5.3.1. Lagrangian function with no explicit dependence on time
Let’s analyse first some properties of systems, whose Lagrangian function are not an explicit function of time,
\[\mathscr{L}(\dot{q}^k(t), q^k(t)) = K(\dot{q}^k(t), q^k(t)) + U(q^k(t)) \ ,\]
and then go back to the most general case.
As the Lagrange equation is not an explicit function of time, relation (5.5) reads
\[\dfrac{d}{dt} \left[ \dot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} - \mathscr{L} \right] = \dot{q}^k Q_k \ .\]
Since the Lagrangian doesn’t expliclty depend on time, and potential is not a function of time, relation (5.6) gives \(\dot{q}^k \frac{\partial \mathscr{L}}{\partial \dot{q}^k} = 2 K\), and thus the content of the braces is the mechanical energy of the system,
\[\dot{q}^k \dfrac{\partial \mathscr{L}}{\partial \dot{q}^k} - \mathscr{L} = 2 K - \mathscr{L} = 2 K - K - U = K - U = E^{mec} \ ,\]
and it becomes clear that the relation is nothhing but the balance equation of mechanical energy
\[\dfrac{d E^{mec}}{dt} = \dot{q}^k Q_k \ ,\]
that becomes conservation of mechanical energy, in absence of non-conservative actions, \(Q_k = 0\),
\[E^{mec} = \overline{E}^{mec} \quad \text{const.}\]
5.3.1.1. Properties of kinetic energy and potential
This section collects some properties of the kinetic energy and potential of systems, where physical coordinates of the system are written as a function of generalized coordinates only, \(q^k(t)\). As an example, coordinates of point masses, material points of rigid bodies and the rotation tensor representing their orientation in space can be written as
\[\vec{r}_P\left(q^k(t)\right) \quad , \quad \mathbb{R} \left( q^k(t) \right) \ ,\]
so that velocities and angular velocities become
\[\begin{split}\begin{aligned}
\vec{v}_P\left(\dot{q}^k(t), q^k(t)\right) & = \dfrac{d \vec{r}_P}{dt} = \dot{q}^k \dfrac{\partial \vec{r}}{\partial q^k}\left(q^k(t)\right) \\
\vec{\omega}_{\times}\left(\dot{q}^k(t), q^k(t)\right) & = \dfrac{d \mathbb{R}}{d t} \cdot \mathbb{R}^T = \dot{q}^k(t) \, \dfrac{\partial \mathbb{R}}{\partial q^k} \cdot \mathbb{R}^T = \dot{q}^k(t) \, \dfrac{\partial \vec{\theta}_{\times}}{\partial q^k}\left(q^k(t)\right) \ ,
\end{aligned}\end{split}\]
As the kinetic energy of a mechanical system is a quadratic function of velocity and angular velocity of its sub-systems, the kinetic energy can be written as
\[K\left( \dot{q}^k(t), q^k(t) \right) = \frac{1}{2} A_{ij}\left(q^k(t)\right) \, \dot{q}^i(t) \, \dot{q}^j(t) \ .\]
Since \(A_{ij}\) is symmetric w.r.t. the swap of indices (or it can be written in a symmetric form), partial derivative of the kinetic energy w.r.t. \(\dot{q}^l\) reads
\[\dfrac{\partial K}{\partial \dot{q}^l} = A_{lj} \dot{q}^j \ ,\]
and
(5.6)\[\dot{q}^l \dfrac{\partial K}{\partial \dot{q}^l} = \dot{q}^l A_{lj} \dot{q}^j = 2 K \ .\]
\[\begin{aligned}
\dfrac{\partial K}{\partial \dot{q}^l}
= \dfrac{\partial }{\partial \dot{q}^l} \left[ \frac{1}{2} A_{ij} \dot{q}^i \dot{q}^j \right]
= \frac{1}{2} A_{ij} \bigg[ \underbrace{ \dfrac{\partial \dot{q}^i}{\partial \dot{q}^l} }_{\delta^i_l} \dot{q}^j + \dot{q}^i \underbrace{ \dfrac{\partial \dot{q}^j}{\partial \dot{q}^l}}_{\delta^j_l} \bigg]
= \frac{1}{2} \left[ A_{lj} \dot{q}^j + A_{il} \dot{q}^i \right]
= A_{lj} \dot{q}^j
\end{aligned}\]
