize that the path length $int_0^infty|gamma'(t)|dt$ must be infinite since $|gamma'(t)|$ is constant.










    share|cite|improve this question



























      2












      2








      2


      1





      Suppose I have a parameterized path $gamma:[0,infty)rightarrowmathbb C$ with initial condition $gamma(0)=1$ and which satisfies the following:
      $$gamma'(t)=uBig(igamma(t)-gamma(t)Big)=expBig(frac{3pi}{4}iBig)frac{gamma(t)}{|gamma(t)|}$$
      where $u:mathbb C^*rightarrowmathbb D$ denotes the unit vector function. In other words, $gamma(t)$ is always moving at constant unit speed towards its rotation in the plane by $pi/2$. I found computationally that
      $$lim_{trightarrowinfty}gamma(t)=0$$



      Questions:




      1. Is there a formula for $gamma(t)$?


      2. Does $gamma$ reach the origin in finite time? In other words, does there exist $T<infty$ such that $lim_{trightarrow T}gamma(t)=0$?



      Edit: Originally we had $gamma'(t)=s(i-1)frac{gamma(y)}{|gamma(t)|}$ but without loss of generality set $s=1/sqrt{2}$ so that $gamma'(t)$ has unit magnitude. Also I realize that the path length $int_0^infty|gamma'(t)|dt$ must be infinite since $|gamma'(t)|$ is constant.










      share|cite|improve this question















      Suppose I have a parameterized path $gamma:[0,infty)rightarrowmathbb C$ with initial condition $gamma(0)=1$ and which satisfies the following:
      $$gamma'(t)=uBig(igamma(t)-gamma(t)Big)=expBig(frac{3pi}{4}iBig)frac{gamma(t)}{|gamma(t)|}$$
      where $u:mathbb C^*rightarrowmathbb D$ denotes the unit vector function. In other words, $gamma(t)$ is always moving at constant unit speed towards its rotation in the plane by $pi/2$. I found computationally that
      $$lim_{trightarrowinfty}gamma(t)=0$$



      Questions:




      1. Is there a formula for $gamma(t)$?


      2. Does $gamma$ reach the origin in finite time? In other words, does there exist $T<infty$ such that $lim_{trightarrow T}gamma(t)=0$?



      Edit: Originally we had $gamma'(t)=s(i-1)frac{gamma(y)}{|gamma(t)|}$ but without loss of generality set $s=1/sqrt{2}$ so that $gamma'(t)$ has unit magnitude. Also I realize that the path length $int_0^infty|gamma'(t)|dt$ must be infinite since $|gamma'(t)|$ is constant.







      differential-equations






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      edited Dec 19 '18 at 3:28

























      asked Dec 9 '18 at 3:01









      M. Nestor

      74913




      74913






















          1 Answer
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          5














          It appears that this equation can be well-addressed via a transformation to polar coordinates; in polars, we write



          $gamma(t) = r(t)e^{itheta(t)}; tag 1$



          then



          $dot gamma = dot r e^{itheta} + r i dot theta e^{itheta}; tag 2$



          also,



          $vert gamma vert = r, tag 3$



          whence



          $dfrac{gamma}{vert gamma vert} = e^{itheta}; tag 4$



          assembling all this into the given equation



          $dot gamma = s(i - 1) dfrac{gamma}{vert gamma vert} tag 5$



          yields



          $dot r e^{itheta} + r i dot theta e^{itheta} = s(i - 1) e^{itheta}; tag 6$



          we divide out $e^{itheta}$:



          $dot r + ridot theta = s(i - 1) = -s + is; tag 7$



          equating real and imaginary parts,



          $dot r = -s, tag 8$



          $ridot theta = is; tag 9$



          $rdot theta = s; tag{10}$



          we may easily integrate (8) 'twixt $t_0$ and $t$, assuming $r(t_0) = r_0$:



          $r - r_0 = -s(t - t_0), tag{11}$



          $r = r_0 - s(t - t_0); tag{12}$



          we observe that the coordinate constraint $r > 0$, along with the condition $s > 0$, implies that $t$ must remain less than $r_0/s + t_0$; we shall soon find that the polar solution circumvents this limit on $t$; inserting (12) into (10):



