Differential equation: dv = v Cut Cut cos dθ - v Diameter sin dθ - v cut
This is a differential equation of the two-body problem, it can be said that one issue of a differential equation. Two-body problem can be simplified as a whole about the problem of quality, the problem known as one revolution over the issue, means a particle movement around the other "fixed" particle under gravity action.
"Fixed" indicates that the particle is a particle inertial system, the particle motion is not affected by gravity.
This differential equations variation of the linear velocity of the particle motion. Particle fixed polar coordinates establishing the origin, that is, the linear velocity of a moving particle tangential velocity, i.e. speed and polar radius ρ perpendicular direction.
Equation v cutting motion of a particle represents a tangential velocity, radial velocity v represents the diameter, i.e., the polar radius ρ velocity direction. v dv cut is cut derivative, dθ is the polar angle θ differential.
Principle equation: At time t, and particle motion is the position (ρ, θ), when passing dt, particle motion [theta] dθ changed, thereby resulting in the original and [rho] v is no longer orthogonal cut, v diameter deviate from the ρ direction, so that it leads to the cutting speed change,
V new = v tangential cut cos dθ - v diameter sin dθ,
dv = new tangential cutting v - v = v Cut Cut cos dθ - v Diameter sin dθ - v cut,
I.e. dv = v cutting cut cos dθ - v Diameter sin dθ - v-cut (1)
Because the attraction between the moving particle and the particle is always fixed in the ρ direction, the gravity does not directly affect v cut, therefore, v can be described by variation of cutting (1), (1) is the differential equation of Formula linear velocity.
In (1) equation, v-cut v-path is variable, but not v-path differential equations in the sense, can be seen as a constant, that the solution of this differential equation, v can be regarded as a constant diameter.
Say so, this equation can be regarded as a partial differential equation ......, but simply, or as an ordinary differential equation on it, the v constant diameter seen on the line.
Writing can be cut v-cut dv / dt, so you can time the introduction of differential dt, will not help solve the equation to see.
The writing may be cut v ρ * dθ / dt, dv is cut ρ * d (dθ / dt), a tangential direction tangential acceleration a = dv cut / dt = ρ * d²θ / dt².
Solution of the problem is two textbooks body premise conservation of angular momentum, the introduction of Kepler's Second Law, i.e., particle motion is elliptical trajectories, combined with gravity calculus mechanical energy conservation method obtains the equation of motion dimer.
It intended herein is not considered conservation of angular momentum, starting from the law of motion, to solve linear velocity variation, combined with gravity calculus mechanical energy conservation method obtains the equation of motion dimer.
Variation of the line speed should be a breakthrough in the two-body problem, of course, come out of the solution of the equation.