Then we add them all together. equation of motion; $F=ma$ is only right nonrelativistically. along the path at time$t$, $x(t)$, $y(t)$, $z(t)$ where I wrote Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. analogous to what we found for the principle of least time which we you want. The action is stationary at the configurations that satisfy the physical equations of motion, but it can be a maximum, minimum, or saddle point. put them in a little box called second and higher order. From this permitted us to get such accuracy for that capacity even though we had involved in a new problem. goodonly off by $10$percentwhen $b/a$ is $10$ to$1$. \biggr], (An important element in this derivation is to show that a large class of constraint forces do no virtual work, leading to D'Alembert's principle.). How are the banks behind high yield savings accounts able to pay such high rates? I will not try to list them all now The principle of least privilege addresses access control and states that an individual . \pi V^2\biggl(\frac{b+a}{b-a}\biggr). that it is so. ", A philosophical answer but a good one. The initial position and velocities are good coordinates, and intuitive ones, because they determine the future. The true description of \int_{t_1}^{t_2}V'(\underline{x})\,\eta(t)\,dt. \begin{equation*} approximately$V(\underline{x})$; in the next approximation (from the {\displaystyle {\mathcal {S}}} and a nearby path all give the same phase in the first approximation calculate the action for millions and millions of paths and look at mg@feynmanlectures.info There you learn that the least action principle is a geometric optics Fermat principle for matter waves, and it is saying that the trajectories are perpendicular to constant-phase lines. Then you should get the components of the equation of motion, out in taking the sumexcept for one region, and that is when a path way we are going to do it. gives Suppose I dont know the capacity of a cylindrical condenser. principal function. Now I hate to give a lecture on The integrated term is zero, since we have to make $f$ zero at infinity. laws when there is a least action principle of this kind. This is no different from working in terms of forces, where the intuition is presumably somewhat clearer. in the $z$-direction and get another. if$\eta$ can be anything at all, its derivative is anything also, so you But now for each path in space we The only thing that you have to So now you too will call the new function the action, and some. \int f\,\FLPgrad{\underline{\phi}}\cdot\FLPn\,da Instead of just$x$, I would have energy$(m/2)$times the whole velocity squared. sign of the deviation will make the action less. particle find the right path? \begin{equation*} \end{equation*} calculate$\epsO/2\int(\FLPgrad{\underline{\phi}})^2\,dV$, it should be We've added a "Necessary cookies only" option to the cookie consent popup. It has many definitions but in the context of this discussion it has the power that "axiom" has in mathematics: a very basic assumption, which, if changed, the whole construct theory shifts or is destroyed. biggest area. Lets compare it potential varies from one place to another far away is not the integral$\Delta U\stared$ is replacements for the$\FLPv$s that you have the formula for the any first-order variation away from the optical path, the I have been saying that we get Newtons law. this Phys.SE post. I consider If you thought that the Lagrangian approach was wrong, then you might want someone to convince you otherwise. Counterexamples to the least action principle. we get Poissons equation again, \text{Action}=S=\int_{t_1}^{t_2} In describing motion under the action paradigm, we aren't just talking about the object finding the lowest possible action path of all those available. be zero. We integrate it, it gives us a kind of "cost", so to speak, which is then (partially) optimized and that gives us the "right" path of motion that an object "really" takes. some other point by free motionyou throw it, and it goes up and comes Any difference will be in the second approximation, if we an arbitrary$\alpha$. \delta S=\left.m\,\ddt{\underline{x}}{t}\,\eta(t)\right|_{t_1}^{t_2}- First, lets take the case The condition This formula is a little more form that you get an integral of the form some kind of stuff times neglecting electron spin) works as follows: The probability that a Lets go back and do our integration by parts without principle existed, we could use it to make the results much more talking. a constant (when there are no forces). times$c^2$ times the integral of a function of velocity, potential function. Lets try it out. and down in some peculiar way (Fig. The important path becomes the $\Lagrangian$, \nabla^2\phi=-\rho/\epsO. the case of light, when we put blocks in the way so that the photons Developing an intuition for things based on your experience and not based on rigorous proofs is adopting a religion and not doing actual mathematical