          $(r_0 - s(t - t_0))dot theta = s; tag{13}$



          $dot theta = dfrac{s}{r_0 - s(t - t_0)}, tag{14}$



          which is also easily integrated 'twixt $t_0$ and $t$:



          $theta - theta_0 = -ln(r_0 - s(t - t_0)) + ln r_0; tag{15}$



          $theta = theta_0 - ln(r_0 - s(t - t_0)) + ln r_0; tag{16}$



          using (12), we may express $r$ in term of $theta$:



          $theta = theta_0 - ln r + ln r_0; tag{17}$



          $e^theta = e^{theta_0} dfrac{r_0}{r}; tag{18}$



          $r = r_0 e^{theta_0 - theta}; tag{19}$



          the above presents the general solution for $t = t_0$, $r(t_0) = r_0$, $theta(t_0) = theta_0$; we specialize to the case at hand:



          $t_0 = 0, ; gamma(0) = 1 + i; tag{20}$



          $r_0 = vert gamma(0) vert = sqrt 2; tag{21}$



          $theta_0 = dfrac{pi}{4}; tag{22}$



          from (19):



          $r = sqrt 2 e^{pi/4 - theta}; tag{23}$



          from (12) and (16):



          $r(t) = sqrt 2 - st; tag{24}$



          $theta(t) = dfrac{pi}{4} - ln(sqrt 2 - s(t - t_0)) + ln sqrt 2. tag{25}$



          The path length is perhaps easiest found by returning to the expressions for $r(t)$ and $theta(t)$, in terms of which



          $vert dot gamma vert = sqrt{ (dot r)^2 + r^2 (dot theta)^2} = sqrt{s^2 + s^2} = sqrt 2 s; tag{26}$



          since $t$ is restricted to $[0, r_0/s]$, the length of the curve is



          $displaystyle int_0^{r_0/s} vert dot gamma vert ; dt = displaystyle int_0^{r_0/s} sqrt 2 s ; dt = sqrt 2 r_0. tag{27}$



          As concerns our OP M. Nestor's final remarks:



          The curve $gamma(t)$ (1), expressed as a function of the independent variable $theta$ is, in polar coordiates, given by (23); it is easy to see this curve is a spiral which approaches the origin as $theta to infty$.



          Now, as for item (1), there is indeed a formula for $gamma(t)$; in polar coordinates the equations are given in (12) and (16); from these we may derive
          cartesian equations for $x(t)$, $y(t)$ by setting



          $x(t) = r(t) cos theta (t), ; y(t) = r(t) sin theta(t); tag{28}$



          and as for item (2), we have shown above ca. (12) that



          $displaystyle lim_{t to t_0 + r_0/s} r(t) = 0, tag{29}$



          where $gamma(t_0) = (r_0, theta_0)$; but it should be noted that $gamma(t)$ never "reaches the origin", though it does become arbitrarily close as $t to (t_0 + r_0 / s)^-$.



          Finally, in response to M. Nestor's final edit, we remark that the integral



          $displaystyle int_{t_0}^infty vert gamma'(t) vert ; dt tag{30}$



          is not defined, since $gamma(t)$ can't be extended past $t = t_0 + r_0/s$; the length may instead be found via (27).






          share|cite|improve this answer



















          • 1




            Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
            – M. Nestor
            Dec 9 '18 at 7:00












          • @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
            – Robert Lewis
            Dec 9 '18 at 7:04








          • 1




            @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
            – Robert Lewis
            Dec 9 '18 at 7:17











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          1 Answer
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          1 Answer
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          active

          oldest

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          active

          oldest

          votes






          active

          oldest

          votes









          5














          It appears that this equation can be well-addressed via a transformation to polar coordinates; in polars, we write



