Chetaev equations

General canonical equations of the mechanics of a holonomic system, presented in terms of a certain Lie algebra of infinitesimal transformations, and equivalent to the Poincaré equations.

If instead of independent variables $ \eta _ {j} $ defining real displacements one introduces the quantities

$$ y _ {j} = \frac{\partial L }{\partial \eta _ {j} } ,\ \ j = 1 \dots k , $$

where $ L ( t ; x _ {1} \dots x _ {n} ; \eta _ {1} \dots \eta _ {k} ) $ is the Lagrangian, then the Poincaré equations take the simpler form of the Chetaev equations

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

$$ j = 1 \dots k , $$

where

$$ H ( t ; x _ {1} \dots x _ {n} ; y _ {1} \dots y _ {k} ) = \sum_{j=1} ^ { k } \eta _ {j} y _ {j} - L $$

is the Hamiltonian. The second group of equations (1) may be replaced by the equations

$$ \tag{2 } \frac{dx _ i}{dt} = \left ( X _ {0} + \sum _ {\alpha = 1 } ^ { k } \frac{\partial H }{\partial y _ \alpha } X _ \alpha \right ) x _ {i} ,\ i = 1 \dots n . $$

(Cf. Poincaré equations for the operators $ X _ {0} , X _ \alpha $.)

Introducing the action function by the formula

$$ V ( t ; x _ {1} \dots x _ {n} ; x _ {1} ^ {0} \dots x _ {n} ^ {0} ) = \int\limits _ {t _ {0} } ^ { t } \left ( \sum _ {\alpha = 1 } ^ { k } y _ \alpha \eta _ \alpha - H \ \right ) dt , $$

where the integration takes place over the real trajectories of the system, one obtains the relations

$$ \tag{3 } y _ \alpha = X _ \alpha V ,\ y _ \alpha ^ {0} = \ - X _ \alpha ^ { 0 } V ,\ \alpha = 1 \dots k . $$

Here the $ X _ \alpha ^ { 0 } $ denote the operators $ X _ \alpha $ applied to the initial momentum at time $ t _ {0} $ and the initial position of the system $ x _ {i} ^ {0} $; and $ y _ \alpha ^ {0} $ are the initial values of $ y _ \alpha $. If the action function is known, then the equations (3) solve the mechanical problem in which the second group of equations (3) defines, in an implicit way, the law of motion of the system.

The action function satisfies the first-order partial differential equation

$$ \tag{4 } X _ {0} V + H ( t , x _ {1} \dots x _ {n} , X _ {1} V \dots X _ {k} V ) = 0 . $$

If a complete integral $ V ( t , x _ {1} \dots x _ {n} ; a _ {1} \dots a _ {n} ) $ of (4) is known, then the solutions to the Chetaev equations are determined by the relations

$$ \frac{\partial V }{\partial a _ {i} } = b _ {i} ,\ \ y _ {j} = X _ {j} V ,\ \ i = 1 \dots n ,\ j = 1 \dots k , $$

where $ a _ {i} $ and $ b _ {i} $ are arbitrary constants, restricted by the $ n - k $ integrability equations of the constraint.

Instead of the variables $ x _ {i} $ one can consider new variables $ \alpha _ {i} $ defining the position of the system. Suppose that $ A _ {0} = \partial / \partial t $, $ A _ {s} $, $ s = 1 \dots k $, represent the Lie algebra of a $ ( k + 1 ) $- parameter Lie group of continuous transformations in the variables $ \alpha _ {i} $ with structure constants $ \gamma _ {srj} $, where $ \gamma _ {0rj} = 0 $; and suppose that $ \pi _ {s} $ and $ \theta _ {s} $ are variables defining possible and actual displacements, so that for some function

$$ f ( t , \alpha _ {1} \dots \alpha _ {n} ), $$

$$ \delta f = \sum_{s=1} ^ { k } \pi _ {s} A _ {s} f ,\ d f = \left ( \frac{\partial f }{\partial t } + \sum_{s=1} ^ { k } \theta _ {s} A _ {s} f \right ) d t . $$

The transformation of variables is determined by the characteristic function

$$ V ( t , x _ {1} \dots x _ {n} , \alpha _ {1} \dots \alpha _ {n} ),\ \ \left \| \frac{\partial ^ {2} V }{\partial x _ {i} \partial \alpha _ {j} } \right \| \neq 0 , $$

and the formulas

$$ y _ {s} = X _ {s} V ,\ \ \beta _ {s} = - A _ {s} V ,\ \ s = 1 \dots k , $$

together with the integrability equations of the constraints. Such transformations are called canonical transformations, they preserve the canonical form of the equations of motion, in which the Hamilton function in the new variables takes the form

$$ H ^ {*} ( t , \alpha , \beta ) = \ \frac{\partial V }{\partial t } + H . $$

(Cf. also Hamiltonian system.) If the characteristic function of the transformation is a complete integral of equation (4) (for $ X _ {0} = \partial / \partial t $), then $ H ^ {*} = 0 $ and the Chetaev equations (1) and (2) in the new variables take the form

$$ \frac{d \beta _ {s} }{dt} = \sum_{r,j=1} ^ { k } \gamma _ {rsj} \theta _ {r} \beta _ {j} ,\ \theta _ {s} = 0 ,\ \ s = 1 \dots k , $$

$$ \frac{d \alpha _ {i} }{dt} = \sum _ { s } \theta _ {s} A _ {s} \alpha _ {i} , $$

that is $ \alpha _ {i} = \textrm{ const } $, $ \beta _ {s} = \textrm{ const } $, $ i = 1 \dots n $, $ s = 1 \dots k $.

The linear form $ \Omega = \sum_{s=1}^ {k} y _ {s} \omega _ {s} $ defines the fundamental relative integral invariant of dynamics.

The condition for $ f ( t , x _ {1} \dots x _ {n} , y _ {1} \dots y _ {k)} $ to be constant is that the first integral of Chetaev's equations has the form

$$ X _ {0} f + ( H , f ) = 0 , $$

where

$$ ( f , g ) = \sum _ {\alpha = 1 } ^ { k } \left ( \frac{\partial f }{\partial y _ \alpha } X _ \alpha g - \frac{\partial g }{\partial y _ \alpha } X _ \alpha f \right ) + $$

$$ + \sum _ {\alpha , \beta , s = 1 } ^ { k } c _ {\alpha \beta s } y _ {s} \frac{\partial f }{\partial y _ \alpha } \frac{\partial g }{\partial y _ \beta } $$

defines the Poisson bracket.

If $ f = a $ and $ g = b $ are first integrals, then $ ( f , g ) = c $ is also an integral (a generalization of the Poisson theorem).

The Chetaev equations were introduced by N.G. Chetaev [1]–[3], who also developed their theory.

References

[1]	N.G. Chetaev, "Sur les équations de Poincaré" C.R. Acad. Sci. Paris , 185 (1927) pp. 1577–1578
[2]	N.G. Chetaev, "Sur les équations de Poincaré" Dokl. Akad. Nauk SSSR Ser A. : 7 (1928) pp. 103–104
[3]	N.G. Chetaev, "On the equations of Poincaré" Prikl. Mat. i Mekh. , 5 : 2 (1941) pp. 253–262 (In Russian)