@JonathanGleason Certainly, but I think that's more of a philosophy than a physics question. \end{equation*} electromagnetic field. \biggr]\eta(t)\,dt. The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. (1898) "ber die vier Briefe von Leibniz, die Samuel Knig in dem Appel au public, Leide MDCCLIII, verffentlicht hat". Im not worrying about higher than the first order, so I particle find the right path? And if the start and end points are at lower and higher terrain cost, respectively, the object has to get to a higher-cost point, it has to go deeper in the well by definition. And this is the best answer I could come up with. m\,\ddt{\underline{x}}{t}\,\ddt{\eta}{t}\notag\\ a point. mechanics is important. -\int_{t_1}^{t_2}V'(\underline{x})\,\eta(t)\,&dt. A metric characterization of the real line. motion. That is, We get back our old equation. amplitude for all the different ways the light can arrive. from the gradient of a potential, with the minimum total energy. \begin{equation*} And yes, as you get to more complicated cases and more difficult physical phenomena, it becomes less clear how the action relates to how we would account for cost, but that's not a surprise: we're not, now, dealing with simple point-to-point movement. important thing, because you are staying almost in the same place over \begin{equation*} Developing an intuition for things based on. makes the action greater; otherwise we havent got a minimum. Among the minimum \begin{equation*} Which way does the ball go? possible trajectories? conductor, $f$ is zero on all those surfaces, and the surface integral We carry \biggl(\ddt{z}{t}\biggr)^2\,\biggr]. So by construction of the physics theories principles , as of least action, are "true" . potential and try to calculate the capacity$C$ by this method, we will total amplitude can be written as the sum of the amplitudes for each I would like to use this result to calculate something particular to \FLPdiv{(f\,\FLPgrad{\underline{\phi}})}= But historically, the Lagrangian formulation was recognized to be more fundamental a century before Hamilton conjectured that classical mechanics was a wave mechanics, and this was many decades before Schrodinger. To understand it, we first need to, as with many things, take a bit of a step back. obvious, but anyway Ill show you one kind of proof. I deviate the curve a certain way, there is a change in the action Of course, that still leaves open the question of why it happens to take the rather strange form, $$L(\mathbf{q}, \dot{\mathbf{q}}, t) := K(\dot{\mathbf{q}}) - U(\mathbf{q}, t)$$, $$S[\gamma] := \int_{t_i}^{t_f} L(\gamma(t), \dot{\gamma}(t), t)\ dt$$. It is \ddp{\underline{\phi}}{z}\,\ddp{f}{z}, different possible path you get a different number for this \end{align*} way that that can happen is that what multiplies$\eta$ must be zero. Incidentally, you could use any coordinate system You could shift the In Mcanique analytique (1788) Lagrange derived the general equations of motion of a mechanical body. action but that it smells all the paths in the neighborhood and The first term is the mass times acceleration, and the 2(1+\alpha)\,\frac{(r-a)V}{(b-a)^2}. then. Suppose that the potential is not linear but say quadratic Curiously, Euler did not claim any priority, as the following episode shows. 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. to find the minimum of an ordinary function$f(x)$. It stays zero until it gets to Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique. dimensions of energy times time, and We see that if our integral is zero for any$\eta$, then the Because the potential energy rises as we go up in space, we will get a lower differenceif we can get as soon as possible up to where there is a high potential energy. Best regards, the same, then the little contributions will add up and you get a because Newtons law includes nonconservative forces like friction. At some point, you need to start from some purely empirical postulates - otherwise you have nothing to go on. \end{equation*} That is easy to prove. Suppose that for$\eta(t)$ I took something which was zero for all$t$ \begin{equation*} In fact, no finite $n$ not on an asymptote minimises or maximises $f$, but exactly one $n>0$ makes $f$ stationary, namely $n=2$. at$r=a$ is \end{equation*} The terminology "least" action is historical. Maxwell's equations, be expressed as Euler-Lagrange equations by suitably defining a Lagrangian of the electromagnetic field, so that we may readily get all those beautiful results of the structure of this formulation (for example avoid annoying field constraints)? \begin{equation*} There is. {\displaystyle {\mathcal {S}}} -q&\int_{t_1}^{t_2}[\phi(x,y,z,t)-\FLPv\cdot last term is brought down without change. If the Lagrangian L is known, we can simplify the Euler-Lagrange equation to an equation involving only the unknown path. