How to calculate the envelope of the trajectory of a double pendulum?Double pendulum probability distributionLagrangian equations on Double Pendulum (Potential and kinetic Energy of double pendulum)Calculating a double pendulumIs this divergence-free? (Double Pendulum)Double Pendulum CuspinessHow does one calculate a re-bounce sling shot trajectory such a Juno?Finding the Equations of Motion in a Classical Mechanics ProblemArnold DiffusionDouble pendulum lagrangianIs there a proof that the double pendulum path reaches every point in a closed region eventually?Periodic solutions of the double pendulum

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How to calculate the envelope of the trajectory of a double pendulum?


Double pendulum probability distributionLagrangian equations on Double Pendulum (Potential and kinetic Energy of double pendulum)Calculating a double pendulumIs this divergence-free? (Double Pendulum)Double Pendulum CuspinessHow does one calculate a re-bounce sling shot trajectory such a Juno?Finding the Equations of Motion in a Classical Mechanics ProblemArnold DiffusionDouble pendulum lagrangianIs there a proof that the double pendulum path reaches every point in a closed region eventually?Periodic solutions of the double pendulum













5












$begingroup$


Consider a double pendulum:



enter image description here



Background



For the angles $varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):



$dotvarphi_1 = 6frac2p_1 - 3p_2cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotvarphi_2 = 6frac8p_2 - 3p_1cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotp_1 = -frac12big( dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +3sin(varphi_1) big)$



$dotp_2 = -frac12big( -dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +sin(varphi_1) big)$



To see the relations more clearly:




$dotvarphi_1 = B(2p_1 + Ap_2)$



$dotvarphi_2 = B(8p_2 + Ap_1)$



$dotp_1 = -C + 3D$



$dotp_2 = +C + D $




with



$A = -3cos(varphi_1 - varphi_2)$



$B = 6/(16 - A^2)$



$C = dotvarphi_1dotvarphi_2sin(varphi_1 - varphi_2)/2$



$D = -sin(varphi_1)/2$



Observations



With initial angles $varphi_1^0 = varphi_2^0 = 0$ and different combinations of small values for $p_1^0$, $p_2^0$ a number of intriguiung patterns can be observed when plotting the tractory of the tip of the pendulum:




  • $p_1^0 = 1$, $p_2^0 = 1$

enter image description here



  • $p_1^0 = 1, p_2^0 = -1$

enter image description hereenter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 1$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 2$

enter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3.7$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 4$

enter image description hereenter image description here



What these patterns have in common: The tip of the pendulum draws a curve which more or less slowly "fills" an area enclosed by a specific envelope, intermediately exhibiting seemingly regular patterns which inevitably eventually vanish.



Questions



  1. Can the envelope be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?


  2. Can the positions of the two inner cusps which can be seen clearly for $p_1=0$, $p_2=2,3$ be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?



[The envelope for $p_1^0=0, p_2^0 = 1,2,dots$ looks like a canoe whose bow and stern bend to each other, eventually amalgamating. Can anyone guess what's the explicit formula for this shape?]










share|cite|improve this question











$endgroup$











  • $begingroup$
    Upvoted just for the animations - how did you do them? Also, the paths have cusps at the ends so they are not smooth everywhere.
    $endgroup$
    – marty cohen
    Oct 25 '18 at 13:34






  • 1




    $begingroup$
    I should give credit to travisdoesmath who uses d3.js and Runge-Kutta.
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 13:39







  • 1




    $begingroup$
    The phase space $(phi_1,phi_2,p_1,p_2)$ is 4-dimensional, and the total energy and the angular momentum about the fixed pivot are conserved over time. These two constraints determine a 2-dimensional slice of the phase space for any given set of initial conditions. Presumably, the projection of this abstract 2-dimensional set of states to physical space gives the desired envelope.
    $endgroup$
    – Rahul
    Oct 25 '18 at 13:45
















5












$begingroup$


Consider a double pendulum:



enter image description here



Background



For the angles $varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):



