System of equations: x ˙ = x − y − x ( x 2 + y 2 ) + x y x 2 + y 2 y ˙ = x + y − y ( x 2 + y 2 ) − x 2 x 2 + y 2 How can I get this in polar coordinates? I know that r 2 = x 2 + y 2 , but how can I find r ˙ or θ ˙ ?

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System of equations:      x    ˙    =  x  −  y  −  x  (      x    2    +      y    2    )  +            x      y                      x        2            +              y        2                    y    ˙    =  x  +  y  −  y  (      x    2    +      y    2    )  −            x      2                      x        2            +              y        2            How can I get this in polar coordinates? I know that       r    2    =      x    2    +      y    2  , but how can I find       r    ˙   or       θ    ˙  ?

szilincsifs · Accepted Answer

Setting  x  =  r  cos  ⁡  θ  ,     y  =  r  sin  ⁡  θ  ,we have                                    r          ˙                cos        ⁡        θ        −        r                        θ          ˙                sin        ⁡        θ                            =                                    x          ˙                =        r        (        cos        ⁡        θ        −        sin        ⁡        θ        )        −                  r          3                cos        ⁡        θ        +        r        cos        ⁡        θ        sin        ⁡        θ        ,                                            r          ˙                sin        ⁡        θ        +        r                        θ          ˙                cos        ⁡        θ                            =                                    y          ˙                =        r        (        cos        ⁡        θ        +        sin        ⁡        θ        )        −                  r          3                sin        ⁡        θ        −        r                  cos          2                ⁡        θ        .            It follows that      r    ˙    =  r  −      r    3    ,        θ    ˙    =  1  −  cos  ⁡  θ  .

sag2y8s · Accepted Answer

When we use polar co-ordinates, we let   x  =  r  cos  ⁡  θ and   y  =  r  sin  ⁡  θ. We can differentiate each of these equations to find       x    ˙   and       y    ˙   (and this will help you solve your question). If you want to find       r    ˙   or       θ    ˙  , just differentiate the equations   tan  ⁡  θ  =  y  /  x and       r      2    =      x      2    +      y      2   implicitly.For example,      x    ˙    =      d          d      t        (  r  cos  ⁡  θ  )  =            d      r              d      t        cos  ⁡  θ  −  r  sin  ⁡  θ            d      θ              d      t        =      r    ˙    cos  ⁡  θ  −  r      θ    ˙    sin  ⁡  θ

System of equations: <mover> x ˙ </mover> = x −<!

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