Subfield: Mechanics

A The Extremum Principle in Mechanics

Consider a horizontal frictionless $x-y$ plane shown in Fig. 1. It is divided into two regions, I and II, by a line $\mathrm{AB}$ satisfying the equation $x=x_{1}$. The potential energy of a point particle of mass $m$ in region I is $V=0$ while it is $V=V_{0}$ in region II. The particle is sent from the origin $\mathrm{O}$ with speed $v_{1}$ along a line making an angle $\theta_{1}$ with the $x$-axis. It reaches point $\mathrm{P}$ in region II traveling with speed $v_{2}$ along a line that makes an angle $\theta_{2}$ with the $x$-axis. Ignore gravity and relativistic effects in this entire task T-2 (all parts).

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Figure 1
Context question:
A.1 Obtain an expression for $v_{2}$ in terms of $m, v_{1}$ and $V_{0}$.
Context answer:
\boxed{$v_{2}=(v_{1}^{2}-\frac{2 V_{0}}{m})^{1 / 2}$}


Context question:
A.2 Express $v_{2}$ in terms of $v_{1}, \theta_{1}$ and $\theta_{2}$.
Context answer:
\boxed{$v_2=\frac{v_1 \sin\theta_1}{\sin \theta_2}$}


Extra Supplementary Reading Materials:

We define a quantity called action $A=m \int v(s) d s$, where $d s$ is the infinitesimal length along the trajectory of a particle of mass $m$ moving with speed $v(s)$. The integral is taken over the path. As an example, for a particle moving with constant speed $v$ on a circular path of radius $R$, the action $A$ for one revolution will be $2 \pi m R v$. For a particle with constant energy $E$, it can be shown that of all the possible trajectories between two fixed points, the actual trajectory is the one on which $A$ defined above is an extremum (minimum or maximum). Historically this is known as the Principle of Least Action (PLA).
Context question:
A.3 PLA implies that the trajectory of a particle moving between two fixed points in a region of constant <br> potential will be a straight line. Let the two fixed points $\mathrm{O}$ and $\mathrm{P}$ in Fig. 1 have coordinates $(0,0)$ and <br> $\left(x_{0}, y_{0}\right)$ respectively and the boundary point where the particle transits from region I to region II have <br> coordinates $\left(x_{1}, \alpha\right)$. Note that $x_{1}$ is fixed and the action depends on the coordinate $\alpha$ only. State the <br> expression for the action $A(\alpha)$. Use PLA to obtain the relationship between $v_{1} / v_{2}$ and these coordinates.
Context answer:
\boxed{$\frac{v_{1}}{v_{2}}=\frac{(y_{0}-\alpha)(x_{1}^{2}+\alpha^{2})^{1 / 2}}{\alpha[(x_{0}-x_{1})^{2}+(y_{0}-\alpha)^{2}]^{1 / 2}}$}


Extra Supplementary Reading Materials:

B The Extremum Principle in Optics

A light ray travels from medium I to medium II with refractive indices $n_{1}$ and $n_{2}$ respectively. The two media are separated by a line parallel to the $x$-axis. The light ray makes an angle $i_{1}$ with the $y$-axis in medium I and $i_{2}$ in medium II (see Fig. 2). To obtain the trajectory of the ray, we make use of another extremum (minimum or maximum) principle known as Fermat's principle of least time.

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Figure 2

Question: B.1 The principle states that between two fixed points, a light ray moves along a path such that time taken between the two points is an extremum. Derive the relation between $\sin i_{1}$ and $\sin i_{2}$ on the basis of Fermat's principle.

Solution:
The speed of light in medium I is $c / n_{1}$ and in medium II is $c / n_{2}$, where $c$ is the speed of light in vacuum. Let the two media be separated by the fixed line $y=y_{1}$. Then time $T(\alpha)$ for light to travel from origin $(0,0)$ and $\left(x_{0}, y_{0}\right)$ is

$$
T(\alpha)=n_{1}\left(\sqrt{y_{1}^{2}+\alpha^{2}}\right) / c+n_{2}\left(\sqrt{\left(x_{0}-\alpha\right)^{2}+\left(y_{0}-y_{1}\right)^{2}}\right) / c
$$

Differentiating w.r.t. $\alpha$ and setting the derivative of $T(\alpha)$ to zero

$$
\begin{gathered}
\frac{n_{1} \alpha}{\left(y_{1}^{2}+\alpha^{2}\right)^{1 / 2}}-\frac{n_{2}\left(y_{0}-\alpha\right)}{\left[\left(x_{0}-\alpha\right)^{2}+\left(y_{0}-y_{1}\right)^{2}\right]^{1 / 2}}=0 \\
\therefore n_{1} \sin i_{1}=n_{2} \sin i_{2}
\end{gathered}
$$

[Note: Derivation is similar to A3. This is Snell's law.]

Final answer: $n_{1} \sin i_{1}=n_{2} \sin i_{2}$