$dotvarphi_1 = 6frac2p_1 - 3p_2cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotvarphi_2 = 6frac8p_2 - 3p_1cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotp_1 = -frac12big( dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +3sin(varphi_1) big)$



$dotp_2 = -frac12big( -dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +sin(varphi_1) big)$



To see the relations more clearly:




$dotvarphi_1 = B(2p_1 + Ap_2)$



$dotvarphi_2 = B(8p_2 + Ap_1)$



$dotp_1 = -C + 3D$



$dotp_2 = +C + D $




with



$A = -3cos(varphi_1 - varphi_2)$



$B = 6/(16 - A^2)$



$C = dotvarphi_1dotvarphi_2sin(varphi_1 - varphi_2)/2$



$D = -sin(varphi_1)/2$



Observations



With initial angles $varphi_1^0 = varphi_2^0 = 0$ and different combinations of small values for $p_1^0$, $p_2^0$ a number of intriguiung patterns can be observed when plotting the tractory of the tip of the pendulum:




  • $p_1^0 = 1$, $p_2^0 = 1$

enter image description here



  • $p_1^0 = 1, p_2^0 = -1$

enter image description hereenter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 1$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 2$

enter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3.7$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 4$

enter image description hereenter image description here



What these patterns have in common: The tip of the pendulum draws a curve which more or less slowly "fills" an area enclosed by a specific envelope, intermediately exhibiting seemingly regular patterns which inevitably eventually vanish.



Questions



  1. Can the envelope be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?


  2. Can the positions of the two inner cusps which can be seen clearly for $p_1=0$, $p_2=2,3$ be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?



[The envelope for $p_1^0=0, p_2^0 = 1,2,dots$ looks like a canoe whose bow and stern bend to each other, eventually amalgamating. Can anyone guess what's the explicit formula for this shape?]










share|cite|improve this question











$endgroup$











  • $begingroup$
    Upvoted just for the animations - how did you do them? Also, the paths have cusps at the ends so they are not smooth everywhere.
    $endgroup$
    – marty cohen
    Oct 25 '18 at 13:34






  • 1




    $begingroup$
    I should give credit to travisdoesmath who uses d3.js and Runge-Kutta.
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 13:39







  • 1




    $begingroup$
    The phase space $(phi_1,phi_2,p_1,p_2)$ is 4-dimensional, and the total energy and the angular momentum about the fixed pivot are conserved over time. These two constraints determine a 2-dimensional slice of the phase space for any given set of initial conditions. Presumably, the projection of this abstract 2-dimensional set of states to physical space gives the desired envelope.
    $endgroup$
    – Rahul
    Oct 25 '18 at 13:45














5












5








5


1



$begingroup$


Consider a double pendulum:



enter image description here



Background



For the angles $varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):



$dotvarphi_1 = 6frac2p_1 - 3p_2cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotvarphi_2 = 6frac8p_2 - 3p_1cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotp_1 = -frac12big( dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +3sin(varphi_1) big)$



$dotp_2 = -frac12big( -dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +sin(varphi_1) big)$



To see the relations more clearly:




$dotvarphi_1 = B(2p_1 + Ap_2)$



$dotvarphi_2 = B(8p_2 + Ap_1)$



$dotp_1 = -C + 3D$



$dotp_2 = +C + D $




with



$A = -3cos(varphi_1 - varphi_2)$



$B = 6/(16 - A^2)$



$C = dotvarphi_1dotvarphi_2sin(varphi_1 - varphi_2)/2$



$D = -sin(varphi_1)/2$



Observations



With initial angles $varphi_1^0 = varphi_2^0 = 0$ and different combinations of small values for $p_1^0$, $p_2^0$ a number of intriguiung patterns can be observed when plotting the tractory of the tip of the pendulum:




  • $p_1^0 = 1$, $p_2^0 = 1$

enter image description here



  • $p_1^0 = 1, p_2^0 = -1$

enter image description hereenter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 1$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 2$

enter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3.7$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 4$

enter image description hereenter image description here



What these patterns have in common: The tip of the pendulum draws a curve which more or less slowly "fills" an area enclosed by a specific envelope, intermediately exhibiting seemingly regular patterns which inevitably eventually vanish.



