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Intuitive concept of Action?

  1. Apr 23, 2014 #1
    I am having problems getting any intuitive idea on the concept of Action (as in the Principle of the Least....) when the Lagrangian is T-V. Every site I visit starts with the mathematical formulation, which is fine, and I understand that not every physical concept comes with an intuitive idea, but most concepts outside of quantum mechanics (and even some of them in quantum physics) have their basis in a historical development that, even if the concepts have evolved into something more abstract, at least gives some conceptual anchor. For example, momentum is an operator and all that, but it began as a more tangible concept ("oomph") that one can at least refer to before heading off into calculations. Energy has always been abstract, but even "caloric fluid" was better than nothing. Time doesn't have an operator as such, but it is like a US Supreme Court judge said: that he couldn't define hard-core porno, but he knew it when he saw it. Often the units give a clue; sometimes a graph or analogy, or occasionally even everyday usage. And so forth. But Action...? It doesn't seem to correspond to the everyday use of the term, and the units of Energy integrated over time, joules-seconds, don't seem to correspond to anything except Planck's constant (which, as a proportionality constant, doesn't give much to cling to) or momentum over distance (also a combination that corresponds to no good intuition), the area under a curve of energy versus time doesn't help, and historically Action seems to have arisen as a purely more convenient method for classical problems (and extended to less classical problems) than the cumbersome contortions of the combination of conservation of energy and momentum. The analogy of the principle of least action with the fact that "lightning follows the path of least resistance" does not seem to be a very good analogy. And so forth. So, can someone forgive my wordiness and supply me with an intuitive idea on which to hang my conceptual hat for Action, without simply stating the equations it works for or how one can use Lagrangian mechanics to derive other (more intuitive) physical principles, or how nice it is in the context of Noether's Theorem, etc.? Thanks.
     
  2. jcsd
  3. Apr 23, 2014 #2

    UltrafastPED

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    Try this: http://www.cs.helsinki.fi/u/ldaniel/mm_cn/FeynmanPrincipleofLeastAction.pdf [Broken]

    It's how Richard Feynman first grasped the concept.
     
    Last edited by a moderator: May 6, 2017
  4. Apr 23, 2014 #3
    Thanks, UltrafastPED. That helps some. Feynman presents a very nice explanation, as always. The approach is to show how the principle of least action makes sense. Outside of the context of this principle, however, I am not sure whether action has any meaning.
     
  5. Apr 24, 2014 #4

    UltrafastPED

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    This is the principle context in which it is used: the principle of least action; Lagrangians & Hamiltonians are the result of this principle, so action it is very important.
     
  6. Apr 24, 2014 #5
    John Baez explains it here (D'Alambert's principle) towards the end:

    http://math.ucr.edu/home/baez/classical/cm05week01.pdf

    I'd explain it slightly differently, myself, but I'm too lazy to dig it out of my memory banks at the moment.

    Another part of it, which Baez touches on there is that you can think of the Lagrangian as a sort of "cost" per unit time and the action is the total cost over time, such that nature chooses the path of least (or stationary) total cost. The idea is that by picking the right cost function, you can guide things through configuration space to follow whatever sort of trajectories you want, which might be ones that imply Newton's laws or some other laws if you picked a different Lagrangian.

    It helps to have a good idea of the meaning of a configuration space, not just the action. An example that you can visualize is the configuration space of a robot arm with one joint, which you can picture as a torus. Each position of the robot arm corresponds to some point on the torus and some movement of the robot arm corresponds to tracing out a path through the torus. So, there's a 1-1 correspondence between points on the torus and possible configurations of the system. The Lagrangian is actually a function of velocities as well, so it's a function on velocity space or what mathematicians call the tangent bundle, which is the space of all velocity vectors, which you can sort of picture as a all the different tangent planes to the torus. The Lagrangian is just saying some of those configurations and velocities are better than others, with the best ones being those corresponding to Newton's laws (for conservative systems) if you choose it to be T-U.
     
  7. Apr 24, 2014 #6
    Thanks very much, homeomorphic. I have long been a fan of Baez's explanations. The ones in that link helped me a lot, and your take on it was also very helpful.
     
  8. Apr 24, 2014 #7
    One last question on this topic, if I may. In the excerpt from Feynman, he mentions that the angle for the phase shift angle is proportional to the action divided by Plank's constant. (Precisely, he states (p. 19-9)"Our action integral tells us what the amplitude for a single path ought to be. The amplitude is proportional to some constant times eiS/[itex]\hbar[/itex], where S is the action for that path......")

    But on the other hand, http://en.wikipedia.org/wiki/Quantum_gate#Phase_shift_gates
    states that "Phase shift gates....... map |1> to e|0>...The probability of measuring a |0> or |1> is unchanged after applying this gate, however it modifies the phase of the quantum state."

    Since this is defined without restrictions on the phase angle in question (whereby the probability amplitudes change but the probabilities do not), then it would seem that this was saying that a shift in phase angle could not affect the probability of the path.... but according to Feynman it does. What am I reading wrong here?
     
  9. Apr 24, 2014 #8
    Amplitude for a SINGLE path. That's one of the keys, I think. The question is what happens when you sum over all possible paths. The phase angles determine how the amplitudes of the different paths interfere with each other, so they do matter for the results of experiments, even if they don't change the amplitude for a single path by itself. In the end, it's the square magnitude that will be the probability, but those magnitudes result from adding up (integrating) the amplitudes of all the different paths, not just one.
     
  10. Apr 24, 2014 #9
    Yes and if a path is at a minimum of the action, all the neighboring paths will have essentially the same action to 1st order which means they have the same phase and those paths add constructively. Other paths that are not close to a minimum of the action will have widely varying phases when compared with neighbor paths and interfere destructively with them. That's why only paths with minimum action survive the transition from quantum mechanics to classic mechanics. That's the path integral formalism explanation for the minimum action principle. It is in a way very similar to Huygens principle that's used to explain why light follows a path of minimum optical length. The optical length plays the role of the action within the theory of optics as a wave.
     
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