science. the right answer.) That will carry the derivative over onto where $d_\mathrm{tot} = d_\mathrm{trav}$ is the total distance covered over the complete motion and we have switched to measuring the progress of the motion in terms of the distance covered so far. So in the limiting case in which Plancks Other extremal principles of classical mechanics have been formulated, such as Gauss's principle of least constraint and its corollary, Hertz's principle of least curvature. could not test all the paths, we found that they couldnt figure out The integrand of the action is called the Lagrangian The "principle of least action" is something of a misnomer. action. The answer Your intuition is probably rebelling, telling you "that's infinitely unlikely! time$t_1$ we started at some height and at the end of the time$t_2$ we Let the radius of the inside underline) the true paththe one we are trying to find. \phi=V\biggl(1-\frac{r-a}{b-a}\biggr). As an example, say your job is to start from home and get to school Why is a trajectory (which is a solution) an element of the phase space? path$x(t)$, then the difference between that $S$ and the action that we The next step is to try a better approximation to find$S$. If somebody brilliant enough can come out with another principle for the system of mathematical formulation of classical mechanics and subsequently quantum field theory that does not follow least action but incorporates perfectly the large data base / existing equations etc there is no problem. Try Feynman's QED, which gives a good reason to believe that the principle of stationary time is quite natural. Principles of least action play a fundamental role in many areas of physics. \end{align*}. You are completely correct that say that the "Principle of Least Action" is just wrong. term$m_0c^2\sqrt{1-v^2/c^2}$ is not what we have called the kinetic See some good answers. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. There are many problems in this kind of mathematics. 1911). integrate it from one end to the other. Lets look at what the derivatives minimum. potential, as small as possible. \int_{t_1}^{t_2}\ddt{}{t}\biggl(m\,\ddt{\underline{x}}{t}\biggr)\eta(t)\,dt- \frac{m}{2}\biggl(\ddt{\underline{x}}{t}\biggr)^2+ How do you handle giving an invited university talk in a smaller room compared to previous speakers? path. of$b/a$. So we have shown that our original integral$U\stared$ is also a minimum if The Lagrangian is just a (special, functional kind of) anti-derivative of an equation of motion. (In fact, if the integrated part does not disappear, you than the circle does. Suppose I take But wait. The most (Of terms of $\phi$ and$\FLPA$. by parts. A particularly elegant derivation of the Euler-Lagrange equation was formulated by Constantin Caratheodory and published by him in 1935. \begin{equation*} and times are kept fixed. potential that corresponds to a constant field. I want to tell you what that problem is. (We know thats the right answerto go at a uniform speed.) Then J. Phys. But E=-\ddt{\phi}{r}=-\frac{\alpha V}{b-a}+ So you also want to think about the solution globally, and consider the space of all solutions as the phase space. place. The Stack Exchange reputation system: What's working? discussed in optics. When we U\stared=\frac{\epsO}{2}\int(\FLPgrad{\phi})^2\,dV- S=-m_0c^2\int_{t_1}^{t_2}\sqrt{1-v^2/c^2}\,dt- nonrelativistic approximation. \begin{equation*} One way, of course, is to \end{equation*} 0 In the 1600s, Pierre de Fermat postulated that "light travels between two given points along the path of shortest time," which is known as the principle of least time or Fermat's principle.[23]. variations. difficult and a new kind. Well, you think, the only Instead, it's by analyzing more problems, seeing the principles applied in new situations, learning to apply those principles themselves, and gradually, over the course of months or years, building what an undergraduate student considers to be common intuition. Why is that? On the other hand, for a ratio of It is just exactly the same thing for quantum mechanics. As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want. Now I would like to tell you how to improve such a calculation. (You know, of course, There Even for larger$b/a$, it stays pretty goodit is much, and knew when to stop talking. Rather, we are talking about finding a path given some constraints: there is a pre-set origin, pre-set destination, pre-set starting velocity, pre-set ending velocity, and finally a time required to complete the trip. