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Difference between revisions of "Two-body problem"

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and  $  m _ {2} $,  
 
and  $  m _ {2} $,  
 
respectively, moving in three-dimensional Euclidean space  $  E  ^ {3} $
 
respectively, moving in three-dimensional Euclidean space  $  E  ^ {3} $
when acted upon by the mutual Newton attracting forces. The problem is a special case of the  $  n $-
+
when acted upon by the mutual Newton attracting forces. The problem is a special case of the  $  n $-body problem, which may be described by a system of ordinary differential equations of order  $  6n $,  
body problem, which may be described by a system of ordinary differential equations of order  $  6n $,  
 
 
and has 10 independent integrals: 6 of motion of the centre of inertia, 3 of law of areas (equivalently, conservation of angular momentum) and 1 of energy conservation [[#References|[1]]]. The two-body problem also has three Laplace integrals (one of which is independent of the preceding ones) and is completely integrable [[#References|[2]]].
 
and has 10 independent integrals: 6 of motion of the centre of inertia, 3 of law of areas (equivalently, conservation of angular momentum) and 1 of energy conservation [[#References|[1]]]. The two-body problem also has three Laplace integrals (one of which is independent of the preceding ones) and is completely integrable [[#References|[2]]].
  
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$$ \tag{1 }
 
$$ \tag{1 }
\mu \dot{x} dot =  - f x r  ^ {-} 3 ,\  \mu \dot{y} dot =  - f y r  ^ {-} 3 ,
+
\mu \ddot{x}  =  - f x r  ^ {- 3} ,\  \mu \ddot{y}  =  - f y r  ^ {- 3} ,
 
$$
 
$$
  
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\frac{1}{2}
 
\frac{1}{2}
 
  \mu ( {\dot{x} } {}  ^ {2} + {\dot{y} }
 
  \mu ( {\dot{x} } {}  ^ {2} + {\dot{y} }
{}  ^ {2} ) - f r  ^ {-} 1 =  h \ roman (energy) ,
+
{}  ^ {2} ) - f r  ^ {- 1}  =  h \textrm{ (energy) } ,
 
$$
 
$$
  
 
$$  
 
$$  
\mu  ^ {2} c \dot{y} - \mu f x r  ^ {-} 1 =  \lambda _ {1} \  \textrm{ and } \ \  
+
\mu  ^ {2} c \dot{y} - \mu f x r  ^ {- 1}  =  \lambda _ {1} \  \textrm{ and } \ \  
\mu  ^ {2} c \dot{x} + \mu f y r  ^ {-} 1 =  - \lambda _ {2} \  roman (Laplace) ,
+
\mu  ^ {2} c \dot{x} + \mu f y r  ^ {- 1}  =  - \lambda _ {2} \  \textrm{ (Laplace) },
 
$$
 
$$
  
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i.e. the orbits of the relative position vector are conical sections with parameter  $  p = c  ^ {2} / \mu $,  
 
i.e. the orbits of the relative position vector are conical sections with parameter  $  p = c  ^ {2} / \mu $,  
 
major semi-axis  $  a = - \mu / ( 2h) $,  
 
major semi-axis  $  a = - \mu / ( 2h) $,  
eccentricity  $  e = \mu  ^ {-} 1 \sqrt {1 + 2hc  ^ {2} } $,  
+
eccentricity  $  e = \mu  ^ {- 1} \sqrt {1 + 2hc  ^ {2} } $,  
longitude of pericentre  $  \omega $(
+
longitude of pericentre  $  \omega $ (
 
$  \lambda _ {1} = \mu e  \cos  \omega $,  
 
$  \lambda _ {1} = \mu e  \cos  \omega $,  
 
$  \lambda _ {2} = \mu e  \sin  \omega $),  
 
$  \lambda _ {2} = \mu e  \sin  \omega $),  
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.L. Siegel,  "Vorlesungen über Himmelmechanik" , Springer  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.K. Abalakin,  et al.,  "Handbook of celestial mechanics and astrodynamics" , Moscow  (1971)  (In Russian)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  C.L. Siegel,  "Vorlesungen über Himmelmechanik" , Springer  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.K. Abalakin,  et al.,  "Handbook of celestial mechanics and astrodynamics" , Moscow  (1971)  (In Russian)</TD></TR>
====Comments====
+
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Poincaré,  "Les méthodes nouvelles de la mécanique céleste" , '''1–3''' , Gauthier-Villars  (1899)</TD></TR>
 
