Poincaré equations

General equations of the mechanics of holonomic systems (cf. Holonomic system) presented with the help of a certain Lie algebra of infinitesimal transformations.

Let $ x _ {i} $, $ i = 1 \dots n $, be variables determining the position of a holonomic mechanical system constrained by ideal constraints depending explicitly on time. If the system has $ k $ degrees of freedom, then there is an intransitive algebra of infinitesimal transformations

$$ X _ {0} = \frac \partial {\partial t } + \sum _ { j= } 1 ^ { n } \xi _ {0} ^ {j} \frac \partial {\partial x _ {j} } ,\ \ X _ {a} = \sum _ { j= } 1 ^ { n } \xi _ \alpha ^ {j} \frac \partial {\partial x _ {j} } , $$

$$ \alpha = 1 \dots k , $$

enabling the system to be carried at the moment of time $ t $ from the position $ x _ {i} $ to the infinitesimally-near actual position $ x _ {i} + d x _ {i} $ and the virtual position $ x _ {i} + \delta x _ {i} $ by infinitesimal transformations $ ( X _ {0} + \sum _ {\alpha = 1 } ^ {k} \eta _ \alpha X _ \alpha ) d t $ of the algebra and $ \sum _ {\alpha = 1 } ^ {k} \omega _ {i} X _ {i} $ of the subalgebra, respectively. Here $ \omega _ \alpha $ and $ \eta _ \alpha $, the independent variables determining the virtual and actual displacements respectively, are connected by the equations

$$ \delta {\eta _ {i} } = \ \frac{d \omega _ {i} }{dt} - \sum _ {\alpha , \beta = 1 } ^ { k } c _ {\alpha \beta i } \omega _ \alpha \eta _ \beta ,\ \ i = 1 \dots k , $$

if the algebra of virtual displacements $ X _ \alpha $ is determined by its structure constants $ c _ {\alpha \beta i } $:

$$ ( X _ \alpha X _ \beta ) = \ X _ \alpha X _ \beta - X _ \beta X _ \alpha = \ \sum _ { i= } 1 ^ { k } c _ {\alpha \beta i } X _ {i} , $$

$$ \alpha , \beta = 1 \dots k , $$

and the operator $ X _ {0} $ commutes with the algebra of virtual displacements

$$ ( X _ {0} X _ \alpha ) = 0 ,\ \alpha = 1 \dots k . $$

Below these conditions are assumed to hold.

The Poincaré equations are ordinary first-order (with respect to $ \eta _ {j} $) differential equations

$$ \tag{1 } \frac{d}{dt} \frac{\partial L }{\partial \eta _ {j} } \ = \sum _ {\alpha , \beta = 1 } ^ { k } c _ {\alpha j \beta } \eta _ \alpha \frac{\partial L }{\partial \eta _ \beta } + X _ {j} L , $$

where $ j = 1 \dots k $,

$$ L ( t , x _ {1} \dots x _ {n} , \eta _ {1} \dots \eta _ {k} ) = \ T + U $$

is the Lagrange function, $ T ( t , x , \eta ) $ is the kinetic energy and $ U ( t , x ) $ is the force function.

The equations (1) were first obtained by H. Poincaré (see [1]) in the case when the algebra of virtual displacements is transitive and the constraints do not depend explicitly on time, and he applied them (see [2]) to investigate the motion of a solid body with an ellipsoidal cavity, entirely filled by a uniformly vortex moving ideal fluid. N.G. Chetaev (see [3]) generalized and developed the theory of the Poincaré equations to the case where the algebra of displacements is intransitive and the constraints depend explicitly on time (see [3]–[5]) and also converted them to a simpler canonical form (see Chetaev equations). In particular, he gave (see [5]) a method for constructing the algebra of virtual and actual displacements when the holonomic constraints are given by a differential form and he introduced the important concept of cyclic displacements.

Displacements $ X _ {r} $, $ r = s + 1 \dots k $, are said to be cyclic if they satisfy the conditions: 1) $ X _ {r} L = 0 $; 2) $ ( X _ {r} X _ \beta ) = 0 $, $ r = s + 1 \dots k $, $ \beta = 1 \dots k $.

According to 2) the cyclic displacements $ X _ {r} $ form an Abelian subalgebra of the algebra of virtual displacements which commutes with all the operators $ X _ \beta $. For cyclic displacements the first integrals of the Poincaré equations,

$$ \frac{\partial L }{\partial \eta _ {r} } \ = a _ {r} = \textrm{ const } ,\ \ r = s + 1 \dots k , $$

exist. From these relations the variables $ \eta _ {r} $ can be expressed in terms of the constants $ a _ {r} $ and the variables $ t $, $ x _ {i} $, $ \eta _ {1} \dots \eta _ {s} $, and one can introduce the Routh function

$$ R ( t , x _ {1} \dots x _ {n} ; \eta _ {1} \dots \eta _ {s} ; \ a _ {s+} 1 \dots a _ {k} ) = $$

$$ = \ L - \sum _ {r = s + 1 } ^ { k } \frac{\partial L }{\partial \eta _ {r} } \eta _ {r} . $$

Then for the non-cyclic displacements the Poincaré equations take the form

$$ \tag{2 } \frac{d}{dt} \frac{\partial R }{\partial \eta _ {j} } \ = \sum c _ {\alpha j \beta } \eta _ \alpha \frac{\partial R }{\partial \eta _ \beta } + \sum c _ {\alpha j \gamma } \eta _ \alpha a _ \gamma + X _ {j} R , $$

$$ \alpha , j , \beta = 1 \dots s ; \ \gamma = s + 1 \dots k . $$

After integrating the equations (2) the values $ \eta _ {r} $ are determined by the equations

$$ \eta _ {r} = - \frac{\partial R }{\partial a _ {r} } ,\ \ r = s + 1 \dots k . $$

If, in addition, $ c _ {\alpha j \gamma } = 0 $, $ \alpha , j = 1 \dots s $, $ \gamma = s + 1 \dots k $, that is, if the non-cyclic displacements $ X _ \beta $, $ \beta = 1 \dots k $, form a subalgebra of the algebra of virtual displacements, then the mechanical system corresponding to this subalgebra forms some sort of independent holonomic system with $ s $ degrees of freedom, described by the equations (1) when $ \alpha , j , \beta = 1 \dots s $, where the role of the function $ L $ is played by the function $ R $.

The Poincaré equations contain as particular cases: the Lagrange equations (in mechanics) when the transformation algebra of infinitesimal increments in one of the variables reduces to a commutative algebra of transformations; and the Euler equation for the rotation of a solid body, when the $ \eta _ {i} $ are taken to be the projections $ p $, $ q $ and $ r $ of the instantaneous angular velocity.

References

[1]	H. Poincaré, C.R. Acad. Sci. Paris , 132 (1901) pp. 369–371
[2]	H. Poincaré, Bull. Astron. , 27 (1910) pp. 321–356
[3]	N.G. Chetaev, Dokl. Akad. Nauk SSSR , 7 (1928) pp. 103–104
[4]	N.G. Chetaev, "Sur les équations de Poincaré" C.R. Acad. Sci. Paris , 185 (1927) pp. 1577–1578
[5]	N.G. Chetaev, Prikl. Mat. i. Mekh. , 5 : 2 (1941) pp. 253–262