Lagrangian function with no explicit dependence on time

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,

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

\frac{d}{d t} [{\dot{q}}^{k} \frac{\partial L}{\partial {\dot{q}}^{k}} - L] = {\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 ${\dot{q}}^{k} \frac{\partial L}{\partial {\dot{q}}^{k}} = 2 K$ , and thus the content of the braces is the mechanical energy of the system,

{\dot{q}}^{k} \frac{\partial L}{\partial {\dot{q}}^{k}} - L = 2 K - L = 2 K - K - U = K - U = E^{m e c},

and it becomes clear that the relation is nothhing but the balance equation of mechanical energy

\frac{d E^{m e c}}{d t} = {\dot{q}}^{k} Q_{k},

that becomes conservation of mechanical energy, in absence of non-conservative actions, $Q_{k} = 0$ ,

E^{m e c} = {\overset{―}{E}}^{m e c} 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} (q^{k} (t)), R (q^{k} (t)),

so that velocities and angular velocities become

\begin{array}{r} \begin{aligned} {\vec{v}}_{P} ({\dot{q}}^{k} (t), q^{k} (t)) & = \frac{d {\vec{r}}_{P}}{d t} = {\dot{q}}^{k} \frac{\partial \vec{r}}{\partial q^{k}} (q^{k} (t)) \\ {\vec{ω}}_{\times} ({\dot{q}}^{k} (t), q^{k} (t)) & = \frac{d R}{d t} \cdot R^{T} = {\dot{q}}^{k} (t) \frac{\partial R}{\partial q^{k}} \cdot R^{T} = {\dot{q}}^{k} (t) \frac{\partial {\vec{θ}}_{\times}}{\partial q^{k}} (q^{k} (t)), \end{aligned} \end{array}

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 ({\dot{q}}^{k} (t), q^{k} (t)) = \frac{1}{2} A_{i j} (q^{k} (t)) {\dot{q}}^{i} (t) {\dot{q}}^{j} (t) .

Since $A_{i j}$ 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

\frac{\partial K}{\partial {\dot{q}}^{l}} = A_{l j} {\dot{q}}^{j},

and

(5.6)#

{\dot{q}}^{l} \frac{\partial K}{\partial {\dot{q}}^{l}} = {\dot{q}}^{l} A_{l j} {\dot{q}}^{j} = 2 K .

Lagrangian function with no explicit dependence on time

Contents

5.3.1. Lagrangian function with no explicit dependence on time#

5.3.1.1. Properties of kinetic energy and potential#