View source for 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 [[#References|[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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571503.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571505.png" /> are the Cartesian coordinates and the velocities of the points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571506.png" /> is the number of points of the system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571507.png" /> is the time, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571508.png" /> is the mass of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571509.png" />-th point, which has coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715012.png" />.

The relations (1) and (2) are assumed to be independent, that is, the ranks of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715014.png" /> are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715016.png" />, respectively. Lagrange's equations of the first kind have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715019.png" /> are undetermined Lagrange multipliers, proportional to the reactions of the constraints, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715020.png" /> are the projections on the coordinate axes of the given active forces, and the force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715021.png" />, acting on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715022.png" />-th point, has the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715025.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715026.png" />.

To the differential equations (3) one must adjoin the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715027.png" /> equations (1) and (2), as a result of which one obtains a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715028.png" /> equations in the same number of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715031.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715032.png" /> independent generalized Lagrange coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715033.png" />, by means of which any possible position of the system can be obtained for certain values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715034.png" /> from the equalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

that convert equations (1) into identities, one can establish for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715036.png" /> a one-to-one correspondence between the possible positions of the system and the points of some region of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715037.png" />-dimensional configuration space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715038.png" />. In the case of stationary constraints (1) it is always possible to choose the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715039.png" /> so that the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715041.png" /> corresponding to all possible displacements of the system:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715042.png" /></td> </tr></table>

and the kinetic energy of the system:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715043.png" /></td> </tr></table>

Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715044.png" /></td> </tr></table>

is the generalized force corresponding to the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715045.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715046.png" /> are homogeneous forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715047.png" /> in the generalized velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715048.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715051.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715052.png" /></td> </tr></table>

In the case of stationary constraints <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715053.png" />.

Lagrange's equations of the second kind have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

Equations (5) form a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715055.png" /> ordinary second-order differential equations with unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715056.png" />. Their form is invariant with respect to the choice of Lagrange coordinates. This system of equations of motion has least possible order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715057.png" />. 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|Newton laws of mechanics]]).

In the case of potential generalized forces, when there is a force function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715060.png" />, equations (5) take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715062.png" /> is called the Lagrange function (formerly, the kinetic potential).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715063.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715064.png" />, then equations (6) have a generalized energy integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715065.png" /></td> </tr></table>

or a cyclic integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715066.png" /></td> </tr></table>

corresponding to the cyclic coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715067.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,   "Mécanique analytique" , '''1–2''' , Blanchard, reprint , Paris  (1965)  ((Also: Oeuvres, Vol. 11.))</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.T. Whittaker,   "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>