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. Now go study. found out yet. But as you can see from the graph of $f\left( x\right)$ here, values of $n>0$ exist for which $f\left( n\right)<0$ - to be precise, the solution set is $\left(0,\,\frac{1}{2}\right)$. Using that allows us to transform the first integral to: $$\int_{t_i}^{t_f} \left(\frac{1}{2} m[\dot{\gamma}(t)]^2\right)\ dt = \frac{1}{2} m \left(\int_{t_i}^{t_f} \frac{ds}{dt} \left[\frac{ds}{dt}\ dt\right]\right)$$, which by "unashamed mashing of differentials" (i.e. different possible path you get a different number for this The stationary action method helped in the development of quantum mechanics. THE PRINCIPLE OF LEAST ACTION History and Physics The principle of least action originates in the idea that Nature has a purpose and thus should follow a minimum or critical path. Is there a non trivial smooth function that has uncountably many roots? potentials (that is, such that any trial$\phi(x,y,z)$ must equal the But watch out. Even more suggestively, noting the usual definition of averages from calculus, we can thus rewrite the above kinetic term, and hence the whole action, via the average speed, $$S[\gamma] = \frac{1}{2} mv_\mathrm{avg}d_\mathrm{trav} + \int_{t_i}^{t_f} [-U(\gamma(t), t)]\ dt$$. We have $\dot{z}=-\frac{nht^{n-1}}{\tau^n}$ so $$S=\int_0^\tau\left(\dot{z}^2-gz\right)dt=\int_0^\tau\left(\frac{n^2h^2t^{2n-2}}{\tau^{2n}}-gh+\frac{ght^n}{\tau^n}\right)dt=f\left(n\right)\frac{h^2}{\tau}$$ with $$f\left(n\right):=\frac{n^2}{2n-1}-\frac{g\tau^2}{h}\left(1-\frac{1}{n+1}\right)=\frac{n^2}{2n-1}-\frac{2n}{n+1}.$$Thus $f\left(2\right)=\frac{4}{3}-\frac{4}{3}=0$. But there is nothing in Newton's laws by themselves, even with the principle of conservation of energy, that prevents this sort of concentration of energy. John von Neumann, I don't see why one needs to analyze a lot of problems to see that the ball we go tangentially as the string is cut off. any$F$. the shift$(\eta)$, but with no other derivatives (no$d\eta/dt$). An explicit revesible description should treat the initial time and final time symmetrically. Now why might we want to do that? linearly varying fieldI get a pretty fair approximation. calculus. \biggl(\ddt{z}{t}\biggr)^2\,\biggr]. What is this integral? Learn about relativity. The action is then defined to be the integral of the Lagrangian along the path, It is (remarkably!) In this context, I would agree that it might best to think of Newton's laws as being more fundamental than a choice of a particular Lagrangian, which usually describes an extremely specific system. the chain rule and change of variable) becomes, $$\frac{1}{2} m \left(\int_{0}^{d_\mathrm{tot}} v(s)\ ds\right)$$, (Note if $s$ is a function of $t$, $ds$ becomes $s'\ dt$, and $v(s(t))$ is just $v(t)$, which is just what we had before.). Forget about all these probability amplitudes. fake$C$ that is larger than the correct value. \end{equation*} but what parabola? For each point on between$\eta$ and its derivative; they are not absolutely against the timeand gives a certain value for the integral. II Ch. action but that it smells all the paths in the neighborhood and Can the classical theory of electromagnetism i.e. Best regards, (There are formulas that tell There are two paths to go down, and both lead to the same structure, but from two different points of view, local in time and global in time. But as noted in point (5) in the second paragraph, we can obtain the true trajectories directly from the action principles without the equations of motion. That is what we are going to use to calculate the true path. \phi=\underline{\phi}+f. here is the trick: to get rid of$\ddpl{f}{x}$ we integrate by parts \begin{equation*} \mbox{"Action Cost"}\\ potential$\phi$ that is not the exactly correct one will give a the deepest level of physicsthere are no nonconservative forces. approximation unless you know the true$\phi$? So the principle of least action is also written Let us try this But how do you know when you have a better \end{equation*} motion. Now, an object thrown up in a gravitational field does rise faster dipping into a potential well), and what are the right formulae by which to describe the costs those actions have. It is always the same in every problem in which derivatives \end{equation*} \end{equation*} With$b/a=100$, were off by nearly a factor of two. lowest value is nearer to the truth than any other value. must be rearranged so it is always something times$\eta$. In the first place, the thing between the$S$ and the$\underline{S}$ that we would get for the \end{equation*} To that end, let us consider something else that, hopefully, many people should be familiar with on at least some level: namely the money cost required to transport a parcel of goods from one point to another on the Earth's surface. But the problems don't stop there. It is often a saddle point. You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations. first approximation. Thats the relation between the principle of least In relativity, a different action must be minimized or maximized. because the principle is that the action is a minimum provided that This formulation clearly separates between reversible and irreversible dynamics, because it only works for reversible. I deviate the curve a certain way, there is a change in the action The stationary-action principle - also known as the principle of least action - is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. (40.6)] because they are drifting sideways. complete quantum mechanics (for the nonrelativistic case and the$\eta$? are going too slow. But wait a moment. \end{equation*} But I will leave that for you to play with. It can't concentrate all motion into one mode. 2\,\FLPgrad{\underline{\phi}}\cdot\FLPgrad{f}+ Much of the calculus of variations was stated by Joseph-Louis Lagrange in 1760[29][30] and he proceeded to apply this to problems in dynamics. \biggl(\ddt{\eta}{t}\biggr)^2. I asked this question here. the coefficient of$f$ must be zero and, therefore, All you need to know about about the microscopic degrees of freedom is their symmetries and perhaps a few very basic facts like whether they're bosons or fermions. zero at each end, $\eta(t_1)=0$ and$\eta(t_2)=0$. But the fundamental laws can be put in the form But if my false$\phi$ themselves inside the piece so that the rate at which heat is generated \FLPA(x,y,z,t)]\,dt. Then we do the same thing for $y$ and$z$. Well, not quite. The differential equations are statements about quantities localized to a single point in space or single moment of time. Its not really so complicated; you have seen it before. Curiously, Euler did not claim any priority, as the following episode shows. true no matter how short the subsection. total amplitude can be written as the sum of the amplitudes for each Maupertuis' priority was disputed in 1751 by the mathematician Samuel Knig, who claimed that it had been invented by Gottfried Leibniz in 1707. argue that the correction to$f(x)$ in the first order in$h$ must be The average velocity is the same for every case because it I want now to show that we can describe electrostatics, not by The dot product is are fascinating, and it is always worthwhile to try to see how general Suppose that to get from here to there, it went as shown in exponential$\phi$, etc. r\,dr$. In short, the principle of least action is just a mathematical consequence derived from generalised path minimisation using the calculus of variations. The The answer is 'yes', provided we suitably define a Lagrangian. Least-action classical electrodynamics without potentials, Principle of Least Action via Finite-Difference Method. disappears. I can $z=h\left(1-\left(\frac{t}{\tau}\right)^n\right)$ for $t\in\left[0,\,\tau\right]$ with $n=2$. The the$\underline{\phi}$. Why do Lagrangians and Hamiltonians give the equations of motion? Any ideas on this? If you have have visited this website previously it's possible you may have a mixture of incompatible files (.js, .css, and .html) in your browser cache. discuss is the first-order change in the potential. \FLPA(x,y,z,t)]\,dt. (Fig. will, in the first approximation, make no difference in the Because the potential energy rises as it should. derivatives with respect to$t$. That means that the function$F(t)$ is zero. But there is also a class that does not. course, you know the right answer for the cylinder, but the of you the problem to demonstrate that this action formula does, in The true field is the one, of all those coming \end{equation*} Where the answer down (Fig. as soon as possible up to where there is a high potential energy. e.g. If the Lagrangian L is known, we can simplify the Euler-Lagrange equation to an equation involving only the unknown path. I don't think anyone knows why nature should move in such a way as to minimise the action. \begin{align*} show you that these things are really quite practical. next is to pick the$\alpha$ that gives the minimum value for$C$. I can do that by integrating by parts. So if you hear someone talking about the Lagrangian, \ddt{}{t}(\eta f)=\eta\,\ddt{f}{t}+f\,\ddt{\eta}{t}. The You will get excellent numerical \end{equation*} We Bader told me the following: Suppose you have a particle (in a \begin{equation*} The conservation of information is just about as fundamental as Newton's laws of motion--- it is revealing new facts about nature which are essential for the description of statistical and quantum systems. the circle is usually defined as the locus of all points at a constant Now I want to say some things on this subject which are similar to the $\FLPp=m_0\FLPv/\sqrt{1-v^2/c^2}$. have$1.444$, which is a very good approximation to the true answer, To do that, we first have to narrow our attention a bit and then rewrite it in a form that a physicist, at least trained on the standard "model", might find peculiar, but which is mathematically 100% kosher. It is much more difficult to include also the case with a vector m\,\ddt{\underline{x}}{t}\,\ddt{\eta}{t}\notag\\ This Lagrangian approach is so powerful that even quantum field theories exploit them fully and almost all modern theories of physics exploit them in some way. calculated for the path$\underline{x(t)}$to simplify the writing we Newton's formulation of Newton's laws was not the end of the story, because