Questions



  1. Can the envelope be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?


  2. Can the positions of the two inner cusps which can be seen clearly for $p_1=0$, $p_2=2,3$ be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?



[The envelope for $p_1^0=0, p_2^0 = 1,2,dots$ looks like a canoe whose bow and stern bend to each other, eventually amalgamating. Can anyone guess what's the explicit formula for this shape?]










share|cite|improve this question











$endgroup$




Consider a double pendulum:



enter image description here



Background



For the angles $varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):



$dotvarphi_1 = 6frac2p_1 - 3p_2cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotvarphi_2 = 6frac8p_2 - 3p_1cos(varphi_1 - varphi_2)16 - 9cos^2(varphi_1 - varphi_2)$



$dotp_1 = -frac12big( dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +3sin(varphi_1) big)$



$dotp_2 = -frac12big( -dotvarphi_1dotvarphi_2 sin(varphi_1 - varphi_2) +sin(varphi_1) big)$



To see the relations more clearly:




$dotvarphi_1 = B(2p_1 + Ap_2)$



$dotvarphi_2 = B(8p_2 + Ap_1)$



$dotp_1 = -C + 3D$



$dotp_2 = +C + D $




with



$A = -3cos(varphi_1 - varphi_2)$



$B = 6/(16 - A^2)$



$C = dotvarphi_1dotvarphi_2sin(varphi_1 - varphi_2)/2$



$D = -sin(varphi_1)/2$



Observations



With initial angles $varphi_1^0 = varphi_2^0 = 0$ and different combinations of small values for $p_1^0$, $p_2^0$ a number of intriguiung patterns can be observed when plotting the tractory of the tip of the pendulum:




  • $p_1^0 = 1$, $p_2^0 = 1$

enter image description here



  • $p_1^0 = 1, p_2^0 = -1$

enter image description hereenter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 1$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 2$

enter image description hereenter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 3.7$

enter image description hereenter image description here




  • $p_1^0 = 0$, $p_2^0 = 4$

enter image description hereenter image description here



What these patterns have in common: The tip of the pendulum draws a curve which more or less slowly "fills" an area enclosed by a specific envelope, intermediately exhibiting seemingly regular patterns which inevitably eventually vanish.



Questions



  1. Can the envelope be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?


  2. Can the positions of the two inner cusps which can be seen clearly for $p_1=0$, $p_2=2,3$ be given in closed form, depending only on the two parameters $p_1^0, p_2^0$?



[The envelope for $p_1^0=0, p_2^0 = 1,2,dots$ looks like a canoe whose bow and stern bend to each other, eventually amalgamating. Can anyone guess what's the explicit formula for this shape?]







ordinary-differential-equations mathematical-physics classical-mechanics ergodic-theory chaos-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









Andrews

1,2641320




1,2641320










asked Oct 25 '18 at 13:14









Hans StrickerHans Stricker

6,47943994




6,47943994











  • $begingroup$
    Upvoted just for the animations - how did you do them? Also, the paths have cusps at the ends so they are not smooth everywhere.
    $endgroup$
    – marty cohen
    Oct 25 '18 at 13:34






  • 1




    $begingroup$
    I should give credit to travisdoesmath who uses d3.js and Runge-Kutta.
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 13:39







  • 1




    $begingroup$
    The phase space $(phi_1,phi_2,p_1,p_2)$ is 4-dimensional, and the total energy and the angular momentum about the fixed pivot are conserved over time. These two constraints determine a 2-dimensional slice of the phase space for any given set of initial conditions. Presumably, the projection of this abstract 2-dimensional set of states to physical space gives the desired envelope.
    $endgroup$
    – Rahul
    Oct 25 '18 at 13:45

















  • $begingroup$
    Upvoted just for the animations - how did you do them? Also, the paths have cusps at the ends so they are not smooth everywhere.
    $endgroup$
    – marty cohen
    Oct 25 '18 at 13:34






  • 1




    $begingroup$
    I should give credit to travisdoesmath who uses d3.js and Runge-Kutta.
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 13:39