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.L. Siegel,  J. Moser,  "Lectures on celestial mechanics" , Springer  (1971)</TD></TR>
====References====
+
<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin  (1978)</TD></TR>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Poincaré,  "Les méthodes nouvelles de la mécanique céleste" , '''1–3''' , Gauthier-Villars  (1899)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.L. Siegel,  J. Moser,  "Lectures on celestial mechanics" , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin  (1978)</TD></TR></table>
+
</table>

Latest revision as of 16:44, 1 November 2023


A problem dealing with the motion of two material points $ P _ {1} $ and $ P _ {2} $ with masses $ m _ {1} $ and $ m _ {2} $, respectively, moving in three-dimensional Euclidean space $ E ^ {3} $ when acted upon by the mutual Newton attracting forces. The problem is a special case of the $ n $-body problem, which may be described by a system of ordinary differential equations of order $ 6n $, and has 10 independent integrals: 6 of motion of the centre of inertia, 3 of law of areas (equivalently, conservation of angular momentum) and 1 of energy conservation [1]. The two-body problem also has three Laplace integrals (one of which is independent of the preceding ones) and is completely integrable [2].

The integration of the two-body problem is more conveniently effected in a special system of coordinates, in which these integrals are employed. If the origin of the Cartesian coordinates $ x , y , z $ is placed at the centre of mass $ ( m _ {1} \vec{r} {} _ {1} + m _ {2} \vec{r} {} _ {2} ) / ( m _ {1} + m _ {2} ) $ and the axis $ z $ is directed along the relative angular momentum vector, then the motion of the relative position vector $ \vec{r} {} _ {1} - \vec{r} {} _ {2} = ( x, y, z) $ takes place in the plane $ z = 0 $ and satisfies the system

$$ \tag{1 } \mu \ddot{x} = - f x r ^ {- 3} ,\ \mu \ddot{y} = - f y r ^ {- 3} , $$

where $ r = \sqrt {x ^ {2} + y ^ {2} } $, $ \mu = m _ {1} m _ {2} /( m _ {1} + m _ {2} ) $ is the reduced mass and $ f $ is the gravitational constant. The system (1) has four integrals:

$$ x \dot{y} - y \dot{x} = c \ \textrm{ (law of areas), } $$

$$ \frac{1}{2} \mu ( {\dot{x} } {} ^ {2} + {\dot{y} } {} ^ {2} ) - f r ^ {- 1} = h \textrm{ (energy) } , $$

$$ \mu ^ {2} c \dot{y} - \mu f x r ^ {- 1} = \lambda _ {1} \ \textrm{ and } \ \ \mu ^ {2} c \dot{x} + \mu f y r ^ {- 1} = - \lambda _ {2} \ \textrm{ (Laplace) }, $$

which are interconnected by the relation

$$ \lambda _ {1} ^ {2} + \lambda _ {2} ^ {2} = \ 2 \mu ^ {3} h c ^ {2} + \mu ^ {2} f ^ { 2 } . $$

Here

$$ \tag{2 } c ^ {2} = \lambda _ {1} x + \lambda _ {2} y + \mu r , $$

i.e. the orbits of the relative position vector are conical sections with parameter $ p = c ^ {2} / \mu $, major semi-axis $ a = - \mu / ( 2h) $, eccentricity $ e = \mu ^ {- 1} \sqrt {1 + 2hc ^ {2} } $, longitude of pericentre $ \omega $ ( $ \lambda _ {1} = \mu e \cos \omega $, $ \lambda _ {2} = \mu e \sin \omega $), and with focus at the coordinate origin. The location of the relative positive vector on the orbit is determined by the true anomaly $ v $, counted from the direction towards the pericentre; (2) then implies that $ r = p / ( 1 + e \cos v ) $. If $ c \neq 0 $, three types of orbits are possible:

I) If $ h < 0 $, they are ellipses.

II) If $ h > 0 $, they are hyperbolas.

III) If $ h = 0 $, they are parabolas.

If $ c = 0 $, the motion is rectilinear. The two-body problem describes an unperturbed Kepler motion of a planet with respect to the Sun or of a satellite with respect to a planet, etc.

References

[1] C.L. Siegel, "Vorlesungen über Himmelmechanik" , Springer (1956)
[2] V.K. Abalakin, et al., "Handbook of celestial mechanics and astrodynamics" , Moscow (1971) (In Russian)
[a1] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1899)
[a2] C.L. Siegel, J. Moser, "Lectures on celestial mechanics" , Springer (1971)
[a3] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a4] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin (1978)
How to Cite This Entry:
Two-body problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-body_problem&oldid=49048
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article