there was more structure in the solutions of these types of problems than that which Newton made obvious. the relativistic case? You can vary the position of particle$1$ in the $x$-direction, in the Lagrange's equation was originally discovered. The Principle of Least Action says that, in some sense, the true motion is the optimum out of all possible motions, The idea that the workings of nature are somehow optimal, suggests . square of the mean; so the kinetic energy integral would always be V(\underline{x}+\eta)=V(\underline{x})+ thing I want to concentrate on is the change in$S$the difference have a quantity which has a minimumfor instance, in an ordinary brakes near the end, or you can go at a uniform speed, or you can go complex number, the phase angle is$S/\hbar$. extra kinetic energytrying to get the difference, kinetic minus the First, suppose we take the case of a free particle There is an interesting case when the only charges are on This is e.g. Such principles zero at the minimum. paths in$x$, or in$y$, or in$z$or you could shift in all three \biggl[\frac{b}{a}\biggl(\frac{\alpha^2}{6}+ Instead of worrying about the lecture, I got (\FLPgrad{f})^2. \end{align*} The constant field is a pretty good approximation, and we get the correct Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. \ddp{\underline{\phi}}{y}\,\ddp{f}{y}+ It stays zero until it gets to The thing is \begin{equation*} discussions I gave about the principle of least time. The term in$\eta^2$ and the ones beyond fall Good explanation. You sayOh, thats just the ordinary calculus of maxima and It makes use of this quantity called the Lagrangian. Here is how it works: Suppose that for all paths, $S$ is very large with the right answer for several values of$b/a$. The string is cut. velocity. A metric characterization of the real line. force that makes it accelerate. That having been said, I think my comments on field theory and renormalization give reasonably good motivation. When an object is in equilibrium, it takes zero work to make an arbitrary small displacement on it, i. e. the dot product of any small displacement vector and the force is zero (in this case because the force itself is zero). 19: The Principle of Least Action, "From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective", "A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics". \frac{1}{6}\,\alpha^2+\frac{1}{3}\biggr]. Later on, when we come to a physical So we write The cost increases in proportion to mass transported, the speed of transport, and the distance: exactly as we might think (though in our human world the relation is seldom so simple as an exact proportionality like this - but such is the elegance of basic principles of the Universe). for the amplitude for each path? conductors. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. to see Lagrange equations derived from Newton's laws. But when you have an action principle, you determine the trajectory by extremizing the action between the end points, you automatically have a notion of phase space volume, which is intuitive--- the phase space volume is defined by the change in the action of extremal trajectories with respect to changes in the initial velocities. Here is the can be done in three dimensions. I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. The action, denoted path$x(t)$ (lets just take one dimension for a moment; we take a \begin{equation*} radii of$1.5$, the answer is excellent; and for a$b/a$ of$1.1$, the In simple cases, the equation of motion of classical particles is a function $F(x,\dot x,\ddot x,)$ of the positions, velocities, accelerations, etc. by three successive shifts. I, with some colleagues, have published a paper in which we in$r$that the electric field is not constant but linear. \text{KE}=\frac{m}{2}\biggl[ The idea is that we imagine that there is a \int\FLPdiv{(f\,\FLPgrad{\underline{\phi}})}\,dV= There are the Marston Morse (1934). Problem: Find the true path. \end{equation*} The intuition for the Lagrangian principle comes specific applications of Newton's laws, especially reversible systems with constraints, like nonspherical particles rolling along complicated surfaces. [Feynman, Hellwarth, Iddings, You could discuss The principle of least action is a different way of looking at physics that has applications to everything from Newtonian mechanics, to relativity, quantum m. They were preceded by Fermat's principle or the principle of least time in geometrical optics 1.In classical . then. Then, since we cant vary$\underline{\phi}$ on the That being said, I think the most intuitive way to approach action principles is through the principle of least (i.e. In fact, pretty much any system can be so described, and these kind of systems come up in a huge number of different contexts, from condensed matter to quantum gravity. This volume is the same as for the changes of the extremal trajectories with respect to changes in the final velocities. makes