  • 1




    $begingroup$
    The phase space $(phi_1,phi_2,p_1,p_2)$ is 4-dimensional, and the total energy and the angular momentum about the fixed pivot are conserved over time. These two constraints determine a 2-dimensional slice of the phase space for any given set of initial conditions. Presumably, the projection of this abstract 2-dimensional set of states to physical space gives the desired envelope.
    $endgroup$
    – Rahul
    Oct 25 '18 at 13:45
















$begingroup$
Upvoted just for the animations - how did you do them? Also, the paths have cusps at the ends so they are not smooth everywhere.
$endgroup$
– marty cohen
Oct 25 '18 at 13:34




$begingroup$
Upvoted just for the animations - how did you do them? Also, the paths have cusps at the ends so they are not smooth everywhere.
$endgroup$
– marty cohen
Oct 25 '18 at 13:34




1




1




$begingroup$
I should give credit to travisdoesmath who uses d3.js and Runge-Kutta.
$endgroup$
– Hans Stricker
Oct 25 '18 at 13:39





$begingroup$
I should give credit to travisdoesmath who uses d3.js and Runge-Kutta.
$endgroup$
– Hans Stricker
Oct 25 '18 at 13:39





1




1




$begingroup$
The phase space $(phi_1,phi_2,p_1,p_2)$ is 4-dimensional, and the total energy and the angular momentum about the fixed pivot are conserved over time. These two constraints determine a 2-dimensional slice of the phase space for any given set of initial conditions. Presumably, the projection of this abstract 2-dimensional set of states to physical space gives the desired envelope.
$endgroup$
– Rahul
Oct 25 '18 at 13:45





$begingroup$
The phase space $(phi_1,phi_2,p_1,p_2)$ is 4-dimensional, and the total energy and the angular momentum about the fixed pivot are conserved over time. These two constraints determine a 2-dimensional slice of the phase space for any given set of initial conditions. Presumably, the projection of this abstract 2-dimensional set of states to physical space gives the desired envelope.
$endgroup$
– Rahul
Oct 25 '18 at 13:45











1 Answer
1






active

oldest

votes


















3












$begingroup$

This is not a complete or rigorous answer, but it should point you in the right direction.



The double pendulum is a Hamiltonian system; energy is conserved. The most extreme points of the trajectory occur when both legs are at standstill ($p_1=p_2=0$) as all the energy is positional. The points where this is the case can be computed as parametrised curves as follows.



  1. The parameter of your curve is the angle of excitation of the first pendulum ($φ_1$).

  2. For each $φ_1$, you can calculate the two angles of the inner pendulum ($φ_2$) such that the potential energy of the system corresponds to your total energy.

  3. From both angles ($φ_1$ and $φ_2$) obtain the position of the inner pendulums end as the point of your curve.

I would conjecture that the outermost of the two curves is the envelope in case of a chaotic dynamics such as your last example.
If the double pendulum is ergodic, the above follows, as all points on the proposed envelope have the same energy and thus have to be visited.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 14:42










  • $begingroup$
    @HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
    $endgroup$
    – Wrzlprmft
    Oct 26 '18 at 6:03










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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

This is not a complete or rigorous answer, but it should point you in the right direction.



The double pendulum is a Hamiltonian system; energy is conserved. The most extreme points of the trajectory occur when both legs are at standstill ($p_1=p_2=0$) as all the energy is positional. The points where this is the case can be computed as parametrised curves as follows.



  1. The parameter of your curve is the angle of excitation of the first pendulum ($φ_1$).

  2. For each $φ_1$, you can calculate the two angles of the inner pendulum ($φ_2$) such that the potential energy of the system corresponds to your total energy.

  3. From both angles ($φ_1$ and $φ_2$) obtain the position of the inner pendulums end as the point of your curve.

I would conjecture that the outermost of the two curves is the envelope in case of a chaotic dynamics such as your last example.
If the double pendulum is ergodic, the above follows, as all points on the proposed envelope have the same energy and thus have to be visited.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 14:42










  • $begingroup$
    @HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
    $endgroup$
    – Wrzlprmft
    Oct 26 '18 at 6:03















3












$begingroup$

This is not a complete or rigorous answer, but it should point you in the right direction.



