5.1.1. Point

Newton dynamical equations - strong form. Dynamical equation governing the motion of a point \(P\) reads

\[m \dot{\vec{v}}_P = \vec{R}^e \ ,\]

being \(m\) the mass of the system, \(\vec{v}_P\) the velocity of point \(P\), \(\vec{a}_P = \dot{\vec{v}}_P\) its acceleration and \(\vec{R}^{e}\) the net external force acting on the system..

Weak form. Weak form of dynamical equations is derived with scalar multiplication of the strong form by an arbitrary test function \(\vec{w}\),

(5.2)\[\vec{0} = \vec{w} \cdot \left( m \dot{\vec{v}} - \vec{R}^e\right) \qquad \forall \vec{w}\]

Lagrange equations. Lagrange equations are derived from a proper choice of the test function. The position of the point \(P\) is written as a function of the generalized coordinates \(q^k(t)\) and time \(t\)

\[\vec{r}_P(t) = \vec{r}(q^k(t),t) \ ,\]

so that its velocity can be written as

\[\vec{v}_P(t) := \frac{d\vec{r}_P}{dt} = \dot{q}^k(t) \underbrace{\frac{\partial \vec{r}}{\partial q^k}}_{\frac{\partial \vec{v}}{\partial \dot{q}^k}}(q^l(t), t) + \frac{\partial \vec{r}}{\partial t}(q^l(t), t) = \vec{v}\left(\dot{q}^k(t), q^k(t), t \right) \ ,\]

from which the relation between partial derivatives

(5.3)\[\dfrac{\partial \vec{r}}{\partial q^k} = \dfrac{\partial \vec{v}}{\partial \dot{q}^k} \ .\]

follows. Choosing the test function \(\vec{w}\) as

\[\vec{w} = \dfrac{\partial \vec{r}}{\partial q^k} = \dfrac{\partial \vec{v}}{\partial \dot{q}^k} \ ,\]

applying the rule of derivative of product, using Schwartz theorem to switch order of derivation, and exploiting relation (5.3) it’s possible to recast weak form (5.2) as

\[\begin{split}\begin{aligned} \vec{0} & = \frac{\partial \vec{v}}{\partial \dot{q}^k} \cdot \left( m \dot{\vec{v}} - \vec{R}^e \right) = \\ & = \frac{d}{dt} \left( \frac{\partial \vec{v}}{\partial \dot{q}^k} \cdot m \vec{v} \right) - \frac{d}{dt} \frac{\partial \vec{r}}{\partial q^k} \cdot m \vec{v} - \frac{\partial \vec{r}}{\partial q^k} \cdot ( \vec{R}^{e,c} + \vec{R}^{e,nc} ) \\ & = \frac{d}{dt} \left( \frac{\partial \vec{v}}{\partial \dot{q}^k} \cdot m \vec{v} \right) - \frac{\partial \vec{v}}{\partial q^k} \cdot m \vec{v} - \frac{\partial \vec{r}}{\partial q^k} \cdot ( \nabla U + \vec{R}^{e,nc} ) \\ & = \frac{d}{dt} \left( \frac{\partial K}{\partial \dot{q}^k} \right) - \frac{\partial K}{\partial q^k} - \frac{\partial U}{\partial q^k} - \underbrace{\frac{\partial \vec{r}}{\partial q^k} \cdot \vec{R}^{e,nc}}_{=: Q^k} \ . \\ \end{aligned}\end{split}\]

Introducing the Lagrangian function

\[\mathscr{L}(\dot{q}^k(t), q^k(t), t) := K(\dot{q}^k(t), q^k(t), t) + U(q^k(t),t) \ ,\]

and recalling that potential function \(U\) is not a function of velocity and thus of time derivatives of the generalized coordinates \(\dot{q}^k\), it’s possible to recast the dynamical equation as the Lagrange equations

\[\frac{d}{dt}\left(\frac{\partial \mathscr{L}}{\partial \dot{q}^k} \right) - \frac{\partial \mathscr{L}}{\partial q^k} = Q^k \ ,\]

being \(Q^k\) the generalized force not included in the gradient of the potential \(\nabla U\) - usually a non conservative contribution -, \(Q^k = \dfrac{\partial \vec{r}}{\partial q^k} \cdot \vec{R}^{e,nc}\).