Appell transformation

A transformation of the equations of plane motion of a material point under the action of a force that varies with the coordinates $ x, y $ of the point only. The Appell transformation is a homographic transformation of the coordinates $ x, y $ into the coordinates $ x _ {1} , y _ {1} $:

$$ x _ {1} = \frac{ax + by + c }{a ^ {\prime\prime} x + b ^ {\prime\prime} y + c ^ {\prime\prime} } ,\ \ y _ {1} = \frac{a ^ \prime x + b ^ \prime y + c ^ \prime }{a ^ {\prime\prime} x + b ^ {\prime\prime} y + c ^ {\prime\prime} } , $$

which involves the substitution of the time $ t _ {1} $ for $ t $ in accordance with the formula

$$ k dt _ {1} = \frac{dt}{( a ^ {\prime\prime} x + b ^ {\prime\prime} y +c ^ {\prime\prime} ) ^ {2} } . $$

As a result of the Appell transformation, the motion of a point $ ( x _ {1} , y _ {1} ) $, acted upon by a force $ F _ {1} ( x _ {1} , y _ {1} ) $ that depends only on the coordinates $ x _ {1} , y _ {1} $, corresponds to the motion of the point $ (x, y) $ acted upon by the force $ F(x, y) $. A transformation of the equations of motion of a point $ (x, y) $ with the aid of formulas of general form

$$ \tag{* } x _ {1} = \phi ( x , y ),\ y _ {1} = \psi ( x , y ),\ \ dt _ {1} = \lambda ( x , y ) dt , $$

yields the equations of motion of the point $ ( x _ {1} , y _ {1} ) $ under the effect of a force that depends only on the coordinates $ x _ {1} , y _ {1} $ only if (*) is a homographic transformation. The Appell transformation is used to solve Bertrand's problem on the determination of the forces under which the trajectories of motion of material points become conic sections. The Appell transformation is named after P.E. Appell.

References

[1]	P.E. Appell, "De l'homographie en mécanique" Amer. J. Math. , 12 (1890) pp. 103–114
[2]	P.E. Appell, "Sur les lois de forces centrales faisant décrire à leur point d'application une conique quelles que soient les conditions initiales" Amer. J. Math. , 13 (1891) pp. 153–158 Zbl 22.0900.02