5.3. Lagrangian functions and time dependence

Some problems may have a Lagrangian function with an explicit dependence on time,

\[\mathscr{L}(\dot{q}^k(t),q^k(t),t) \ .\]

Using the general form (5.1) of Lagrange equations, the time derivative of the Lagrange function reads

\[\begin{split}\begin{aligned} \dfrac{d \mathscr{L}}{dt} & = \ddot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} + \dot{q}^k \dfrac{\partial \mathscr{L}}{\partial q^k} + \dfrac{\partial \mathscr{L}}{\partial t} = && \qquad \text{(IxP)} \\ & = \dfrac{d}{dt} \left( \dot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} \right) - \dot{q}^k \dfrac{d}{dt} \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} + \dot{q}^k \dfrac{\partial \mathscr{L}}{\partial q^k} + \dfrac{\partial \mathscr{L}}{\partial t} = && \qquad \text{(Lagrange eq.)} \\ & = \dfrac{d}{dt} \left( \dot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} \right) - \dot{q}^k Q_k + \dfrac{\partial \mathscr{L}}{\partial t} \ . \end{aligned}\end{split}\]

This latter relation can be recast as

(5.5)\[\dfrac{d}{dt} \left[ \dot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} - \mathscr{L} \right] = \dot{q}^k Q_k - \dfrac{\partial \mathscr{L}}{\partial t} \ ,\]

i.e. time derivative of a physical quantity equals the power of actions not included in the potential, \(\dot{q}^k Q_k\) and a contribution of partial derivative of the Lagrangian function, \(\partial_t \mathscr{L}\).

As it’s discusseed in the section for systems with Lagrangian function with no explicit dependence on time when the Lagrangian function of the system is not an explicit function of time todo *discuss the cases when \(\partial_t \mathscr{L} \ne 0\), the equation (5.5) is nothing but the balance equation of mechanical energy.

When there is no generalized force, that can’t be included in the potential, \(Q_k = 0\), and no explicit dependence of Lagrangian function on time, \(\partial_t \mathscr{L} = 0\), the equation (5.5) can be recast as an Euler-Beltrami equation,

\[\dot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} - \mathscr{L} = \overline{E} \quad \text{const.}\]

describing the conservation of mechanical energy w.r.t. an inertial reference frame, in absence of non-conservative forces, as discussed in the .