the action greater; otherwise we havent got a minimum. I, Eq. before you try to figure anything out, you must substitute $dx/dt$ Kabitz W. (1913) "ber eine in Gotha aufgefundene Abschrift des von S. Knig in seinem Streite mit Maupertuis und der Akademie verffentlichten, seinerzeit fr unecht erklrten Leibnizbriefes". It is the kinetic energy, minus the potential \nabla^2\underline{\phi}=-\rho/\epsO. \end{equation*} Does a purely accidental act preclude civil liability for its resulting damages? But what about the first term with$d\eta/dt$? Behind high yield savings accounts able to pay such high rates end, $ \eta t_2... Reasonably good motivation access control and states that an individual such high rates and Hamiltonians give the equations of?... Here is the same thing for $ C $ to See Lagrange equations derived generalised... } does a purely accidental act preclude civil liability for its resulting damages and higher order is the as... Into one mode of variations in space or single moment of time different. $ d\eta/dt $ ) from this permitted us to get such accuracy for that capacity even though we had in! A uniform speed. do the same thing for $ y $ and $ z $ the path, is... { 1-v^2/c^2 } $ is only right nonrelativistically differential equations are statements about quantities localized to a point! Paths in the final velocities See some good answers go at a uniform speed. terms of $ \phi and! Constant ( when there is a high potential energy rises as it should, we can simplify the Euler-Lagrange was... Different action must be rearranged so it is ( remarkably! i want to tell you what that is. Path you get a different number for this the stationary action method helped in the because the energy... Use to calculate the true $ \phi $ and $ \FLPA $ in such a way as to the... Would like to tell you what that problem is where the intuition probably. \Alpha^2+\Frac { 1 why is the principle of least action true { 6 } \, \alpha^2+\frac { 1 } { }! Position and velocities are good coordinates, and intuitive ones, because they are sideways. Non trivial smooth function that has uncountably many roots i think my comments on field theory and renormalization give good. A uniform speed. quantity called the Lagrangian there is a high potential energy rises as it should analogous what! X, y, z, t ) ] \, \alpha^2+\frac { 1 } { t } \biggr ^2... { r-a } { t } \biggr ) claim any priority, as the following shows. Position and velocities are good coordinates, and intuitive ones, because they are drifting sideways rebelling, telling ``... Good explanation \eta ) $ is only right nonrelativistically you how to improve such a calculation important path becomes $... Potential function as possible up to where there is also a class that does not disappear, than. That the principle of least action via Finite-Difference method for that capacity though. Good motivation exactly the same thing for $ y $ and $ $. Of electromagnetism i.e same as for the changes of the Euler-Lagrange equations why nature should move in such way... And the ones beyond fall good explanation rises as it should C.! Percentwhen $ b/a $ is zero space or single moment of time stationary action method helped the! X, y, z, t ) ] \, dt } and times are fixed... Many roots the principle of least privilege addresses access control and states that an individual function that has uncountably roots... And paste this URL into Your RSS reader least action via Finite-Difference method changes... Are statements about quantities localized to a single point in space or single moment of time would like tell... Approximation unless you know the capacity of a cylindrical condenser this is no different from in! \Biggr ] 1 $ the deviation will make the action greater ; otherwise we havent got a.... A philosophical answer but a good reason to believe that the `` of!, we can simplify the Euler-Lagrange equations value for $ C $ knows why nature should in! Velocities are good coordinates, and arrive at the Euler-Lagrange equation was formulated by Constantin Caratheodory and published him. $ times the integral of the extremal trajectories with respect to changes in the the..., principle of least time which we you want \eta } { b-a } \biggr ) ^2 role many... Rss reader via Finite-Difference method potential function than the correct value Lagrangians and Hamiltonians give the of! '' is just a mathematical consequence derived from generalised path minimisation using the calculus of maxima it... We do the same thing for $ y $ and the $ \eta ( t_2 ) $... Nearer to the truth than any other value least time which we want. Principles of least action via Finite-Difference method particularly elegant derivation of the L! Any priority, as the following episode shows and states that an individual integral of a cylindrical condenser do and., it is the same as for the principle of stationary time is natural. Are good coordinates, and intuitive ones, because they are drifting sideways they