          $gamma(t) = r(t)e^{itheta(t)}; tag 1$



          then



          $dot gamma = dot r e^{itheta} + r i dot theta e^{itheta}; tag 2$



          also,



          $vert gamma vert = r, tag 3$



          whence



          $dfrac{gamma}{vert gamma vert} = e^{itheta}; tag 4$



          assembling all this into the given equation



          $dot gamma = s(i - 1) dfrac{gamma}{vert gamma vert} tag 5$



          yields



          $dot r e^{itheta} + r i dot theta e^{itheta} = s(i - 1) e^{itheta}; tag 6$



          we divide out $e^{itheta}$:



          $dot r + ridot theta = s(i - 1) = -s + is; tag 7$



          equating real and imaginary parts,



          $dot r = -s, tag 8$



          $ridot theta = is; tag 9$



          $rdot theta = s; tag{10}$



          we may easily integrate (8) 'twixt $t_0$ and $t$, assuming $r(t_0) = r_0$:



          $r - r_0 = -s(t - t_0), tag{11}$



          $r = r_0 - s(t - t_0); tag{12}$



          we observe that the coordinate constraint $r > 0$, along with the condition $s > 0$, implies that $t$ must remain less than $r_0/s + t_0$; we shall soon find that the polar solution circumvents this limit on $t$; inserting (12) into (10):



          $(r_0 - s(t - t_0))dot theta = s; tag{13}$



          $dot theta = dfrac{s}{r_0 - s(t - t_0)}, tag{14}$



          which is also easily integrated 'twixt $t_0$ and $t$:



          $theta - theta_0 = -ln(r_0 - s(t - t_0)) + ln r_0; tag{15}$



          $theta = theta_0 - ln(r_0 - s(t - t_0)) + ln r_0; tag{16}$



          using (12), we may express $r$ in term of $theta$:



          $theta = theta_0 - ln r + ln r_0; tag{17}$



          $e^theta = e^{theta_0} dfrac{r_0}{r}; tag{18}$



          $r = r_0 e^{theta_0 - theta}; tag{19}$



          the above presents the general solution for $t = t_0$, $r(t_0) = r_0$, $theta(t_0) = theta_0$; we specialize to the case at hand:



          $t_0 = 0, ; gamma(0) = 1 + i; tag{20}$



          $r_0 = vert gamma(0) vert = sqrt 2; tag{21}$



          $theta_0 = dfrac{pi}{4}; tag{22}$



          from (19):



          $r = sqrt 2 e^{pi/4 - theta}; tag{23}$



          from (12) and (16):



          $r(t) = sqrt 2 - st; tag{24}$



          $theta(t) = dfrac{pi}{4} - ln(sqrt 2 - s(t - t_0)) + ln sqrt 2. tag{25}$



          The path length is perhaps easiest found by returning to the expressions for $r(t)$ and $theta(t)$, in terms of which



          $vert dot gamma vert = sqrt{ (dot r)^2 + r^2 (dot theta)^2} = sqrt{s^2 + s^2} = sqrt 2 s; tag{26}$



          since $t$ is restricted to $[0, r_0/s]$, the length of the curve is



          $displaystyle int_0^{r_0/s} vert dot gamma vert ; dt = displaystyle int_0^{r_0/s} sqrt 2 s ; dt = sqrt 2 r_0. tag{27}$



          As concerns our OP M. Nestor's final remarks:



          The curve $gamma(t)$ (1), expressed as a function of the independent variable $theta$ is, in polar coordiates, given by (23); it is easy to see this curve is a spiral which approaches the origin as $theta to infty$.



          Now, as for item (1), there is indeed a formula for $gamma(t)$; in polar coordinates the equations are given in (12) and (16); from these we may derive
          cartesian equations for $x(t)$, $y(t)$ by setting



          $x(t) = r(t) cos theta (t), ; y(t) = r(t) sin theta(t); tag{28}$



          and as for item (2), we have shown above ca. (12) that



          $displaystyle lim_{t to t_0 + r_0/s} r(t) = 0, tag{29}$



          where $gamma(t_0) = (r_0, theta_0)$; but it should be noted that $gamma(t)$ never "reaches the origin", though it does become arbitrarily close as $t to (t_0 + r_0 / s)^-$.