The double pendulum is a Hamiltonian system; energy is conserved. The most extreme points of the trajectory occur when both legs are at standstill ($p_1=p_2=0$) as all the energy is positional. The points where this is the case can be computed as parametrised curves as follows.



  1. The parameter of your curve is the angle of excitation of the first pendulum ($φ_1$).

  2. For each $φ_1$, you can calculate the two angles of the inner pendulum ($φ_2$) such that the potential energy of the system corresponds to your total energy.

  3. From both angles ($φ_1$ and $φ_2$) obtain the position of the inner pendulums end as the point of your curve.

I would conjecture that the outermost of the two curves is the envelope in case of a chaotic dynamics such as your last example.
If the double pendulum is ergodic, the above follows, as all points on the proposed envelope have the same energy and thus have to be visited.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 14:42










  • $begingroup$
    @HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
    $endgroup$
    – Wrzlprmft
    Oct 26 '18 at 6:03













3












3








3





$begingroup$

This is not a complete or rigorous answer, but it should point you in the right direction.



The double pendulum is a Hamiltonian system; energy is conserved. The most extreme points of the trajectory occur when both legs are at standstill ($p_1=p_2=0$) as all the energy is positional. The points where this is the case can be computed as parametrised curves as follows.



  1. The parameter of your curve is the angle of excitation of the first pendulum ($φ_1$).

  2. For each $φ_1$, you can calculate the two angles of the inner pendulum ($φ_2$) such that the potential energy of the system corresponds to your total energy.

  3. From both angles ($φ_1$ and $φ_2$) obtain the position of the inner pendulums end as the point of your curve.

I would conjecture that the outermost of the two curves is the envelope in case of a chaotic dynamics such as your last example.
If the double pendulum is ergodic, the above follows, as all points on the proposed envelope have the same energy and thus have to be visited.






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$endgroup$



This is not a complete or rigorous answer, but it should point you in the right direction.



The double pendulum is a Hamiltonian system; energy is conserved. The most extreme points of the trajectory occur when both legs are at standstill ($p_1=p_2=0$) as all the energy is positional. The points where this is the case can be computed as parametrised curves as follows.



  1. The parameter of your curve is the angle of excitation of the first pendulum ($φ_1$).

  2. For each $φ_1$, you can calculate the two angles of the inner pendulum ($φ_2$) such that the potential energy of the system corresponds to your total energy.

  3. From both angles ($φ_1$ and $φ_2$) obtain the position of the inner pendulums end as the point of your curve.

I would conjecture that the outermost of the two curves is the envelope in case of a chaotic dynamics such as your last example.
If the double pendulum is ergodic, the above follows, as all points on the proposed envelope have the same energy and thus have to be visited.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Oct 25 '18 at 13:57









WrzlprmftWrzlprmft

3,14111335




3,14111335











  • $begingroup$
    Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 14:42










  • $begingroup$
    @HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
    $endgroup$
    – Wrzlprmft
    Oct 26 '18 at 6:03
















  • $begingroup$
    Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
    $endgroup$
    – Hans Stricker
    Oct 25 '18 at 14:42










  • $begingroup$
    @HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
    $endgroup$
    – Wrzlprmft
    Oct 26 '18 at 6:03















$begingroup$
Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
$endgroup$
– Hans Stricker
Oct 25 '18 at 14:42




$begingroup$
Thanks! Do you think this approach will eventually give rise to a closed formula for the envelope with $p_1^0$ and $p_2^0$ as parameters?
$endgroup$
– Hans Stricker
Oct 25 '18 at 14:42












$begingroup$
@HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
$endgroup$
– Wrzlprmft
Oct 26 '18 at 6:03




$begingroup$
@HansStricker: I consider it likely. It certainly can’t hurt to calculate the proposed curve and compare it with your simulation results.
$endgroup$
– Wrzlprmft
Oct 26 '18 at 6:03

















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