- 1.2 Dimensional analysis
1.2 Dimensional analysis
Mathematics does not know units. Experimental physicists hate large sets of parameters because the sampling of high-dimensional parameter space is tiresome. A remedy to both issues is offered by the Buckingham-Pi-Theorem. We state it here in a form accessible with our present level of mathematical refinement. The discussion of a more advanced formulation may appear as a homework problem later on on this course.
Theorem 1.1 Buckingham-Pi-Theorem
A dynamics with $n$ parameters,
where the positions $\mathbf q$ and the parameters involve the three units meter, seconds and kilogram,
can be rewritten in terms of a dimensionless dynamics
with $n-3$ parameters,
where the positions $\boldsymbol\xi$, velocities $\boldsymbol\zeta$,
and parameters $\pi_j$ with $j \in \{ 1, \dots, n-3\}$
are given solely by numbers.
Figure 1.2: Pendulum discussed in Example 1.8.
Example 1.8 Non-dimensionalization for a pendulum
Let $\mathbf x$ denote the position of a pendulum of mass $M$
that is attached to a chord of length $L$
and swinging in a gravitational field $\mathbf g$ of strength $g$
(see Figure 1.2).
The units of these quantities are
$[\mathbf x] = \text{m}$,
$[M] = \text{kg}$,
$[L] = \text{m}$, and
$[g] = \text{m/s$^2$}$, respectively.
The position $\mathbf x$ describes the position of the system.
Its evolution will depend (potentially) on the three parameters,
$M$, $L$, and $g$, plus the direction of $\mathbf g$.
In this problem we choose $L$ as length scale and $\sqrt{L/g}$ as time scale.
Then the dimensionless positions will be $\boldsymbol\xi = \mathbf x /L$,
the dimensionless velocities will be $\boldsymbol\zeta = \mathbf{\dot x} / \sqrt{g\, L}$.
There is no way to turn $M$ into a dimensionless parameter.
Therefore, the evolution of $(\xi, \zeta)$ can not depend on $M$.
The only dimensionless parameter that remains in the model is the direction of$\mathbf g$.
Example 1.9 Non-dimensionalization for the flight of a piece of chalk
Let $\mathbf x_1$ and $\mathbf x_2$ denote the position of the tip and the tail of a model for a piece of chalk,
where tip and tail are associated to masses $m_1$ and $m_2$.
The piece of chalk has a length $L$.
It performs a free flight in a gravitational field with acceleration $\mathbf g$ of strength $g$.
The units of these quantities are
$[\mathbf x_i] = \text{m}$,
$[m_i] = \text{kg}$,
$L = \text{m}$, and
$[g] = \text{m/s$^2$}$, respectively.
There are four parameters, $n=4$, plus the direction of $\mathbf g$.
In this problem we choose $L$ as length scale and $\sqrt{L/g}$ as time scale.
Then the dimensionless positions will be $\boldsymbol\xi_i = \mathbf x_i /L$,
the dimensionless velocities will be $\boldsymbol\zeta = \mathbf{\dot x}_i / \sqrt{g\, L}$.
The two masses $m_1$ and $m_2$ give rise to the dimensionless parameter $\pi_1 = m_1/m_2$,
and in three dimensions the direction of $\mathbf g$ must be characterized by another two dimensionless parameters.
Proof of the Buckingham-Pi-Theorem
We first look for combinations of the parameters with the following units \begin{align*} \text{m} &= \left[ p_1^{\alpha_1} \right]\:\left[ p_2^{\alpha_2}\right] \: \dots \:\left[ p_n^{\alpha_n} \right] \\ \text{s} &= \left[ p_1^{\beta_1} \right]\:\left[ p_2^{\beta_2}\right] \: \dots \:\left[ p_n^{\beta_n} \right] \\ \text{kg} &= \left[ p_1^{\gamma_1} \right]\:\left[ p_2^{\gamma_2}\right] \: \dots \:\left[ p_n^{\gamma_n} \right] \end{align*} Each of these equations involves constraints on the exponents in order to match the exponents of the three units that can be expressed as a system of linear equations. The solvability conditions for such systems imply that they conditions can always be met by an appropriately chosen set of three parameters. Without loss of generality we denote them as $p_1$, $p_2$ and $p_3$, and we have \begin{align} \text{m} &= \left[ p_1^{\alpha_1} \right]\:\left[ p_2^{\alpha_2}\right] \: \left[ p_3^{\alpha_n} \right] \nonumber\\ \text{s} &= \left[ p_1^{\beta_1} \right]\:\left[ p_2^{\beta_2}\right] \: \left[ p_3^{\beta_n} \right] \label{eq:epressSI}\\ \nonumber \text{kg} &= \left[ p_1^{\gamma_1} \right]\:\left[ p_2^{\gamma_2}\right] \: \left[ p_3^{\gamma_n} \right] \end{align} Thus we use the parameters $p_1, \dots, p_3$ to remove the units from our description. In its dimensionless form it will involve the positions and velocities \begin{align*} \boldsymbol\xi &= \mathbf q \: p_1^{-\alpha_1} \: p_2^{-\alpha_2} \: p_3^{-\alpha_n} \\ \b