Wikipedia:Reference desk/Archives/Mathematics/2012 February 16

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February 16

drawbacks to riemann & riemann stieltjes integral

why we need to introduce a genralisation of riemann integral ? also what are the drawbacks to RS integration ? and what we introduce to overcome that drawbacks? — Preceding unsigned comment added by 14.139.120.178 (talk) 09:09, 16 February 2012 (UTC)

The main drawback of both the Riemann and Riemann-Stieltjes integrals is the lack of easy theorems that allow the interchange of limits with integration. The Lebesgue integral has better properties from this perspective, in part because it is able to integrate more functions (possible limit functions of sequences). The Riemann-Stieltjes integral is still useful in many situations. In probability theory, for instance, sometimes it is necessary to take a Stieltjes integral with respect to a CDF (a PDF may not exist). Sławomir Biały (talk) 10:53, 16 February 2012 (UTC)

Differential equations describing Euler's three-body problem

In Grapher on Mac OS X, explicit two-dimensional differential equations take the form of ${d \over dt}{\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}{\mbox{expression1}}\\{\mbox{expression2}}\end{bmatrix}},{\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}{\mbox{value1}}\\{\mbox{value2}}\end{bmatrix}},t={\mbox{from value}}...{\mbox{to value}}$. What would be the general form describing a restricted three-body system containing $m_{1}$, $m_{2}$, $m_{3}$, $x_{m1}$, $y_{m1}$, $x_{m2}$, $y_{m2}$, $x_{m3}$, $y_{m3}$, $v_{x}$, and $v_{y}$, where

• $m_{1}$ and $m_{2}$ are the masses of two objects fixed in space,
• $m_{3}$ is the mass of the movable object,
• $x_{m1}$, $y_{m1}$, $x_{m2}$, and $y_{m2}$ are the $x$ and $y$ coordinates of their respective fixed objects,
• $x_{m3}$ and $y_{m3}$ are the initial $x$ and $y$ coordinates of the movable object,
• and $v_{x}$ and $v_{y}$ are the component vectors of the movable object's initial velocity?

--Melab±1 ☎ 20:24, 16 February 2012 (UTC)

The equations of motion are second order differential equations (i.e. they involve the second derivatives of x and y with respect to time), so if Grapher can only plot first order ODEs (as your description suggests) then you have hit a fundamental obstacle. Gandalf61 (talk) 09:27, 17 February 2012 (UTC)

They are second order as well. --Melab±1 ☎ 21:33, 17 February 2012 (UTC)

Not sure what you mean by "They are second order as well", but here is an outline of how you find the equations of motion. First you write down expressions for the forces on the movable object due to the gravitational attraction between it and the other two objects - let's call these forces F¹ and F². Note that these are vectors, and the magnitude of each one will be a function of the square of the distance between the movable object and each of the other objects i.e.

$r_{13}^{2}=(x-x_{m1})^{2}+(y-y_{m1})^{2}$

$r_{23}^{2}=(x-x_{m2})^{2}+(y-y_{m2})^{2}$

Then you resolve these forces into x and y components, so you have F¹_x, F¹_y, F²_x and F²_y. Then your equation of motion in the x direction is

${\frac {d^{2}x}{dt^{2}}}={\frac {F_{x}^{1}(x,y)+F_{x}^{2}(x,y)}{m_{3}}}$

and you have a similar equation of motion in the y direction, involving the y components of the force instead of the x components. I am not going to write out all these expressions for you because that is very tedious, but it is conceptually straightforward once you understand the underlying physics.

As you see, this is not in the format that you requested because it is a pair of second order ODEs, and your format was a pair of first order ODEs, as I pointed out above. Gandalf61 (talk) 10:33, 18 February 2012 (UTC)