# Poincaré equations

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 ) 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 ) 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 ) 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 ) and also converted them to a simpler canonical form (see Chetaev equations). In particular, he gave (see ) 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.

How to Cite This Entry:
Poincaré equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_equations&oldid=48204
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article