Two bodies under mutual gravitational attraction as system of first order ODEsThe Runge - Kutta method and two-body problemReplace the second-order ODE as two first-order ODEs and use Runge-Kutta method to solveSolve system of first order ODEs when values are a matrixSolving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLABHow to find the period of motion in MATLAB?Reducing a 2nd order system of ODEs to a 1st order systemConverting two nonlinear DEs into a system of four first order ODEs4th order Runge-Kutta method to solve a system of 8 coupled ODEsSolving a system of ODE's with RK4 correctlyTwo couple equation first order Runge Kutta Matlab troubles?How to change the system in 4x4 system of first-order equations?
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Two bodies under mutual gravitational attraction as system of first order ODEs
The Runge - Kutta method and two-body problemReplace the second-order ODE as two first-order ODEs and use Runge-Kutta method to solveSolve system of first order ODEs when values are a matrixSolving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLABHow to find the period of motion in MATLAB?Reducing a 2nd order system of ODEs to a 1st order systemConverting two nonlinear DEs into a system of four first order ODEs4th order Runge-Kutta method to solve a system of 8 coupled ODEsSolving a system of ODE's with RK4 correctlyTwo couple equation first order Runge Kutta Matlab troubles?How to change the system in 4x4 system of first-order equations?
$begingroup$
I'm trying to model the motion of one mass $m$ about another larger mass $M$.
I have the following relation:
beginalign
ddot x_0 &= -GMx_0/r^3, \
ddot x_1 &= -GMx_1/r^3,
endalign
where $G$ is the gravitational constant and $r = (x_0^2 + x_1^2)^1/2 = ||x||$
is the distance of the orbiting body from the center of mass of the two bodies.
Now, to solve computationally, I want to express the above relations as a system of first order ODEs, so that I can use computational methods (e.g. Matlab or Scipy's integrate.solve_ivp) to find a solution.
However I'm not sure how to set this up when we have two equations that depend on each other. I know the process for equations with a single variable is to re-write the system by defining something like $y_1(t) = x_0'$ and $y_2(t) = x_0'' = -GMx_o/r^3''$.
How can I put this into a system of first order ODEs that can be solved via Runge-Kutta?
ordinary-differential-equations differential
asked Mar 15 at 6:26
Jake CustyJake Custy
11
$endgroup$
add a comment |
$begingroup$
I'm trying to model the motion of one mass $m$ about another larger mass $M$.
I have the following relation:
beginalign
ddot x_0 &= -GMx_0/r^3, \
ddot x_1 &= -GMx_1/r^3,
endalign
where $G$ is the gravitational constant and $r = (x_0^2 + x_1^2)^1/2 = ||x||$
is the distance of the orbiting body from the center of mass of the two bodies.
Now, to solve computationally, I want to express the above relations as a system of first order ODEs, so that I can use computational methods (e.g. Matlab or Scipy's integrate.solve_ivp) to find a solution.
However I'm not sure how to set this up when we have two equations that depend on each other. I know the process for equations with a single variable is to re-write the system by defining something like $y_1(t) = x_0'$ and $y_2(t) = x_0'' = -GMx_o/r^3''$.
How can I put this into a system of first order ODEs that can be solved via Runge-Kutta?
ordinary-differential-equations differential
asked Mar 15 at 6:26
Jake CustyJake Custy
11
$endgroup$
add a comment |
$begingroup$
I'm trying to model the motion of one mass $m$ about another larger mass $M$.
I have the following relation:
beginalign
ddot x_0 &= -GMx_0/r^3, \
ddot x_1 &= -GMx_1/r^3,
endalign
where $G$ is the gravitational constant and $r = (x_0^2 + x_1^2)^1/2 = ||x||$
is the distance of the orbiting body from the center of mass of the two bodies.
Now, to solve computationally, I want to express the above relations as a system of first order ODEs, so that I can use computational methods (e.g. Matlab or Scipy's integrate.solve_ivp) to find a solution.
However I'm not sure how to set this up when we have two equations that depend on each other. I know the process for equations with a single variable is to re-write the system by defining something like $y_1(t) = x_0'$ and $y_2(t) = x_0'' = -GMx_o/r^3''$.
How can I put this into a system of first order ODEs that can be solved via Runge-Kutta?
ordinary-differential-equations differential
asked Mar 15 at 6:26
Jake CustyJake Custy
11
$endgroup$
I'm trying to model the motion of one mass $m$ about another larger mass $M$.
I have the following relation:
beginalign
ddot x_0 &= -GMx_0/r^3, \
ddot x_1 &= -GMx_1/r^3,
endalign
where $G$ is the gravitational constant and $r = (x_0^2 + x_1^2)^1/2 = ||x||$
is the distance of the orbiting body from the center of mass of the two bodies.
Now, to solve computationally, I want to express the above relations as a system of first order ODEs, so that I can use computational methods (e.g. Matlab or Scipy's integrate.solve_ivp) to find a solution.
However I'm not sure how to set this up when we have two equations that depend on each other. I know the process for equations with a single variable is to re-write the system by defining something like $y_1(t) = x_0'$ and $y_2(t) = x_0'' = -GMx_o/r^3''$.
How can I put this into a system of first order ODEs that can be solved via Runge-Kutta?
ordinary-differential-equations differential
ordinary-differential-equations differential
asked Mar 15 at 6:26
Jake CustyJake Custy
11
asked Mar 15 at 6:26
Jake CustyJake Custy
11
asked Mar 15 at 6:26
Jake CustyJake Custy
11
asked Mar 15 at 6:26
Jake CustyJake Custy
11
asked Mar 15 at 6:26
Jake CustyJake Custy
11
11
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This has been asked before.
The two-body problem can be solved almost completely in closed form. The equations of motion can be integrated to obtain the equations of the orbits, which are always conic sections (Kepler's first law generalised to include the unbounded cases of the parabola and hyperbola). However, to express the motions of the bodies as function of time, it's necessary to solve Kepler's equation, for which there is no known simple solution in closed form.
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
This has been asked before.
The two-body problem can be solved almost completely in closed form. The equations of motion can be integrated to obtain the equations of the orbits, which are always conic sections (Kepler's first law generalised to include the unbounded cases of the parabola and hyperbola). However, to express the motions of the bodies as function of time, it's necessary to solve Kepler's equation, for which there is no known simple solution in closed form.
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
$endgroup$
add a comment |
$begingroup$
This has been asked before.
The two-body problem can be solved almost completely in closed form. The equations of motion can be integrated to obtain the equations of the orbits, which are always conic sections (Kepler's first law generalised to include the unbounded cases of the parabola and hyperbola). However, to express the motions of the bodies as function of time, it's necessary to solve Kepler's equation, for which there is no known simple solution in closed form.
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
$endgroup$
add a comment |
$begingroup$
This has been asked before.
The two-body problem can be solved almost completely in closed form. The equations of motion can be integrated to obtain the equations of the orbits, which are always conic sections (Kepler's first law generalised to include the unbounded cases of the parabola and hyperbola). However, to express the motions of the bodies as function of time, it's necessary to solve Kepler's equation, for which there is no known simple solution in closed form.
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
$endgroup$
This has been asked before.
The two-body problem can be solved almost completely in closed form. The equations of motion can be integrated to obtain the equations of the orbits, which are always conic sections (Kepler's first law generalised to include the unbounded cases of the parabola and hyperbola). However, to express the motions of the bodies as function of time, it's necessary to solve Kepler's equation, for which there is no known simple solution in closed form.
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
answered Mar 15 at 21:45
lonza leggieralonza leggiera
1,14128
1,14128
add a comment |
add a comment |
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