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Is there a general way of solving the Maxwell equations?
Special Relativity: Transforming Maxwell's equationsSolving 2nd-order ODEsDifference between action-at-a-distance and a field according to Maxwell?Skyrmions - Solving a second-order nonlinear ODE with a singularity on x=0Maxwell's equations for electromagnetic waveGeneral Maxwell equations (moving media)A simple proof covariance of Maxwell equationsElectromagnetic wave equation: can we ignore the constant of integration?
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Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics of the problem.
Let's say you want to design a series of steps that you can handle to a student and he will be able to obtain E and B for any problem. The instructions don't have to be simple or understandable to someone without proper background but, is it possible?
electromagnetism maxwell-equations differential-equations
asked 8 hours ago
ChegonChegon
1206 bronze badges
$endgroup$
add a comment
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$begingroup$
Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics of the problem.
Let's say you want to design a series of steps that you can handle to a student and he will be able to obtain E and B for any problem. The instructions don't have to be simple or understandable to someone without proper background but, is it possible?
electromagnetism maxwell-equations differential-equations
asked 8 hours ago
ChegonChegon
1206 bronze badges
$endgroup$
add a comment
|
$begingroup$
Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics of the problem.
Let's say you want to design a series of steps that you can handle to a student and he will be able to obtain E and B for any problem. The instructions don't have to be simple or understandable to someone without proper background but, is it possible?
electromagnetism maxwell-equations differential-equations
asked 8 hours ago
ChegonChegon
1206 bronze badges
$endgroup$
Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics of the problem.
Let's say you want to design a series of steps that you can handle to a student and he will be able to obtain E and B for any problem. The instructions don't have to be simple or understandable to someone without proper background but, is it possible?
electromagnetism maxwell-equations differential-equations
electromagnetism maxwell-equations differential-equations
asked 8 hours ago
ChegonChegon
1206 bronze badges
asked 8 hours ago
ChegonChegon
1206 bronze badges
asked 8 hours ago
ChegonChegon
1206 bronze badges
asked 8 hours ago
ChegonChegon
1206 bronze badges
asked 8 hours ago
ChegonChegon
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1206 bronze badges
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4 Answers
4
active
oldest
votes
$begingroup$
Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).
The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
$endgroup$
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
$endgroup$
add a comment
|
$begingroup$
Exact solution of Maxwells equations does not exist (for whole space and time). Maxwells equations describe behaviour of the fields localy (in a neighbourhood of a point) because to calculate any field in a point, all you need is the information about it's neighbourhood. If you place random objects that modify the $E$ field or $B$ field, you have to take those into account, but there is no way to embed them to those equations other that to say that those fields have to satisfy some additional conditions (i.e boundary conditions) on those objects.
Analytical solving often assumes special cases (e.g free space, $rho=0, J=0$) to simplify the equations. Numerically, you can allways find better and better approximation of the distribution of fields in space by using smaller steps in space and time during discretization. Finate difference method and it's variants are often used for solving differential equations in general.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
No, there is not. The three particle problem is chaotic and does not have an analytic solution. They are a subset of Maxwell equations. https://en.wikipedia.org/wiki/Three-body_problem
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
$endgroup$
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).
The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
$endgroup$
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).
The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
$endgroup$
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).
The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
$endgroup$
Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).
The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
edited 7 hours ago
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
answered 7 hours ago
G. SmithG. Smith
23.3k2 gold badges41 silver badges76 bronze badges
23.3k2 gold badges41 silver badges76 bronze badges
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
$begingroup$
This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
$endgroup$
add a comment
|
$begingroup$
You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
$endgroup$
add a comment
|
$begingroup$
You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
$endgroup$
You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
answered 3 hours ago
tparkertparker
25.9k1 gold badge55 silver badges137 bronze badges
25.9k1 gold badge55 silver badges137 bronze badges
add a comment
|
add a comment
|
$begingroup$
Exact solution of Maxwells equations does not exist (for whole space and time). Maxwells equations describe behaviour of the fields localy (in a neighbourhood of a point) because to calculate any field in a point, all you need is the information about it's neighbourhood. If you place random objects that modify the $E$ field or $B$ field, you have to take those into account, but there is no way to embed them to those equations other that to say that those fields have to satisfy some additional conditions (i.e boundary conditions) on those objects.