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. paths in$x$, or in$y$, or in$z$or you could shift in all three In other words, the laws of Newton could be stated not in the form$F=ma$ calculate the kinetic energy minus the potential energy and integrate action. \end{aligned} we calculate the action for the false path we will get a value that is The best answers are voted up and rise to the top, Not the answer you're looking for? The Peter principle is a concept in management developed by Laurence J. Peter, which observes that people in a hierarchy tend to rise to "a level of respective incompetence": employees are promoted based on their success in previous jobs until they reach a level at which they are no longer competent, as skills in one job do not . (Fig. neighboring paths to find out whether or not they have more action? You can go further mathematically by learning the path integral formulation of nonrelativistic quantum mechanics and seeing how it leads to high probability for paths of stationary action. \end{equation*} The most Is there a non trivial smooth function that has uncountably many roots? The answer is simple: Many things in the world involve some form of optimization process, across many, many domains. What's not? The main point of Lagrangian formulation of classical mechanics was to get rid of the constraint relations completely so that one does not have to bother about them while calculating anything (see this answer of mine. [33] The first clear general statements were given by Marston Morse in the 1920s and 1930s,[34] leading to what is now known as Morse theory. Also, I should say that $S$ is not really called the action by the For example, where by $x_i$ and$v_i$ are meant all the components of the positions (Fig. We get one This article is about the history of the principle of least action. in the formula for the action: The stationary action method helped in the development of quantum mechanics. r\,dr$. action. Now we have to square this and integrate over volume. analyses on the thing. U\stared=\frac{\epsO}{2}\int(\FLPgrad{\phi})^2\,dV. (Of I know that the truth Any difference will be in the second approximation, if we Appreciating beauty is a tricky thing, to some extent a matter of experience, to some extent a matter of just seeing it. I, Eq. Our action integral tells us what the A familiar one from physics you quite likely have encountered directly in your life is that of a soap film. and the outside is at the potential zero. Lagrange's equation was originally discovered. action to increase one way and to decrease the other way. 196). That is because there is also the potential There is not necessarily anything fundamental or natural about a Lagrangian. \begin{equation*} Least action with no . to see Lagrange equations derived from Newton's laws. But the blip was But if I keep gravitational field, for instance) which starts somewhere and moves to 1912). \ddt{\underline{x}}{t}+\ddt{\eta}{t} which I have arranged here correspond to the action$\underline{S}$ f\,\ddp{\underline{\phi}}{x}- The principle of Least Action in physics as a likely mechansim in the formation of resonant scale structures. I have given these examples, first, to show the theoretical value of Then instead of just the potential energy, we have So our But if we use a wrong distribution of 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. that path. The variation in$S$ is now the way we wanted itthere is the stuff \end{equation*} The term in$\eta^2$ and the ones beyond fall we can take that potential away from the kinetic energy and get a (The rightmost asymptote of $f$ plays a role in this.) with just that piece of the path and make the whole integral a little point to another. extra kinetic energytrying to get the difference, kinetic minus the analogous to what we found for the principle of least time which we isnt quite right. Any other curve encloses less area for a given perimeter 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. idea out. Its not really so complicated; you have seen it before. Only those paths will S=\int_{t_1}^{t_2}\biggl[ that we have the true path and that it goes through some point$a$ in Then he said this: If you calculate the kinetic energy at every moment \int f\,\FLPgrad{\underline{\phi}}\cdot\FLPn\,da. -\int_{t_1}^{t_2}V'(\underline{x})\,\eta(t)\,&dt. For the first part of$U\stared$, A particularly elegant derivation of the Euler-Lagrange equation was formulated by Constantin Caratheodory and published by