          Finally, in response to M. Nestor's final edit, we remark that the integral



          $displaystyle int_{t_0}^infty vert gamma'(t) vert ; dt tag{30}$



          is not defined, since $gamma(t)$ can't be extended past $t = t_0 + r_0/s$; the length may instead be found via (27).






          share|cite|improve this answer



















          • 1




            Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
            – M. Nestor
            Dec 9 '18 at 7:00












          • @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
            – Robert Lewis
            Dec 9 '18 at 7:04








          • 1




            @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
            – Robert Lewis
            Dec 9 '18 at 7:17
















          5














          It appears that this equation can be well-addressed via a transformation to polar coordinates; in polars, we write



          $gamma(t) = r(t)e^{itheta(t)}; tag 1$



          then



          $dot gamma = dot r e^{itheta} + r i dot theta e^{itheta}; tag 2$



          also,



          $vert gamma vert = r, tag 3$



          whence



          $dfrac{gamma}{vert gamma vert} = e^{itheta}; tag 4$



          assembling all this into the given equation



          $dot gamma = s(i - 1) dfrac{gamma}{vert gamma vert} tag 5$



          yields



          $dot r e^{itheta} + r i dot theta e^{itheta} = s(i - 1) e^{itheta}; tag 6$



          we divide out $e^{itheta}$:



          $dot r + ridot theta = s(i - 1) = -s + is; tag 7$



          equating real and imaginary parts,



          $dot r = -s, tag 8$



          $ridot theta = is; tag 9$



          $rdot theta = s; tag{10}$



          we may easily integrate (8) 'twixt $t_0$ and $t$, assuming $r(t_0) = r_0$:



          $r - r_0 = -s(t - t_0), tag{11}$



          $r = r_0 - s(t - t_0); tag{12}$



          we observe that the coordinate constraint $r > 0$, along with the condition $s > 0$, implies that $t$ must remain less than $r_0/s + t_0$; we shall soon find that the polar solution circumvents this limit on $t$; inserting (12) into (10):



          $(r_0 - s(t - t_0))dot theta = s; tag{13}$



          $dot theta = dfrac{s}{r_0 - s(t - t_0)}, tag{14}$



          which is also easily integrated 'twixt $t_0$ and $t$:



          $theta - theta_0 = -ln(r_0 - s(t - t_0)) + ln r_0; tag{15}$



          $theta = theta_0 - ln(r_0 - s(t - t_0)) + ln r_0; tag{16}$



          using (12), we may express $r$ in term of $theta$:



          $theta = theta_0 - ln r + ln r_0; tag{17}$



          $e^theta = e^{theta_0} dfrac{r_0}{r}; tag{18}$



          $r = r_0 e^{theta_0 - theta}; tag{19}$



          the above presents the general solution for $t = t_0$, $r(t_0) = r_0$, $theta(t_0) = theta_0$; we specialize to the case at hand:



          $t_0 = 0, ; gamma(0) = 1 + i; tag{20}$



          $r_0 = vert gamma(0) vert = sqrt 2; tag{21}$



          $theta_0 = dfrac{pi}{4}; tag{22}$



          from (19):



          $r = sqrt 2 e^{pi/4 - theta}; tag{23}$



          from (12) and (16):



          $r(t) = sqrt 2 - st; tag{24}$



          $theta(t) = dfrac{pi}{4} - ln(sqrt 2 - s(t - t_0)) + ln sqrt 2. tag{25}$



          The path length is perhaps easiest found by returning to the expressions for $r(t)$ and $theta(t)$, in terms of which



          $vert dot gamma vert = sqrt{ (dot r)^2 + r^2 (dot theta)^2} = sqrt{s^2 + s^2} = sqrt 2 s; tag{26}$



          since $t$ is restricted to $[0, r_0/s]$, the length of the curve is



          $displaystyle int_0^{r_0/s} vert dot gamma vert ; dt = displaystyle int_0^{r_0/s} sqrt 2 s ; dt = sqrt 2 r_0. tag{27}$



          As concerns our OP M. Nestor's final remarks:



          The curve $gamma(t)$ (1), expressed as a function of the independent variable $theta$ is, in polar coordiates, given by (23); it is easy to see this curve is a spiral which approaches the origin as $theta to infty$.



