Galilean Transformation

From bio-physics-wiki

Jump to: navigation, search

The analysis of a mechanical system is in general dependent on the chosen coordinate system. In some coordinate systems equations of motion of a simple movement might look complicated. In a general reference frame, space is whether homogeneous in space and time, nor isotropic. \begin{align} \tag{1} f(t) &\rightarrow \text{time dependence: non-homogeneous in time}\\ f(\mathbf{x}) &\rightarrow \text{dependent on location: non-homogeneous in space}\\ f \left(\frac{\mathbf{x}}{|\mathbf{x}|},\frac{\mathbf{v}}{|\mathbf{v}|} \right) &\rightarrow \text{dependent on the direction of space: non-isotropic} \end{align} Non-homogeneity can result in the acceleration of a particle in a certain direction of space, like we have shown in the article on inertial systems, non-inertial frames can include fictitious forces that make particles accelerate relative to this non-inertial frame. An inertial frame can always be found, in fact there are even infinitely many inertial frames moving with constant speed relative to another. None of these inertial systems is unique. The transformation, that transforms from one inertial frame of reference to another is called Galilean transformation. \begin{align} \mathbf{r} &\rightarrow \mathbf{r}'\\ \mathbf{v} &\rightarrow \mathbf{v}'\\ t &\rightarrow t' \end{align}

Galilei.png

In the first inertial system (gray) a particle has the coordinates $\mathbf{r}'=\mathbf{r}+\mathbf{u} \cdot t$. In the second inertial system (black) the same particle has coordinates $\mathbf{r}$. The system moves with velocity $\mathbf{u}$ with respect to the black system. Both systems are inertial systems, this means Newtons laws are valid in both systems \begin{align} m \cdot \ddot{\mathbf{r}}_i = \sum_j \mathbf{F}_{ij}(\mathbf{r}_i-\mathbf{r}_j) \hspace{1cm} \text{ and } \hspace{1cm} m \cdot \ddot{\mathbf{r}}_i' = \sum_j \mathbf{F}_{ij}(\mathbf{r}_i'-\mathbf{r}_j') \end{align} Since only the distance between particles are relevant for the force, the transformation term cancels \begin{align} \mathbf{r}'=\mathbf{r}+\mathbf{u} \cdot t \, \Rightarrow \, \mathbf{r}_i-\mathbf{r}_j=\mathbf{r}_i'-\mathbf{r}_j' \end{align} From (1) follows that the Lagrangian of an inertial system is independent of the location and direction of $\mathbf{r}$, the time and the direction of $\mathbf{v}$. Thus the Lagrangian is a function of the scalar $\mathbf{v}^2=v^2$ only and is independent of $\mathbf{r}$ \begin{align} \frac{\partial L}{ \partial \mathbf{r}}=0 \end{align} thereafter \begin{align} \frac{d}{dt}\frac{\partial L}{ \partial \mathbf{v}}=0 \end{align} and it must hold that $\frac{\partial L}{ \partial \mathbf{v}}=const.$, which means $v$ must be constant. Hence, the only allowed transformation that satisfy (1) are those between coordinate frames moving with constant velocity $v$ with respect to one another and are called Galilean transformation.



Further Reading:

  • L. D. Landau & E. M. Lifschitz - Lehrbuch der Theoretischen Physik I: Mechanik