Lagrange equations (in mechanics)

Ordinary second-order differential equations which describe the motions of mechanical systems under the action of forces applied to them. The equations were established by J.L. Lagrange [1] in two forms: Lagrange's equations of the first kind, or equations in Cartesian coordinates with undetermined Lagrange multipliers, and of the second kind, or equations in generalized Lagrange coordinates.

Lagrange's equations of the first kind describe motions of both holonomic systems, constrained only by geometrical relations of the form

$$ \tag{1 } f _ {s} ( x _ {1}, \dots, x _ {3N} , t ) = 0 ,\ \ s = 1, \dots, k ,\ f _ {s} ( x , t ) \in C ^ {2} , $$

and non-holonomic systems, on which one imposes, as well as relations (1), kinematic relations of the form

$$ \tag{2 } \phi _ {r} ( x _ {1}, \dots, x _ {3N} , \dot{x} _ {1} \dots \dot{x} _ {3N} , t ) = 0 , $$

$$ r = 1, \dots, m ,\ \phi _ {r} ( x , \dot{x} , t ) \in C ^ {1} , $$

where $ x _ \nu $ and $ \dot{x} _ \nu = d x _ \nu / d t $ are the Cartesian coordinates and the velocities of the points, $ N $ is the number of points of the system, $ t $ is the time, and $ m _ {3p-2} = m _ {3p-1} = m _ {3p} $ is the mass of the $ p $-th point, which has coordinates $ x _ {3p-2} $, $ x _ {3p-1} $, $ x _ {3p} $.

The relations (1) and (2) are assumed to be independent, that is, the ranks of the matrices $ \| \partial f _ {s} / \partial x _ \nu \| $ and $ \| \partial \phi _ {r} / \partial \dot{x} _ \nu \| $ are equal to $ k $ and $ m $, respectively. Lagrange's equations of the first kind have the form

$$ \tag{3 } m _ \nu \ddot{x} _ \nu = X _ \nu + \sum _ { s= 1} ^ { k } \lambda _ {s} \frac{\partial f _ {s} }{\partial x _ \nu } + \sum _ { r= 1} ^ { m } \mu _ {r} \frac{\partial \phi _ {r} }{\partial \dot{x} _ \nu } , $$

where $ \lambda _ {s} $ and $ \mu _ {r} $ are undetermined Lagrange multipliers, proportional to the reactions of the constraints, $ X _ \nu $ are the projections on the coordinate axes of the given active forces, and the force $ F _ {p} $, acting on the $ p $-th point, has the projections $ X _ {3p-2} $, $ X _ {3p-1} $, $ X _ {3p} $; $ \ddot{x} _ \nu = d \dot{x} _ \nu / d t $.

To the differential equations (3) one must adjoin the $ k + m $ equations (1) and (2), as a result of which one obtains a system of $ 3 N + k + m $ equations in the same number of variables $ x _ \nu $, $ \lambda _ {s} $, $ \mu _ {r} $. In practice, Lagrange's equations of the first kind are usually applied to systems with a small number of unknowns.

Lagrange's equations of the second kind describe only motions of holonomic systems restricted by constraints of the form (1). By introducing $ n = 3 N - k $ independent generalized Lagrange coordinates $ q _ {i} $, by means of which any possible position of the system can be obtained for certain values of the $ q _ {i} $ from the equalities

$$ \tag{4 } x _ \nu = x _ \nu ( q _ {1} \dots q _ {n} , t ) ,\ \ x _ \nu ( q _ {i} , t ) \in C ^ {2} , $$

that convert equations (1) into identities, one can establish for every $ t $ a one-to-one correspondence between the possible positions of the system and the points of some region of the $ n $- dimensional configuration space $ ( q _ {1}, \dots, q _ {n} ) $. In the case of stationary constraints (1) it is always possible to choose the variables $ q _ {i} $ so that the time $ t $ does not occur in (4). Also, by means of (4) one can write down expressions for the sum of the elementary works of all the active forces $ F _ {p} $ corresponding to all possible displacements of the system:

$$ \sum _ { p= 1} ^ { N } F _ {p} \cdot \delta r _ {p} = \ \sum _ {\nu = 1 } ^ { 3N } X _ \nu \delta x _ \nu = \ \sum _ { i= 1} ^ { n } Q _ {i} \delta {q _ {i} } , $$

and the kinetic energy of the system:

$$ T ( q _ {i} , \dot{q} _ {i} , t ) = \frac{1}{2} \sum _ {\nu = 1 } ^ { 3N } m _ \nu \dot{x} _ \nu ^ {2} = \ T _ {2} + T _ {1} + T _ {0} . $$

