Well, now I knew that one day or another I did not cut off and it happened this weekend. I was preparing to question a long time, collecting educational explanations, demonstrations more or less comprehensible and historical exegesis on the subject. This weekend, my numberOne asked me the killer question: "where it leaves the law with sinuses on the refraction of light?". I am therefore the answer today and took the opportunity to present the "principle of least action", a little known law of physics also fascinating that explains both the laws of optics that of classical mechanics, relativistic or quantum, no less! Ticket X-rated course, but if you're allergic to math you can still follow this post (hopefully) simply by skipping paragraphs reserved furious. Go! A little reflection to begin.
been known or almost always (since Heron of Alexandria at least) one ray of light is reflected from a surface by an angle of reflection (i2 in the figure) equal to its angle of incidence (i1 ), just like a billiard ball bounces off a surface perpendicular to it.
The radius he always chooses the shortest path connecting two points? You have to see, but it could explain why it moves in a straight line between two obstacles. There would be a universal law?
How does the lifeguard is not a light?
How does the lifeguard is not a light?
When a stick is dipped in water it look like broken image (source here) In the seventeenth century, Scientists struggled to find a rigorous explanation of this phenomenon of refraction. We suspected that the speed of light was different in the air and water and Pierre de Fermat made up his mind to demonstrate this by itself on the assumption that light always follows the trajectory faster. To understand the problem, it usually takes the following image: imagine a lifeguard on the beach, which sees a person in the water drowning. What path should it take to rescue her as soon as possible, knowing that it is faster when running on the beach only swim in the sea? 
 (For furious only) In triangles and OFH OFK, was HF = OF sin i and sin r = OF OK deduced that HF / OK = sin i / sin r = V air / V water If there XY} {travel time from X to Y was so {HF} = {} OK Moreover AL> AO and LF> HF ( ALO as triangles and rectangles are HLF) therefore AF> AO + HF we can rewrite in terms of journey times {AF}> {AO} + {HF} Similarly {FB}> {KB} Adding these two inequalities we obtain {AF} {FB} +> {AO} + {HF}} + {KB But we have seen that {HF} = {OK} {this inequality is then written AF} {FB} +> {AO} +} + {{OK} KB therefore {AF} {FB} +> {AO} + {OB} which is what we wanted to show ... |
When Fermat succeeded in demonstrating this result in 1661 from the single principle of "least time" of the light he is fascinated: "The fruit of my work has been most extraordinary, most unexpected and that was the happiest ever. " He feels that there is a law of physics very general he calls "principle of natural economy ", that" nature always acts by the shortest ways "*. The story would prove him right beyond anything he could imagine ...
From the perspective in mechanics: the principle of least action
Fermat's approach is obviously disturbing to the rational mind: how light she know that this path is the shortest and not another? Its principle also appears to violate principle of causality: how the department knows it the way to go from its final destination? minds thinking of the time saw in this principle very simple and a little magical expression of a certain perfection God who ruled all laws of physics. As Leibniz wrote eg
"For given that the bill is the whole world that is more perfect and has been run by the creator wiser, it does absolutely nothing in the world, which manifests itself in a certain method of maximum or minimum, so there can be no doubt that all effects of the world can be easily deduced as final causes, through the method of maximum and minima, as efficient causes themselves "(quoted by Jacques Bouveresse here).
other words, Leibniz defended a teleological argument, that is to say that could explain natural phenomena rather than from their causes (forces suffered etc..) But from their purpose. The principle is that the result must be the "best" possible, ie a maximum or minimum. This explanation upside Voltaire brought out of its hinges and it inspired the famous tirade Pangloss in Candide (image source: here ):
"Pangloss taught metaphysical -theologo-cosmolonigologie. He proved admirably that there is no effect without cause, and that in this best of possible worlds, the castle of the Baron was the most beautiful castles and Madam best possible baronesses. "There is evidence, he said, that things can not be otherwise: for all being created for a purpose, everything is necessarily for the best end. Note that noses were made to wear spectacles, as did our glasses. The legs are visibly designed for stockings, and we have stockings. " (Candide chap1) "Now here this principle, so wise, so worthy of the supreme being: when a change occurs in nature, the amount of action used to change is always the smallest it is possible. "
Rather than explain Newton as the movement of a body describing every moment its variations of speed and position with the laws of such acceleration = force / mass, Maupertuis conjectured that one can directly find its global trajectory once one knows the points of departure and arrival. His approach is bold: among all conceivable paths between these two points, one that chooses the nature is one that maximizes (or minimizes) the famous "action". A small scheme is better than a long explanation:
What was this "action" that nature strives to extremal? For Maupertuis was accumulating momentum of the particle along its trajectory. Its side Euler, Maupertuis who helped to formalize the theory itself, rather bent on the accumulation of potential energy (that which gives the altitude for instance), all bodies seeking to voluntarily put themselves in the state of the lowest potential energy. It was Lagrange which put everyone agreed in 1788 with a general formulation still in force today. In the particular case of a body subjected to a potential (gravity, electromagnetic forces, for example), the amount and the actual path minimizes time interval, is the average the difference between kinetic energy (T) and potential energy (U).