          Now, as for item (1), there is indeed a formula for $gamma(t)$; in polar coordinates the equations are given in (12) and (16); from these we may derive
          cartesian equations for $x(t)$, $y(t)$ by setting



          $x(t) = r(t) cos theta (t), ; y(t) = r(t) sin theta(t); tag{28}$



          and as for item (2), we have shown above ca. (12) that



          $displaystyle lim_{t to t_0 + r_0/s} r(t) = 0, tag{29}$



          where $gamma(t_0) = (r_0, theta_0)$; but it should be noted that $gamma(t)$ never "reaches the origin", though it does become arbitrarily close as $t to (t_0 + r_0 / s)^-$.



          Finally, in response to M. Nestor's final edit, we remark that the integral



          $displaystyle int_{t_0}^infty vert gamma'(t) vert ; dt tag{30}$



          is not defined, since $gamma(t)$ can't be extended past $t = t_0 + r_0/s$; the length may instead be found via (27).






          share|cite|improve this answer



















          • 1




            Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
            – M. Nestor
            Dec 9 '18 at 7:00












          • @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
            – Robert Lewis
            Dec 9 '18 at 7:04








          • 1




            @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
            – Robert Lewis
            Dec 9 '18 at 7:17














          5












          5








          5






          It appears that this equation can be well-addressed via a transformation to polar coordinates; in polars, we write



          $gamma(t) = r(t)e^{itheta(t)}; tag 1$



          then



          $dot gamma = dot r e^{itheta} + r i dot theta e^{itheta}; tag 2$



          also,



          $vert gamma vert = r, tag 3$



          whence



          $dfrac{gamma}{vert gamma vert} = e^{itheta}; tag 4$



          assembling all this into the given equation



          $dot gamma = s(i - 1) dfrac{gamma}{vert gamma vert} tag 5$



          yields



          $dot r e^{itheta} + r i dot theta e^{itheta} = s(i - 1) e^{itheta}; tag 6$



          we divide out $e^{itheta}$:



          $dot r + ridot theta = s(i - 1) = -s + is; tag 7$



          equating real and imaginary parts,



          $dot r = -s, tag 8$



          $ridot theta = is; tag 9$



          $rdot theta = s; tag{10}$



          we may easily integrate (8) 'twixt $t_0$ and $t$, assuming $r(t_0) = r_0$:



          $r - r_0 = -s(t - t_0), tag{11}$



          $r = r_0 - s(t - t_0); tag{12}$



          we observe that the coordinate constraint $r > 0$, along with the condition $s > 0$, implies that $t$ must remain less than $r_0/s + t_0$; we shall soon find that the polar solution circumvents this limit on $t$; inserting (12) into (10):



          $(r_0 - s(t - t_0))dot theta = s; tag{13}$



          $dot theta = dfrac{s}{r_0 - s(t - t_0)}, tag{14}$



          which is also easily integrated 'twixt $t_0$ and $t$:



          $theta - theta_0 = -ln(r_0 - s(t - t_0)) + ln r_0; tag{15}$



          $theta = theta_0 - ln(r_0 - s(t - t_0)) + ln r_0; tag{16}$



          using (12), we may express $r$ in term of $theta$:



          $theta = theta_0 - ln r + ln r_0; tag{17}$



          $e^theta = e^{theta_0} dfrac{r_0}{r}; tag{18}$



          $r = r_0 e^{theta_0 - theta}; tag{19}$



          the above presents the general solution for $t = t_0$, $r(t_0) = r_0$, $theta(t_0) = theta_0$; we specialize to the case at hand:



          $t_0 = 0, ; gamma(0) = 1 + i; tag{20}$



          $r_0 = vert gamma(0) vert = sqrt 2; tag{21}$



          $theta_0 = dfrac{pi}{4}; tag{22}$



          from (19):



          $r = sqrt 2 e^{pi/4 - theta}; tag{23}$



          from (12) and (16):



          $r(t) = sqrt 2 - st; tag{24}$



          $theta(t) = dfrac{pi}{4} - ln(sqrt 2 - s(t - t_0)) + ln sqrt 2. tag{25}$



          The path length is perhaps easiest found by returning to the expressions for $r(t)$ and $theta(t)$, in terms of which



