Mathematics of Physics and Engineering

Lagrange's Equations 85

2.3.1 Lagrange's Equations

We first illustrate Lagrange's method for a point mass moving in the plane.
The modern methods of multi-variable calculus reduce the derivation of the
main result (equation (2.3.17)) to a succession of simple applications of the
chain rule.
Consider an inertial frame in M^2 with origin O and basis vectors i, j.
Let r = xi + yjbe the position of point mass m moving under the action
of force F = F 2 i + F 2 j. By Newton's Second Law (2.1.1),

Fi = mi, F 2 = my. (2.3.1)

The state of m at any time t is given by the four-dimensional vector
(x,y,x,y), that is, by the position and velocity. Knowledge of the state
at a given time allows us to determine the state at all future times by
solving equations (2.3.1).
Now consider a different pair ((71,(72) of coordinates in the plane, for
example, for example, polar coordinates, so that

x = x(q 1 ,q 2 ), y = y{qi,q 2 ); q\ = qi(x, y), q 2 = q 2 {x, y), (2.3.2)

and all the functions are sufficiently smooth. Differentiating (2.3.2),

dx. dx.. dy. dy. ,„ „ „,

x = —qi + —q 2 , y=~-qi + ^-q 2. (2.3.3)

dqi dq 2 dqi dq 2

We call qi,q 2 ,q\,q 2 the generalized coordinates, since their values de-
termine the state (x,y,x,y) by (2.3.2) and (2.3.3). Note that the partial
derivatives dx/dqi, dy/dqi, i = 1,2, depend only on q\ and q 2. We then
differentiate (2.3.3) to find

dx _ dx dx _ dx dy dy dy dy

dq\ dq\' dq 2 dq 2 ' dq\ dqi' dq 2 dq 2 '

Next, we apply the chain rule to the function dx(qi(t),q 2 (t))/dqi to find

_d_ / dx\ _ d^2 x dq\ d^2 x dq 2 _ d^2 x. d^2 x.

Jt\dTi)^Wi^ + d~q~ 2 ~bTlltt^Wiqi + ^dTiq2' }

Prom (2.3.3), differentiating with respect to the variable q\,

dx d^2 x. d^2 x. d (dx\ ,n „ „,

— = -jr-jgi + T-j^q 2 = -£;,-• 2.3.6)

dqi dq{ dqidq 2 dt \dqi J