Here

$$ Q _ {i} = \sum _ {\nu = 1 } ^ { 3N } X _ \nu \frac{\partial x _ \nu }{\partial q _ {i} } $$

is the generalized force corresponding to the coordinate $ q _ {i} $, the $ T _ {s} ( q _ {i} , \dot{q} _ {i} , t ) $ are homogeneous forms of degree $ s $ in the generalized velocities $ \dot{q} _ {i} $, and

$$ T _ {2} = \frac{1}{2} \sum _ { i,j= 1} ^ { n } a _ {ij} \dot{q} _ {i} \dot{q} _ {j} ,\ \ T _ {1} = \sum _ { i= 1} ^ { n } a _ {i} \dot{q} _ {i} , $$

$$ T _ {0} = \frac{1}{2} \sum _ {\nu = 1 } ^ { 3N } m _ \nu \left ( \frac{\partial x _ \nu }{\partial t } \right ) ^ {2} , $$

$$ a _ {ij} = \sum _ {\nu = 1 } ^ { 3N } m _ \nu \frac{\partial x _ \nu }{\partial q _ {i} } \frac{\partial x _ \nu }{\partial q _ {j} } , $$

$$ a _ {i} = \sum _ {\nu = 1 } ^ { 3N } m _ \nu \frac{\partial x _ \nu }{\partial q _ {i} } \frac{\partial x _ \nu }{\partial t } . $$

In the case of stationary constraints $ T = T _ {2} $.

Lagrange's equations of the second kind have the form

$$ \tag{5 } \frac{d}{dt} \frac{\partial T }{\partial \dot{q} _ {i} } - \frac{\partial T }{\partial q _ {i} } = \ Q _ {i} ,\ i = 1, \dots, n . $$

Equations (5) form a system of $ n $ ordinary second-order differential equations with unknowns $ q _ {i} $. Their form is invariant with respect to the choice of Lagrange coordinates. This system of equations of motion has least possible order $ 2n $. In this respect, and also in the absence of reactions of the constraints in (5), equations (5) have a great advantage compared to Lagrange's equations of the first kind (3). After integrating (5) one can determine the reactions of the constraints from the equations that express Newton's second law for the points of the system (cf. also Newton laws of mechanics).

In the case of potential generalized forces, when there is a force function $ U ( q _ {1}, \dots, q _ {n} , t ) $ such that $ Q _ {i} = \partial U / \partial q _ {i} $, $ i = 1, \dots, n $, equations (5) take the form

$$ \tag{6 } \frac{d}{dt} \frac{\partial L }{\partial \dot{q} _ {i} } - \frac{\partial L }{\partial q _ {i} } = \ 0 ,\ i = 1, \dots, n , $$

where $ L ( q _ {i} , \dot{q} _ {i} , t ) = T + U $ is called the Lagrange function (formerly, the kinetic potential).

If $ \partial L / \partial t \equiv 0 $, or $ \partial L / \partial q _ \alpha \equiv 0 $, then equations (6) have a generalized energy integral

$$ \sum _ { i= 1} ^ { n } \frac{\partial L }{\partial \dot{q} _ {i} } \dot{q} _ {i} - L = T _ {2} - T _ {0} - U = \textrm{ const } , $$

or a cyclic integral

$$ \frac{\partial L }{\partial \dot{q} _ \alpha } = \ \beta _ \alpha = \textrm{ const } , $$

corresponding to the cyclic coordinate $ q _ \alpha $.

References

[1]	J.L. Lagrange, "Mécanique analytique" , 1–2 , Blanchard, reprint , Paris (1965) ((Also: Oeuvres, Vol. 11.)) Zbl 0137.24101
[a1]	E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)
[a2]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a3]	F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)