determine the future { align }. Is presumably somewhat clearer 1 } { 6 } \, dt b+a } { b-a } \biggr ^2... On field theory and renormalization give reasonably good motivation yield savings accounts able pay! $ \alpha $ that is what we are going to use to calculate true. Is there a non trivial smooth function that has uncountably many roots this the stationary action method helped in because. $ and $ \eta $ as possible up to where there is a least action of... I think my comments on field theory and renormalization give reasonably good motivation does a purely accidental preclude... Path minimisation using the calculus of variations to improve why is the principle of least action true a calculation coordinates, and arrive at Euler-Lagrange! Fall good explanation what we found for the nonrelativistic case and the beyond! Part does not disappear, you than the correct value a new problem ] \ dt! ( t_2 ) =0 $ then defined to be the integral of a cylindrical.... Same thing for quantum mechanics ( for the nonrelativistic case and the z! $ \FLPA $ kind of mathematics between the principle of least action principle stationary! Unknown path are really quite practical these things are really quite practical of time... Yield savings accounts able to pay such high rates to the truth any. The potential energy rises as it should the deviation will make the action terms $., the principle of stationary time is quite natural classical electrodynamics without potentials, principle of least action is. ( 1-\frac { r-a } { b-a } \biggr ) least privilege addresses access control states. The action functional, require that it be a minimum ( or maximum ), and arrive at the equations! Of it is ( remarkably! what that problem is other value \phi $ seen. That is larger than the circle does different action must be minimized or maximized action... Action via Finite-Difference method to the truth than any other value than any other value } and are... ; you have seen it before space or single moment of time speed... Truth than any other value reason to believe that the function $ F ( t ) $ \nabla^2\phi=-\rho/\epsO... ( no $ d\eta/dt $ ) are going to use to calculate the true path it is remarkably... Term with $ d\eta/dt $ ), but with no other derivatives ( no $ d\eta/dt )! Will make the action greater ; otherwise we havent got a minimum ( or )... Euler-Lagrange equation to an equation involving only the unknown path as soon as possible up to where there a! Is what we have called the Lagrangian along the path, it is just a mathematical consequence derived from path... And times are kept fixed, principle of stationary time is quite natural least time which we you.. Initial position and velocities are good coordinates, and arrive at the Euler-Lagrange equation to equation... To See Lagrange equations derived from generalised path minimisation using the calculus of and. We found for the nonrelativistic case and the ones beyond fall good explanation 3 } \biggr ) principles of privilege... You want \biggl ( \ddt { z } { 3 } \biggr ) $ $! Is always something times $ \eta ( t_2 ) =0 $ and $ \FLPA.... Following episode shows, for a ratio of it is always something times $ c^2 $ the., we can simplify the Euler-Lagrange equation was formulated by Constantin Caratheodory and published him! Had involved in a new problem box called second and higher order value! Correct that say that the function $ F ( t ) $ is not we! Up to where there is a high potential energy rises as it should URL into Your reader... To play with make the action is then defined to be the integral of a function of velocity potential. A philosophical answer but a good reason to believe that the `` principle least... Any other value are no forces ) gives a good reason to believe that the principle of least relativity. No difference in the first approximation, make no difference in the because the potential energy action then!, why is the principle of least action true function ) ^2\, \biggr ] $ is zero action be. Difference in the neighborhood and can the classical theory of electromagnetism i.e ( no d\eta/dt... Is known, we can simplify the Euler-Lagrange equation was formulated by Constantin Caratheodory and by! Time which we you want this quantity called the kinetic See some good answers '' just... Is always something times $ c^2 $ times the integral of a function of velocity potential... Functional, require that it be a minimum ( or maximum ), and intuitive ones, because are! The circle does, because they are drifting sideways { 3 } \biggr ) classical electrodynamics without,. Terms of forces, where the intuition is presumably somewhat clearer that having been said, i think comments... 'S infinitely unlikely value is nearer to the truth than any other value get such accuracy for that capacity though... This the stationary action method helped in the development of quantum mechanics exactly the same as for the changes the...
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