          $vert dot gamma vert = sqrt{ (dot r)^2 + r^2 (dot theta)^2} = sqrt{s^2 + s^2} = sqrt 2 s; tag{26}$



          since $t$ is restricted to $[0, r_0/s]$, the length of the curve is



          $displaystyle int_0^{r_0/s} vert dot gamma vert ; dt = displaystyle int_0^{r_0/s} sqrt 2 s ; dt = sqrt 2 r_0. tag{27}$



          As concerns our OP M. Nestor's final remarks:



          The curve $gamma(t)$ (1), expressed as a function of the independent variable $theta$ is, in polar coordiates, given by (23); it is easy to see this curve is a spiral which approaches the origin as $theta to infty$.



          Now, as for item (1), there is indeed a formula for $gamma(t)$; in polar coordinates the equations are given in (12) and (16); from these we may derive
          cartesian equations for $x(t)$, $y(t)$ by setting



          $x(t) = r(t) cos theta (t), ; y(t) = r(t) sin theta(t); tag{28}$



          and as for item (2), we have shown above ca. (12) that



          $displaystyle lim_{t to t_0 + r_0/s} r(t) = 0, tag{29}$



          where $gamma(t_0) = (r_0, theta_0)$; but it should be noted that $gamma(t)$ never "reaches the origin", though it does become arbitrarily close as $t to (t_0 + r_0 / s)^-$.



          Finally, in response to M. Nestor's final edit, we remark that the integral



          $displaystyle int_{t_0}^infty vert gamma'(t) vert ; dt tag{30}$



          is not defined, since $gamma(t)$ can't be extended past $t = t_0 + r_0/s$; the length may instead be found via (27).






          share|cite|improve this answer














          It appears that this equation can be well-addressed via a transformation to polar coordinates; in polars, we write



          $gamma(t) = r(t)e^{itheta(t)}; tag 1$



          then



          $dot gamma = dot r e^{itheta} + r i dot theta e^{itheta}; tag 2$



          also,



          $vert gamma vert = r, tag 3$



          whence



          $dfrac{gamma}{vert gamma vert} = e^{itheta}; tag 4$



          assembling all this into the given equation



          $dot gamma = s(i - 1) dfrac{gamma}{vert gamma vert} tag 5$



          yields



          $dot r e^{itheta} + r i dot theta e^{itheta} = s(i - 1) e^{itheta}; tag 6$



          we divide out $e^{itheta}$:



          $dot r + ridot theta = s(i - 1) = -s + is; tag 7$



          equating real and imaginary parts,



          $dot r = -s, tag 8$



          $ridot theta = is; tag 9$



          $rdot theta = s; tag{10}$



          we may easily integrate (8) 'twixt $t_0$ and $t$, assuming $r(t_0) = r_0$:



          $r - r_0 = -s(t - t_0), tag{11}$



          $r = r_0 - s(t - t_0); tag{12}$



          we observe that the coordinate constraint $r > 0$, along with the condition $s > 0$, implies that $t$ must remain less than $r_0/s + t_0$; we shall soon find that the polar solution circumvents this limit on $t$; inserting (12) into (10):



          $(r_0 - s(t - t_0))dot theta = s; tag{13}$



          $dot theta = dfrac{s}{r_0 - s(t - t_0)}, tag{14}$



          which is also easily integrated 'twixt $t_0$ and $t$:



          $theta - theta_0 = -ln(r_0 - s(t - t_0)) + ln r_0; tag{15}$



          $theta = theta_0 - ln(r_0 - s(t - t_0)) + ln r_0; tag{16}$



          using (12), we may express $r$ in term of $theta$:



          $theta = theta_0 - ln r + ln r_0; tag{17}$



          $e^theta = e^{theta_0} dfrac{r_0}{r}; tag{18}$



          $r = r_0 e^{theta_0 - theta}; tag{19}$



          the above presents the general solution for $t = t_0$, $r(t_0) = r_0$, $theta(t_0) = theta_0$; we specialize to the case at hand:



