solving a system of 2nd order differential equations with 3 variables
I'm creating a simulator for spacecraft that models orbital mechanics. It's simple enough to find the magnitude and direction of gravitational force, but I want to be able to time accelerate without losing accuracy like in Kerbal Space Program, so I need direct functions of time rather than calculating frame-by-frame.
So, ultimately, I need to solve this system:
$$
frac{mathrm{d}^2x}{mathrm{d}t^2}=-frac{Gmx}{(x^2+y^2)^frac{3}{2}}
$$
$$
frac{mathrm{d}^2y}{mathrm{d}t^2}=-frac{Gmy}{(x^2+y^2)^frac{3}{2}}
$$
where $G$ and $m$ are constants.
From these two equations, I need to find x and y as functions of t (there will obviously be some initial values to plug in). I've taken multivariable calc and diff eq., but I don't remember having done anything quite like this. Any suggestions on how to approach this, or equations of this form in general, for that matter? I could do this easily if it was only one dimension.
And maybe there's an better way to do this that uses conics instead of calculus, but that's more of a programming question.
It has been half a year since I've had a calculus course, so it is quite possible I've forgotten something. I have been unable to find any help on the internet, however, though I know I'm not the first one to do this.
differential-equations multivariable-calculus systems-of-equations physics
asked Nov 26 at 1:33
Nathanael Vetters
111
add a comment |
I'm creating a simulator for spacecraft that models orbital mechanics. It's simple enough to find the magnitude and direction of gravitational force, but I want to be able to time accelerate without losing accuracy like in Kerbal Space Program, so I need direct functions of time rather than calculating frame-by-frame.
So, ultimately, I need to solve this system:
$$
frac{mathrm{d}^2x}{mathrm{d}t^2}=-frac{Gmx}{(x^2+y^2)^frac{3}{2}}
$$
$$
frac{mathrm{d}^2y}{mathrm{d}t^2}=-frac{Gmy}{(x^2+y^2)^frac{3}{2}}
$$
where $G$ and $m$ are constants.
From these two equations, I need to find x and y as functions of t (there will obviously be some initial values to plug in). I've taken multivariable calc and diff eq., but I don't remember having done anything quite like this. Any suggestions on how to approach this, or equations of this form in general, for that matter? I could do this easily if it was only one dimension.
And maybe there's an better way to do this that uses conics instead of calculus, but that's more of a programming question.
It has been half a year since I've had a calculus course, so it is quite possible I've forgotten something. I have been unable to find any help on the internet, however, though I know I'm not the first one to do this.
differential-equations multivariable-calculus systems-of-equations physics
asked Nov 26 at 1:33
Nathanael Vetters
111
1
It's common to use Verlet integration or another symplectic integrator for Newton's equations.
– K B Dave
Nov 26 at 1:47
This problem can be better solved in polar coordinates. You can easily see that the angular momentum is conserved, so all you need to solve is the radial equation.
– Andrei
Nov 29 at 18:33
add a comment |
I'm creating a simulator for spacecraft that models orbital mechanics. It's simple enough to find the magnitude and direction of gravitational force, but I want to be able to time accelerate without losing accuracy like in Kerbal Space Program, so I need direct functions of time rather than calculating frame-by-frame.
So, ultimately, I need to solve this system:
$$
frac{mathrm{d}^2x}{mathrm{d}t^2}=-frac{Gmx}{(x^2+y^2)^frac{3}{2}}
$$
$$
frac{mathrm{d}^2y}{mathrm{d}t^2}=-frac{Gmy}{(x^2+y^2)^frac{3}{2}}
$$
where $G$ and $m$ are constants.
From these two equations, I need to find x and y as functions of t (there will obviously be some initial values to plug in). I've taken multivariable calc and diff eq., but I don't remember having done anything quite like this. Any suggestions on how to approach this, or equations of this form in general, for that matter? I could do this easily if it was only one dimension.
And maybe there's an better way to do this that uses conics instead of calculus, but that's more of a programming question.
It has been half a year since I've had a calculus course, so it is quite possible I've forgotten something. I have been unable to find any help on the internet, however, though I know I'm not the first one to do this.
differential-equations multivariable-calculus systems-of-equations physics
asked Nov 26 at 1:33
Nathanael Vetters
111
I'm creating a simulator for spacecraft that models orbital mechanics. It's simple enough to find the magnitude and direction of gravitational force, but I want to be able to time accelerate without losing accuracy like in Kerbal Space Program, so I need direct functions of time rather than calculating frame-by-frame.
So, ultimately, I need to solve this system:
$$
frac{mathrm{d}^2x}{mathrm{d}t^2}=-frac{Gmx}{(x^2+y^2)^frac{3}{2}}
$$
$$
frac{mathrm{d}^2y}{mathrm{d}t^2}=-frac{Gmy}{(x^2+y^2)^frac{3}{2}}
$$
where $G$ and $m$ are constants.
From these two equations, I need to find x and y as functions of t (there will obviously be some initial values to plug in). I've taken multivariable calc and diff eq., but I don't remember having done anything quite like this. Any suggestions on how to approach this, or equations of this form in general, for that matter? I could do this easily if it was only one dimension.
And maybe there's an better way to do this that uses conics instead of calculus, but that's more of a programming question.
It has been half a year since I've had a calculus course, so it is quite possible I've forgotten something. I have been unable to find any help on the internet, however, though I know I'm not the first one to do this.
differential-equations multivariable-calculus systems-of-equations physics
differential-equations multivariable-calculus systems-of-equations physics
asked Nov 26 at 1:33
Nathanael Vetters
111
asked Nov 26 at 1:33
Nathanael Vetters
111
asked Nov 26 at 1:33
Nathanael Vetters
111
asked Nov 26 at 1:33
Nathanael Vetters
111
asked Nov 26 at 1:33
Nathanael Vetters
111
111
1
It's common to use Verlet integration or another symplectic integrator for Newton's equations.
– K B Dave
Nov 26 at 1:47
This problem can be better solved in polar coordinates. You can easily see that the angular momentum is conserved, so all you need to solve is the radial equation.
– Andrei
Nov 29 at 18:33
add a comment |
1
It's common to use Verlet integration or another symplectic integrator for Newton's equations.
– K B Dave
Nov 26 at 1:47
This problem can be better solved in polar coordinates. You can easily see that the angular momentum is conserved, so all you need to solve is the radial equation.
– Andrei
Nov 29 at 18:33
1
1
It's common to use Verlet integration or another symplectic integrator for Newton's equations.
– K B Dave
Nov 26 at 1:47
It's common to use Verlet integration or another symplectic integrator for Newton's equations.
– K B Dave
Nov 26 at 1:47
This problem can be better solved in polar coordinates. You can easily see that the angular momentum is conserved, so all you need to solve is the radial equation.
– Andrei
Nov 29 at 18:33
This problem can be better solved in polar coordinates. You can easily see that the angular momentum is conserved, so all you need to solve is the radial equation.
– Andrei
Nov 29 at 18:33
add a comment |
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1
It's common to use Verlet integration or another symplectic integrator for Newton's equations.
– K B Dave
Nov 26 at 1:47
This problem can be better solved in polar coordinates. You can easily see that the angular momentum is conserved, so all you need to solve is the radial equation.
– Andrei
Nov 29 at 18:33