In Pursuit of the Unknown by Ian Stewart Page B

digging tunnels through hills rather than taking the train over the top. One reason was that trains tend to slip on steep gradients, but the main one was energy. Climbing a hill, against the force of gravity, costs energy, which shows up as increased fuel consumption, which costs money.
    It’s much the same with interplanetary travel. Imagine a spacecraft moving through space. Where it goes next does not depend solely on where it is now: it also depends on how fast it is moving and in which direction. It takes three numbers to specify the spacecraft’s position – for example its direction from the Earth, which requires two numbers (astronomers use right ascension and declination, which are analogous to longitude andlatitude on the celestial sphere, the apparent sphere formed by the night sky), and its distance from the Earth. It takes a further three numbers to specify its velocity in those three directions. So the spacecraft travels through a mathematical landscape that has six dimensions rather than two.
    A natural landscape is not flat: it has hills and valleys. It takes energy to climb a hill, but a train can gain energy by rolling down into a valley. In fact, two types of energy come into play. The height above sea-level determines the train’s potential energy, which represents work done against the force of gravity. The higher you go, the more potential energy you must create. The second kind is kinetic energy, which corresponds to speed. The faster you go, the greater your kinetic energy becomes. When the train rolls downhill and accelerates, it trades potential energy for kinetic. When it climbs a hill and slows down, the trade is in the reverse direction. The total energy is constant, so the train’s trajectory is analogous to a contour line in the energy landscape. However, trains have a third source of energy: coal, diesel, or electricity. By expending fuel, a train can climb a gradient or speed up, freeing itself from its natural free-running trajectory. The total energy still cannot change, but all else is negotiable.
    It is much the same with spacecraft. The combined gravitational fields of the Sun, planets, and other bodies of the Solar System provide potential energy. The speed of the spacecraft corresponds to kinetic energy. And its motive power – be it rocket fuel, ions, or light-pressure – adds a further energy source, which can be switched on or off as required. The path followed by the spacecraft is a kind of contour line in the corresponding energy landscape, and along that path the total energy remains constant. And some types of contour line are surrounded by tubes, corresponding to nearby energy levels.
    Those Victorian railway engineers were also aware that the terrestrial landscape has special features – peaks, valleys, mountain passes – which have a big effect on efficient routes for railway lines, because they constitute a kind of skeleton for the overall geometry of the contours. For instance, near a peak or a valley bottom the contours form closed curves. At peaks, potential energy is locally at a maximum; in a valley, it is at a local minimum. Passes combine features of both, being at a maximum in one direction, but a minimum in another. Similarly, the energy landscape of the Solar System has special features. The most obvious are the planets and moons themselves, which sit at the bottom of gravity wells, like valleys. Equally important, but less visible, are the peaks and passes of theenergy landscape. All these features organise the overall geometry, and with it, the tubes.
    The energy landscape has other attractive features for the tourist, notably Lagrange points. Imagine a system consisting only of the Earth and the Moon. In 1772 Joseph-Louis Lagrange discovered that at any instant there are precisely five places where the gravitational fields of the two bodies, together with centrifugal force, cancel out exactly. Three are in line with both Earth