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Shortest path between two points in polar coordinates

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Show that the shortest path between two given points in a plane is a straight line, using plane polar coordinates.

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title{Shortest path between two points}
section{Polar coordinates}
Before we start, let's first derive the formula for the path length of an arbitrary path in polar coordinates. Consider two nearby points on a path at coordinates $(r,theta)$ and $(r + dr, theta + dtheta)$. The distance between these two points is $sqrt{dr^{2} + r^{2}dtheta^{2}}$. If we parametrize the path by specifying $theta$ as a function of $r$, the path length $S$, can then be written as:
S = int_{theta_{1}}^{theta_{2}}sqrt{r^{2}+haak{frac{dr}{dtheta}}^{2}}dtheta
Note that when we ask for the minimum path length we need to specify $r$ at $theta_{1}$ and $theta_{2}$ and ask which path which has these fixed end points is the shortest.
If we instead parametrize the path by specifying $r$ as a function of $theta$, then we can write:
S = int_{r_{1}}^{r_{2}}sqrt{1+r^{2}haak{frac{dtheta}{dr}}^{2}}dr
And when we minimize this, we need to specify $theta$ at the two end points at $r_{1}$ and $r_{2}$.

Since we already know that the solution is a straight line, let's find out how a straight line looks like in polar coordinates. For a given line, there will be a point on that line that is closest to the origin. Let's call this point $P$. So, at the point $P$, $r$ is minimal, say $r=r_{0}$ and $theta$ will have some value $theta_{0}$. Because $r$ is minimal on the line at $P$, the line must be at right angles to the line connecting $P$ to the Origin at $r=0$. For some arbitrary point $Q$ on the line, consider the right triangle formed by the Origin, the point $P$ and the point $Q$. If the point has coordinates $(r,theta)$, then:

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Solution Summary

We show that the shortest path between two given points in a plane is a straight line,
using calculus of variations in polar coordinates. We give two derivations: using the Lagrangian, and the Hamiltonian. In the first case we show that the solution follows from conservation of the conjugate momentum, while in the latter case it follows from conservation of the Hamiltonian (analogous to conservation of energy).

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Dimensional analysis, vectors, etc.

1) the rectangular and polar coordinates of a point are(x,y) and (r,θ), where x=6 and θ =27 degrees. Find value of r and value of y.

2) Assume the water has uniform velocity represented by the vector Q in the diagram below. The shore lines are on the left and right hand side of the diagram. A river is crossed by a girl rowing a boat. The rowing speed of the girls's boat and a set of possible orientations of her boat(relative to still water) are shown in the diagram.

(see attachment)

For an observer on shore, the speed of the boat for direction H is > than for direction W.
a) true
b) false

3) To land directly across the river, she must row in direction Y.
a) true
b) false

4) To get across the river in the shortest time, she must row in direction Y.
a) true
b) false

5) Time to row across for direction H is equal to that for direction P.
a) true
b) false

6) Time to row across for direction Y is less that for direction P.
a) true
c) false

7) The total distance traveled in crossing for direction P is greater that for direction H.
a) true
b) false

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a) What is the magnitude of the hummingbird's total displacement? Answer in units of m.
b) How many degrees below the horizontal is this total displacement? Answer in degrees.

8) A plane travels 3.8km at an angle of 26 degrees to the ground, then changes direction and travels 7.4km at an angle of 13 degrees to the ground.

a) What is the magnitude of the plane's total displacement? Answer in km.
b) At what angle above the horizontal is the plane's total displacement? Answer in degrees.

9) The pilot of an aircraft wishes to fly due west in a 48.6km/h wind blowing toward the south. The speed of the aircraft in the absence of a wind is 174 km/h.

a) How many degrees from west should the aircraft head? Let clockwise be positive. Answer in degrees.
b) What should the planes speed be relative to the ground? Answer in units of km/h.

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