Mkd

A similar relation holds for partial derivatives of with respect to <72 and q2. Hence Eq. (A.9) can be written d ( dK \

Sq 1

dK dq2_

dq 1

Sq2.

Since <71 and <72 are independent coordinates they can be varied arbitrarily. Hence, we can conclude that dt ^S^iy) dqi 1 ' dq\ ' dq\ ' dq\

dK dq 1

dq 1 dK

The right-hand side of Eq. (A.20) is the work done by all the forces on the system when the coordinates of the z'th particle undergo the small displacement Sx,, by,, and Sz, due to changes ¿<71 and Sq2 in the generalized coordinates <71 and q2. The coefficients of ¿<71 and Sq2 are known as the generalized forces Q \ and Q2, since they are the quantities by which the variations of the generalized coordinates must be multiplied to calculate the virtual work done by all the forces acting on the system. Hence,

Zi^ dqi

This is one form of Lagrange's equations of motion. They apply to any system that is completely described by two and only two generalized coordinates, whether or not the system is conservative. It can be shown by slightly more extended calculation that they apply to systems of any finite number of degrees of freedom.

A.5 Lagrange's Equations for Conservative Systems

If a system is conservative, the work done by the forces can be calculated from the potential energy P. We define the change in potential energy during a small displacement as the negative of the work done by the forces of the system during the displacement. Because Q\8q\ + Q2Sq2 is the work done by the forces, we have

We have emphasized that <71 and q2 are independent and, hence, can be varied arbitrarily. If

aq 1

Similarly, it can be seen that dP

Replacing Q\ and Qi in Eqs. (A.23) by these expressions we have d dK 9P -0

d 9K dP

dt \dq2J dq2 3q2

These are Lagrange's equations of motion for a conservative system. As before, they hold for systems of any finite number of degrees of freedom.

Example 4

Find the equations of motion of a particle of weight W moving in space under the force of gravity.

Solution: We need three coordinates to describe the position of the particle and can therefore take x, y, z as our generalized coordinates. Taking x and y in the horizontal place and z vertically upward with the origin at the earth's surface and taking the origin as the zero position for potential energy, one obtains

dx g ' 3y g ' ' 3z g dx 3y 3z d/dK\_W.. d/dK\_W.. d{dK\_w-

Hence, Lagrange's equation, Eq. (A.27), gives

0 0