          $t_0 = 0, ; gamma(0) = 1 + i; tag{20}$



          $r_0 = vert gamma(0) vert = sqrt 2; tag{21}$



          $theta_0 = dfrac{pi}{4}; tag{22}$



          from (19):



          $r = sqrt 2 e^{pi/4 - theta}; tag{23}$



          from (12) and (16):



          $r(t) = sqrt 2 - st; tag{24}$



          $theta(t) = dfrac{pi}{4} - ln(sqrt 2 - s(t - t_0)) + ln sqrt 2. tag{25}$



          The path length is perhaps easiest found by returning to the expressions for $r(t)$ and $theta(t)$, in terms of which



          $vert dot gamma vert = sqrt{ (dot r)^2 + r^2 (dot theta)^2} = sqrt{s^2 + s^2} = sqrt 2 s; tag{26}$



          since $t$ is restricted to $[0, r_0/s]$, the length of the curve is



          $displaystyle int_0^{r_0/s} vert dot gamma vert ; dt = displaystyle int_0^{r_0/s} sqrt 2 s ; dt = sqrt 2 r_0. tag{27}$



          As concerns our OP M. Nestor's final remarks:



          The curve $gamma(t)$ (1), expressed as a function of the independent variable $theta$ is, in polar coordiates, given by (23); it is easy to see this curve is a spiral which approaches the origin as $theta to infty$.



          Now, as for item (1), there is indeed a formula for $gamma(t)$; in polar coordinates the equations are given in (12) and (16); from these we may derive
          cartesian equations for $x(t)$, $y(t)$ by setting



          $x(t) = r(t) cos theta (t), ; y(t) = r(t) sin theta(t); tag{28}$



          and as for item (2), we have shown above ca. (12) that



          $displaystyle lim_{t to t_0 + r_0/s} r(t) = 0, tag{29}$



          where $gamma(t_0) = (r_0, theta_0)$; but it should be noted that $gamma(t)$ never "reaches the origin", though it does become arbitrarily close as $t to (t_0 + r_0 / s)^-$.



          Finally, in response to M. Nestor's final edit, we remark that the integral



          $displaystyle int_{t_0}^infty vert gamma'(t) vert ; dt tag{30}$



          is not defined, since $gamma(t)$ can't be extended past $t = t_0 + r_0/s$; the length may instead be found via (27).







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 16 '18 at 2:09

























          answered Dec 9 '18 at 5:23









          Robert Lewis

          43.7k22963




          43.7k22963








          • 1




            Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
            – M. Nestor
            Dec 9 '18 at 7:00












          • @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
            – Robert Lewis
            Dec 9 '18 at 7:04








          • 1




            @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
            – Robert Lewis
            Dec 9 '18 at 7:17














          • 1




            Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
            – M. Nestor
            Dec 9 '18 at 7:00












          • @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
            – Robert Lewis
            Dec 9 '18 at 7:04








          • 1




            @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
            – Robert Lewis
            Dec 9 '18 at 7:17








          1




          1




          Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
          – M. Nestor
          Dec 9 '18 at 7:00






          Thanks! I see that $r=sqrt{2}exp(pi/4-theta)$ for $thetageqpi/4$ is a good polar expression for this curve. However this curve never reaches the origin, but $r(t)=sqrt{2}-st$ on line 24 seems to imply $r=0$ for $t=sqrt{2}/s$. Does the image of $tin[0,sqrt{2}/s)$ under $gamma$ contain the entire curve?
          – M. Nestor
          Dec 9 '18 at 7:00














          @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
          – Robert Lewis
          Dec 9 '18 at 7:04






          @M.Nestor: well, a quick answer which I think adequate is that the polar coordinate system isn't really defined at $r = 0$, so we should really say $t < sqrt 2 / s$. I should probably edit that part of my answer.
          – Robert Lewis
          Dec 9 '18 at 7:04






          1




          1




          @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
          – Robert Lewis
          Dec 9 '18 at 7:17




          @M.Nestor: you should be able to get answers to your edited question by taking $s = 1/sqrt 2$ in mine,
          – Robert Lewis
          Dec 9 '18 at 7:17


















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