Next year will be the 20th anniversary of the Oregon Math Summit of 1997. One innovation from that summit was proposed by Keith Devlin, to better integrate the language of calculus and that of film. That made sense to me at the time and I adopted this education reform for my Oregon Curriculum Network on-line resources.
The Lagrangian and Hamiltonian functions are both about anticipating "what's next" given information about what is moving in what direction how fast in a current frame, and assuming conservation laws, of energy of course, but momentum too.
Life coming at us in 3D, so to speak, gives more inertia values than a linear process might process, but who said we're not concurrent under the hood. Certainly language refuses to settle down into a one dimensional meaning half the time.
You can check Youtube for lectures on these two functions, Shrodinger's too. The concept to grasp is "phase space" meaning license to extend the metaphor, of space, to whatever. Mathematics is poetry with special permissions and licenses that make it worth bitcoin in some cases.
Whatever happens next, to stay within the realm of possibility, and not waste time, means staying on a surface in phase place, whereon the energy E remains constant. In other words, keep it real. Don't expect physics to change its mind about the rules, just for you, just for me. Our best strategy is to use the rules to good advantage.
Notice how time as a parameter to the Lagrangian always creeps in as entirely optional, to where some mentors seem to deliberately forget about it, only to have their students put it back in (in red, but then it goes away).
How have I continued Keith Devlin's proposal? Going with "action" in a "time tunnel", I picture film frames as framing action. What is action? In Newtonian units, discounting angular versus linear (say it's all angular): mvd. Per time slice (d/dt), that's mvd/t which is E. E = hf, or h-bar (h/tau) or whatever. I'm just reconciling at the unit / dimension level, not shooting for numerics. Numerics come later. h is the unit of action attributed to Planck.
Then comes power: E/t. Never mind how much total Energy we mean, how quickly did we spend it? That's a delta. "He blew through twenty million in an hour" is an assertion of some power level, for an hour attained. Electricity generators have the same measure: wattage, i.e. energy per time.
Think of what you mean by "rich" and the different kinds of "rich". A busy farm may spend millions a year meaning there's high throughput, high volume. Picture a lot of computers in a rack, mining bitcoin, using energy. Other enterprises may just sit there and hold assets, potential energy, savings. So we have the kinetic rich and the potential rich. The concept of volatility enters in, as if there's nothing fluctuating, how is it measured? Use it or lose it has meaning even in finance.
Couple "energy expense" with "bang for buck" in another dimension: efficiency or "attainment of goals per potential burned". Where quantum entanglement may be involved, one is in theory able to achieve more bang for the same energy, in terms of computation, exploiting features at this level. Some people think that translates into telepathy, other spooky action at a distance. Or would telepathy just add to the chaos? The Internet keeps a more readable record.
We tend to tease apart Work from Energy precisely where the concept of Entropy occurs. We need a concept of "waste", including inevitable waste. Industrialists who want "everything for nothing" because they read about ephemeralization, need to remember no one is boss, praise Bob.
In geek lingo, a virtue is made of "staying lazy" (Subgenius: getting slack) out of a sense that running to catch up is more likely the precursor to lame outcomes. The concept is not the same as "doing nothing" as work is being done, just keeping it precise means attending to the local physics and actual energy conservation laws.
No one made us superheros all of a sudden. The rules of physics still rule.
The Lagrangian and Hamiltonian functions are both about anticipating "what's next" given information about what is moving in what direction how fast in a current frame, and assuming conservation laws, of energy of course, but momentum too.
Life coming at us in 3D, so to speak, gives more inertia values than a linear process might process, but who said we're not concurrent under the hood. Certainly language refuses to settle down into a one dimensional meaning half the time.
You can check Youtube for lectures on these two functions, Shrodinger's too. The concept to grasp is "phase space" meaning license to extend the metaphor, of space, to whatever. Mathematics is poetry with special permissions and licenses that make it worth bitcoin in some cases.
Whatever happens next, to stay within the realm of possibility, and not waste time, means staying on a surface in phase place, whereon the energy E remains constant. In other words, keep it real. Don't expect physics to change its mind about the rules, just for you, just for me. Our best strategy is to use the rules to good advantage.
Notice how time as a parameter to the Lagrangian always creeps in as entirely optional, to where some mentors seem to deliberately forget about it, only to have their students put it back in (in red, but then it goes away).
How have I continued Keith Devlin's proposal? Going with "action" in a "time tunnel", I picture film frames as framing action. What is action? In Newtonian units, discounting angular versus linear (say it's all angular): mvd. Per time slice (d/dt), that's mvd/t which is E. E = hf, or h-bar (h/tau) or whatever. I'm just reconciling at the unit / dimension level, not shooting for numerics. Numerics come later. h is the unit of action attributed to Planck.
Then comes power: E/t. Never mind how much total Energy we mean, how quickly did we spend it? That's a delta. "He blew through twenty million in an hour" is an assertion of some power level, for an hour attained. Electricity generators have the same measure: wattage, i.e. energy per time.
Think of what you mean by "rich" and the different kinds of "rich". A busy farm may spend millions a year meaning there's high throughput, high volume. Picture a lot of computers in a rack, mining bitcoin, using energy. Other enterprises may just sit there and hold assets, potential energy, savings. So we have the kinetic rich and the potential rich. The concept of volatility enters in, as if there's nothing fluctuating, how is it measured? Use it or lose it has meaning even in finance.
Couple "energy expense" with "bang for buck" in another dimension: efficiency or "attainment of goals per potential burned". Where quantum entanglement may be involved, one is in theory able to achieve more bang for the same energy, in terms of computation, exploiting features at this level. Some people think that translates into telepathy, other spooky action at a distance. Or would telepathy just add to the chaos? The Internet keeps a more readable record.
We tend to tease apart Work from Energy precisely where the concept of Entropy occurs. We need a concept of "waste", including inevitable waste. Industrialists who want "everything for nothing" because they read about ephemeralization, need to remember no one is boss, praise Bob.
In geek lingo, a virtue is made of "staying lazy" (Subgenius: getting slack) out of a sense that running to catch up is more likely the precursor to lame outcomes. The concept is not the same as "doing nothing" as work is being done, just keeping it precise means attending to the local physics and actual energy conservation laws.
No one made us superheros all of a sudden. The rules of physics still rule.