| For furious action Lagrange written:  S = action between points a (at time t a) and b (in time t b ) £ = Lagrangian system, depending on the position, velocity and time For Maupertuis action rather spelled: T = kinetic energy (½ mv ²) U = potential energy (dependent on position)  If the total energy is constant and can be written as E = T + U, the two approaches coincide. One can indeed write that TU-2T = E = E + E = TU-2U Since E is a constant, it is the same look for the extremum of UT (like the Lagrangian), 2T (as Maupertuis) or U (as Euler)! can also demonstrate the principle of Fermat, starting from the action of Lagrange, but it is much more complicated! |
Not only does it work, but he showed more perfect equivalence between the principle of least action and reformulated all laws (causal) of Newton. (See demo here example). Fortiche this Lagrange. But then, I do not know about you, but me the story of minimizing the "difference between kinetic and potential energy" does not speak to me at all. As much as I can see what is the sum of these energies, the total energy (E)-which keeps both the physical sense of their difference (TU) escapes me. Besides, I do not have to be alone, because this principle of least action is often seen as a kind of mathematical trick more or less abstruse. By dint of searching, I finally found one yet interpretation physics at each of the three expressions of this principle of least action as expressed by Lagrange, Euler and Maupertuis. You choose the one that speaks to you most!
A reluctance to change gears
expression Maupertuis based momentum finally seems easiest to understand: to change the momentum of a body should be a strength, it seems logical that the trajectory "chooses" to be one that avoids the most to suffer the action of these forces. I sees a similar path of least resistance, resistance to change etc.. I request advance the reader's indulgence if I say a donkey, but I also see why the golf ball has the unfortunate tendency to bypass the hole rather than to go and come out eventually.
Why difference between kinetic and potential energy?
Leave a ball vertically and let it fall into your hand. It rises very quickly, slows, stops, then falls accelerating. If we measure the evolution of its altitude and time, we obtain a parabolic curve that looks like this:
course these considerations do not have much interest if we remain in case as simple. But in real life, we often deal with forms of fields (including electromagnetic) horribly complicated that we can possibly know the approximate value in every point in space but certainly not the whole equation. Therefore impossible to apply Newton's laws if we do not know the field equation! For cons, the previous approach of seeking optimal trajectory is super simple: just simulate computationally different trajectories for fixed and calculate the total amount for each UT. accumulated. The actual path will be that which minimizes the quantity: easy!
geodesics in space?
Returning for a moment on the approach of Fermat. Light minimizes the travel time in the medium, ie it follows a geodesic : a straight line when the index of the medium is constant, a curve when it changes. As the travel time is proportional to the index, the path actually traveled by light is that which minimizes the average of the index along the trajectory. For light, a high index has exactly the same effect as a greater distance, ie the index variations indicate a deformation of the metric space for the beam.
In the case of a body subjected to forces derived from a potential (gravity, electric field or magnetic ...), we saw that the principle of least action can be expressed as the minimization of average potential energy along the real trajectory (this is the approach of Euler). So there is a formal analogy between the index of the medium for light and potential energy U to the body in motion. We can consider the trajectories of the coup as physical geodesic space deformed by a potential energy field. In other words, according to the principle of least action sauce Euler, there is equivalence between the motion of a particle subject to a potential independent of time in a Euclidean space and the trajectory of a free particle in a curved space. According Basdevant, Einstein would have had this idea in mind in 1908 when he built his theory of general relativity, a theory which rightly concludes that gravity bends the light path in the same way that a change in refractive index.
However, Jacques Leon with whom I had the pleasure to discuss this issue told me the limits of this analogy in the case of physical bodies. The intensity of the potential energy for an object depends not only on its position but also its mass. So the faux "gravitational refractive index 'is not an intrinsic property of space as it depends on the mass of moving bodies. That problem does not arise with the light because photons are massless: the trajectories of light are real geodesic space-time so that the trajectories of massive bodies are not really.
A unifying principle fascinating
Since Hamilton Jacobi and gave it its modern formulation, the principle of least action has found applications in all areas. The Lagrangian formalism is very convenient as it applies any coordinate system (spherical, cylindrical or composite). And when the motion is constrained by barriers or links between elements of a system is a breeze to integrate these constraints in the Lagrange equations (as multipliers Lagrange for those interested).
to my knowledge there is no area of physics whose evolution can be described as a maximization or minimization of something: the shape of soap bubbles, the cells of honeycomb , spirals of nature etc.. can always be explained by the maximization of some function of the system. The principle of least action is the only theory which has so far never been found wanting. Better still, it can find almost all the laws of physics! Thanks to him, Emmy Noether showed that behind every symmetry of the laws of nature lies a conservation law of a certain physical magnitude. David Hilbert found the equations of general relativity using this principle. Finally, this strange principle proved perfectly compatible with the quirks of quantum physics. At that point Richard Feynman has done with his concept of "path integral", the basis of its quantum electrodynamics, a theory which he says allows "to describe all the phenomena of the physical world, excluding gravitational effects.
short, 300 years since the principle of least action has ceased to inspire all the history of physics and I'm fascinated by the amount and strength of its applications from a simple statement, even simplistic. Is it because its power of unification continues to scare us a bit that is not taught at school or in preparation? At a time when we deplore the lack of interest young people in science, we would lose nothing by showing them this little jewel of the laws of nature.
Sources:
R Feynman: The Principle of least action, special reading (pdf )
The Wikipedia article on the subject
An excellent summary (ppt format ) on the subject
Florence Robine conference in 2007 (pdf )
Related posts:
Noether's theorem: Swiss Army knife of physics
classified Ticket (power) X why we always find the Fibonacci sequence in heart of sunflowers, pine cones etc. ...
0 comments:
Post a Comment