Forces are conservative then there exists a function V called potential energy such that

$$ \vec F_α^{(e)} = - \vec \nabla_α V \hspace{7em} (25) $$

$$ = - \left( \frac {\partial V} {\partial x_α}, \frac {\partial V} {\partial y_α}, \frac {\partial V} {\partial z_α} \right) \hspace{7em} (26) $$

where \( (x_\alpha, y_\alpha, z_\alpha) \equiv \vec r_\alpha \) are the components of position for the particle \( \alpha \). The generalized force, defined by l.h.s of (22) becomes

$$ Q_j = \sum _α \vec F_α . \frac {\partial \vec r_α} {\partial q_j} = - \sum _α \left( \frac {\partial \nu} {\partial x_α} \frac {\partial x_α} {\partial q_j} + \frac {\partial \nu} {\partial y_α} \frac {\partial y_α} {\partial q_j} + \frac {\partial \nu} {\partial z_α} \frac {\partial z_α} {\partial q_j} \right) \hspace{7em} (27) $$

The right-hand side of (27) takes a simple form and

$$ \hspace{7em} Q_j = - \frac {\partial V} {\partial q_j} \hspace{7em} (28) $$

Using (28) in

$$ \hspace{7em} \left( \frac d {dt} \frac {\partial T} {\partial \dot q_k} - \frac {\partial T} {\partial q_k} \right) = Q_k \hspace{7em} (29) $$

we get

$$ \hspace{7em} \left( \frac d {dt} \frac {\partial T} {\partial \dot q_k} - \frac {\partial T} {\partial q_k} \right) = - \frac {\partial V} {\partial q_k} \hspace{7em} (30) $$

Thus (30) can be rewritten as

$$ \hspace{7em} \frac d {dt} \frac {\partial T} {\partial \dot q_k} - \frac {\partial T} {\partial q_k} = \frac d {dt} \frac {\partial V} {\partial \dot q_k} - \frac {\partial V} {\partial q_k} \hspace{7em} (31) $$

because \( V \) is a function of \( q’s \) alone and \( \frac {\partial V} {\partial \dot q_k} = 0 \).

5.1 Introduce Lagrangian

Therefore, rearranging the above equation we get

$$ \hspace{7em} \frac d {dt} \frac {\partial L} {\partial \dot q_k} - \frac {\partial L} {\partial q_k} = 0 \hspace{7em} (32) $$

where \( L = T − V \), is called the Lagrangian of the system. Eq.(32) are called Euler Lagrange equation of motion.