Analytical solving often assumes special cases (e.g free space, $rho=0, J=0$) to simplify the equations. Numerically, you can allways find better and better approximation of the distribution of fields in space by using smaller steps in space and time during discretization. Finate difference method and it's variants are often used for solving differential equations in general.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
Exact solution of Maxwells equations does not exist (for whole space and time). Maxwells equations describe behaviour of the fields localy (in a neighbourhood of a point) because to calculate any field in a point, all you need is the information about it's neighbourhood. If you place random objects that modify the $E$ field or $B$ field, you have to take those into account, but there is no way to embed them to those equations other that to say that those fields have to satisfy some additional conditions (i.e boundary conditions) on those objects.
Analytical solving often assumes special cases (e.g free space, $rho=0, J=0$) to simplify the equations. Numerically, you can allways find better and better approximation of the distribution of fields in space by using smaller steps in space and time during discretization. Finate difference method and it's variants are often used for solving differential equations in general.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
Exact solution of Maxwells equations does not exist (for whole space and time). Maxwells equations describe behaviour of the fields localy (in a neighbourhood of a point) because to calculate any field in a point, all you need is the information about it's neighbourhood. If you place random objects that modify the $E$ field or $B$ field, you have to take those into account, but there is no way to embed them to those equations other that to say that those fields have to satisfy some additional conditions (i.e boundary conditions) on those objects.
Analytical solving often assumes special cases (e.g free space, $rho=0, J=0$) to simplify the equations. Numerically, you can allways find better and better approximation of the distribution of fields in space by using smaller steps in space and time during discretization. Finate difference method and it's variants are often used for solving differential equations in general.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Exact solution of Maxwells equations does not exist (for whole space and time). Maxwells equations describe behaviour of the fields localy (in a neighbourhood of a point) because to calculate any field in a point, all you need is the information about it's neighbourhood. If you place random objects that modify the $E$ field or $B$ field, you have to take those into account, but there is no way to embed them to those equations other that to say that those fields have to satisfy some additional conditions (i.e boundary conditions) on those objects.
Analytical solving often assumes special cases (e.g free space, $rho=0, J=0$) to simplify the equations. Numerically, you can allways find better and better approximation of the distribution of fields in space by using smaller steps in space and time during discretization. Finate difference method and it's variants are often used for solving differential equations in general.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
answered 7 hours ago
Luka MandićLuka Mandić
32 bronze badges
32 bronze badges
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Luka Mandić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
$begingroup$
Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
No, there is not. The three particle problem is chaotic and does not have an analytic solution. They are a subset of Maxwell equations. https://en.wikipedia.org/wiki/Three-body_problem
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
$endgroup$
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
No, there is not. The three particle problem is chaotic and does not have an analytic solution. They are a subset of Maxwell equations. https://en.wikipedia.org/wiki/Three-body_problem
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
$endgroup$
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
No, there is not. The three particle problem is chaotic and does not have an analytic solution. They are a subset of Maxwell equations. https://en.wikipedia.org/wiki/Three-body_problem
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
$endgroup$
No, there is not. The three particle problem is chaotic and does not have an analytic solution. They are a subset of Maxwell equations. https://en.wikipedia.org/wiki/Three-body_problem
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
answered 8 hours ago
Wolphram jonnyWolphram jonny
11.4k3 gold badges27 silver badges55 bronze badges
11.4k3 gold badges27 silver badges55 bronze badges
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
add a comment
|
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
I agree, but what about a numeric solution? Someone found one to make the gif in that site.
$endgroup$
– Chegon
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
oh, you can always find numerical solutions to any level of precision you want (in principle, because the computer needed for your needs might need more particles the ones in the universe)
$endgroup$
– Wolphram jonny
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
thanks but my question is how to do it with a fixed set of steps. LIke: "always use the Frobenius method" or "Always do this and then this and later this"
$endgroup$
– Chegon
8 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
yes, it exists.
$endgroup$
– Wolphram jonny
7 hours ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
$begingroup$
In electrodynamics (and GR too) even two-body problem is not analytically solvable due to radiation.
$endgroup$
– Ruslan
1 hour ago
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