Hello RSS subscribers!
It has been over a year since my last post, and since then, I have started developing my own blog. (I was just a guest author here.) It is called Coffeeshop Physics, and it is a relaxed presentation of physics topics that I think are interesting. The name was inspired by my experiences with Cafe Scientifique, a series of coffeeshop presentations about the sciences— I want to replicate that kind of atmosphere online.
Whereas the Everything Seminar was intended for mathematical audiences, Coffeeshop Physics is for general audiences, much like Cafe Sci. Therefore, I don’t assume that the readers know what a derivative is (relevant for yesterday’s article), but you might find it interesting anyway. Many of the topics that I’m writing about are things that I struggled to understand as an undergrad and even a grad student, the intuition behind the mathematical formalism.
For instance, my favorite article so far is about curved surfaces and gravitation. When I studied general relativity, I could push Christoffel symbols around, but I was frustrated by the fact that I couldn’t visualize the problems that we were working on.
I got a better appreciation for curved surfaces by learning to sew, and after making about a dozen little models, the picture came into focus. Here is a photo of a model of space-time at the surface of the earth, in which we can see that a freefall is a shorter path through space-time than just standing on the ground. It doesn’t have Minkowski structure, so it is not quantitatively accurate (it should open up at the top, not the bottom), but it is a picture to keep at the back of one’s mind.
I’ve also turned the spectrum of resonances in electron-positron collisions into a sound, so that we can hear what it sounds like when the collide, and found a nice demonstration of entropy in a story about a leprechaun tying ribbons on trees in a forest. If you enjoyed the What Killed Madame Curie? detective serial that I started on this blog, I am expanding it into a novel, with links on the site.
Cheers,
— Jim
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in many cases these are very important discussions and must exposed thus are observed by prof dr mircea orasanu and prof drd horia orasanu and followed that Hamilton recast Lagrange’s equations of motion in these more natural variables , positions and momenta, instead of . The ‘s and ‘s are called phase space coordinates.
So phase space is the same identical underlying space as state space, just with a different set of coordinates. Any particular state of the system can be completely specified either by giving all the variables or by giving the values of all the .
Going From State Space to Phase Space
Now, the momenta are the derivatives of the Lagrangian with respect to the velocities, . So, how do we get from a function of positions and velocities to a function of positions and the derivatives of that function with respect to the velocities?
How It’s Done in Thermodynamics
To see how, we’ll briefly review a very similar situation in thermodynamics: recall the expression that naturally arises for incremental energy, say for the gas in a heat engine, is
where is the entropy and is the temperature. But is not a handy variable in real life — temperature is a lot easier to measure! We need an energy-like function whose incremental change is some function of rather than The early thermodynamicists solved this problem by introducing the concept of the free energy,
so that This change of function (and variable) was important: the free energy turns out to be more practically relevant than the total energy, it’s what’s available to do work.
So we’ve transformed from a function to a function (ignoring , which are passive observers here).
Math Note: the Legendre Transform
The change of variables described above is a standard mathematical routine known as the Legendre transform. Here’s the essence of it, for a function of one variable.
Suppose we have a function that is convex, which is math talk for it always curves upwards, meaning is positive. Therefore its slope, we’ll call it
,
is a monotonically increasing function of . For some physics (and math) problems, this slope , rather than the variable is the interesting parameter. To shift the focus to , Legendre introduced a new function, , defined by
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indeed are favorite situations on science as observed prof dr mircea orasasnu and prof drd horia orasanu and followed and therefore we find It is clear that is a scalar function whose value depends on the position of (and the fixed point ), but not on the path taken between and . Thus, if is the origin of our coordinate system, and an arbitrary point whose position vector is , then we have effectively defined a scalar field .
Consider a point that is sufficiently close to that the velocity is constant along . Let be the position vector of relative to . It then follows that (see Section A.18)
(4.88)
The previous equation becomes exact in the limit that . Because is arbitrary (provided that it is sufficiently close to ), the direction of the vector is also arbitrary, which implies that
(4.89)
We, thus, conclude that if the motion of a fluid is irrotational then the associated velocity field can always be expressed as minus the gradient of a scalar function of position, . This scalar function is called the velocity potential, and flow which is derived from such a potential is known as potential flow. Note that the velocity potential is undefined to an arbitrary additive constant.
We have demonstrated that a velocity potential necessarily exists in a fluid whose velocity field is irrotational. Conversely, when a velocity potential exists the flow is necessarily irrotational. This follows because [see Equation (A.176)]
(4.90)
Incidentally, the fluid velocity at any given point in an irrotational fluid is normal to the constant- surface that passes through that point.
If a flow pattern is both irrotational and incompressible then we have
(4.91)
and