In a general three-body system, the motion equation of each celestial body under the gravity of the other two celestial bodies can be expressed as three second-order ordinary differential equations or six first-order ordinary differential equations.
Therefore, the motion equation of a general three-body is an 18th-order equation, and 18 integrals must be obtained to get a complete solution. But only the 16 integral of the three-body can be obtained, so it is still far from solving the three-body problem.
Extended data:
Three-body problem is a basic mechanical model in celestial mechanics.
It refers to the motion law of three celestial bodies whose mass, initial position and initial velocity are arbitrary and can be regarded as particles under the action of universal gravitation.
Methods for studying three-body problems can be roughly divided into three categories:
The first kind is analytical method, whose basic principle is to expand the coordinates and velocity of celestial bodies into approximate analytical expressions in the form of series of time or other small parameters, so as to discuss the changes of coordinates or orbital elements of celestial bodies with time;
The second kind is qualitative method, which uses the qualitative theory of differential equations to study the macro-laws and global properties of three-body motion in a long time.
The third is the numerical method, which directly obtains the specific position and velocity of celestial bodies at a certain moment according to the calculation method of differential equations.
These three methods have their own advantages and disadvantages, and the exploration of new integrals and the improvement of various methods are very important topics in the study of three-body problems.
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