Walter Thirring
- Classical Dynamical Systems
In physics, space and time are defined by the way yardsticks and clocks behave, which in turn is determined by the equations of motion. It is this reasoning that gives a concrete significance to the mathematical structure of our formalism.
Walter Thirring
- Classical Dynamical Systems
The theory of special relativity replaces the Galilean group with the Poincare group. This makes the equations of motion of a particle in an external field only slightly more complicated. However, physics at high velocities looks quite different from its nonrelativistic limit.
Walter Thirring
- Classical Dynamical Systems
The study of free particles is the foundation of kinematics, and can be used as a basis of comparison for realistic systems. The canonical flow for free particles is linear.
Walter Thirring
- Classical Dynamical Systems
A 2-form is canonically defined on the cotangent bundle of a manifold. Diffeomorphisms leaving this 2-form invariant are called canonical transformations.
Walter Thirring
- Classical Dynamical Systems
The intuitive picture of a smooth surface becomes analytic with the concept of a manifold. On the small scale a manifold looks like a Euclidean space, so that infinitesimal operations like differentiation may be defined on it.
Richard Feynman
- Lectures on Physics - Volume II
When the wire is pushed to the left, we would expect that the magnet must feel a push to the right. (Otherwise we could put the whole thing on a wagon and have a propulsion system that didn't conserve momentum!) Although the force is too small to make movement of the bar magnet visible, a more sensitively supported magnet, like a compass needle, will show the movement.
Richard Feynman
- Lectures on Physics - Volume II
Equation (1.9) tells us that for a fixed current through the wire the circulation of B is the same for any curve that surrounds the wire. For curves - say circles - that are farther away from the wire, the circumference is larger, so the tangential component of B must decrease. You can see that we would, in fact, expect B to decrease linearly with the distance from a long straight wire.
Richard Feynman
- Lectures on Physics - Volume II
We have said that the electrical force, like a gravitational force, decreases inversely as the square of the distance between charges. This relationship is called Coulomb's law. But it is not precisely true when charges are moving - the electrical forces depend also on the motions of the charges in a complicated way. One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect. That is why we call the subject "electromagnetism."
Richard Feynman
- Lectures on Physics - Volume II
Consider a force like gravitation which varies predominantly inversely as the square of the distance, but which is about a billion-billion-billion-billion times stronger. And with another difference. There are two kinds of "matter," which we can call positive and negative. Like kinds repel and unlike kinds attract - unlike gravity where there is only attraction. What would happen?
Norton Juster
- The Phantom Tollbooth
People laughed less and grumbled more, sang less and shouted more, and the sounds they made grew louder and uglier. It became difficult to hear even the birds or the breeze, and soon everyone stopped listening for them.
Norton Juster
- The Phantom Tollbooth
So each one of you agrees to disagree with whatever the other one agrees with, but if you both disagree with the same thing, aren't you really in agreement?
Norton Juster
- The Phantom Tollbooth
"That was all many years ago," she continued; "but they never appointed a new Which, and that explains why today people use as many words as they can and think themselves very wise for doing so. For always remember that while it is wrong to use too few, it is often far worse to use too many.
Norton Juster
- The Phantom Tollbooth
"Do you think it will rain?" "I thought you were the Weather Man," said Milo, very confused. "Oh no," said the little man, "I'm the Whether Man, not the Weather Man, for after all it's more important to know whether there will be weather than what the weather will be."
Norton Juster
- The Phantom Tollbooth
When he was in school he longed to be out, and when he was out he longed to be in. On the way he thought about coming home, and coming home he thought about going. Wherever he was he wished he were somewhere else, and when he got there he wondered why he'd even bothered.
Norton Juster
- The Phantom Tollbooth
"I don't like to get wet," moaned the unhappy bug, and he shuddered at the thought. "Neither do they," said Canby sadly. "That's what keeps them here. But I wouldn't worry too much about it, for you can swim all day in the Sea of Knowledge and still come out completely dry. Most people do. But you must excuse me now. I have to greet the new arrivals. As you know, I'm as friendly as can be."
Norton Juster
- The Phantom Tollbooth
We're right here on this very spot. Besides, being lost is never a matter of not knowing where you are; it's a matter of not knowing where you aren't - and I don't care at all about where I'm not.
Norton Juster
- The Phantom Tollbooth
"You may not see it now," said the Princess of Pure Reason, looking knowingly at Milo's puzzled face, "but whatever we learn has a purpose and whatever we do affects everything and everyone else, if even in the tiniest way."
Norton Juster
- The Phantom Tollbooth
"I never knew words could be so confusing," Milo said to Tock as he bent down to scratch the dog's ear. "Only when you use a lot to say a little," answered Tock. Milo thought this was quite the wisest thing he'd heard all day.
Norton Juster
- The Phantom Tollbooth
You see... it's really quite strenuous doing nothing all day, so once a week we take a holiday and go nowhere, which was just where we were going when you came along. Would you care to join us?
Norton Juster
- The Phantom Tollbooth
The Mathemagician nodded knowingly and stroked his chin several times. "You'll find," he remarked gently, "that the only thing you can do easily is be wrong, and that's hardly worth the effort."