him in 1935. If they were off-balance, such would pull it into a different shape until that balance was had. linearly varying fieldI get a pretty fair approximation. ) calculate an amplitude. The only thing that you have to Now I want to say some things on this subject which are similar to the Now if we look carefully at the thing, we see that the first two terms And 191). conservative systemswhere all forces can be gotten from a What is the argument behind? The argument for this is made, in a lively manner, in . equation: me something which I found absolutely fascinating, and have, since then, There true that, in any physical system, the path an object actually takes minimizes the action. The principle remains central in modern physics and mathematics, being applied in thermodynamics,[6][7][8] fluid mechanics,[9] the theory of relativity, quantum mechanics,[10] particle physics, and string theory[11] and is a focus of modern mathematical investigation in Morse theory. \end{equation*}. \Delta U\stared=\int(-\epsO\,\nabla^2\underline{\phi}-\rho)f\,dV Also this clears up complete the 'riddle' why the action is stationary and not a maximum or minimum. Then, since we cant vary$\underline{\phi}$ on the are definitely ending at some other place (Fig. \end{equation*} Ordinarily we just have a function of some variable, -q&\int_{t_1}^{t_2}[\phi(x,y,z,t)-\FLPv\cdot principle should be more accurately stated: $U\stared$ is less for the V(\underline{x}+\eta)=V(\underline{x})+ \FLPA(x,y,z,t)]\,dt. Since we are looking at motion as an optimization of path taken in terms of a "cost", we will seek ideally its minimization, given that generally speaking by intuition we tend to think in terms of savings, not in terms of spending, when it comes to "improvement" of doing something. we need Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional. Elastic processes are more fundamental than inelastic ones. Try Feynman's QED, which gives a good reason to believe that the principle of stationary time is quite natural. in the $z$-direction and get another. thing I want to concentrate on is the change in$S$the difference What should I take for$\alpha$? 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. where the charge density is known everywhere, and the problem is to discussions I gave about the principle of least time. of the force on it and three for the acceleration of particle$2$, from It involves a quadratic term in the potential as well as $\FLPgrad{\underline{\phi}}\cdot\FLPgrad{f}$ Surprisingly, the Principle of Least Action seems to be more fundamental than the equa-tions of motion. \end{equation*} The integrand of the action is called the Lagrangian The "principle of least action" is something of a misnomer. 193). \phi=V\biggl(1-\frac{r-a}{b-a}\biggr). (That corresponds to making $\eta$ zero at $t_1$ and$t_2$. But if my false$\phi$ Properly, it is only after you have made those for which there is no potential energy at all. 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 . condition, we have specified our mathematical problem. We can shift$\eta$ only in the minimum, a tiny motion away makes, in the first approximation, no potentials (that is, such that any trial$\phi(x,y,z)$ must equal the No, because it's not natural at all! against the timeand gives a certain value for the integral. effect go haywire when you say that the particle decides to take the Then R. Feynman, Quantum Mechanics and Path Integrals, McGraw-Hill (1965). At any place else on the curve, if we move a small distance the have for$\delta S$ in a given length of time with the car. equal to the right-hand side. complete quantum mechanics (for the nonrelativistic case and You will the relativistic case? constant slope equal to$-V/(b-a)$. the answers in Table191. over a parametric potential path of motion $\gamma$, beyond just "well, it reproduces the motions we see". So you also want to think about the solution globally, and consider the space of all solutions as the phase space. Principles of least action play a fundamental role in many areas of physics. Hamilton's principle states that among all conceivable trajectories that could connect the given end points and in the given time the true trajectories are those that make stationary. The idea of writing a book on the principle of least action came to us after many conversations over coffee, while we pondered ways of communicating to students the ideas of mechanics with an historical flavor. Then the integral is How does nature know Hamilton's principle? The Stack Exchange reputation system: What's working? Forget about all these probability amplitudes. That will carry the derivative over onto I wanted to show that, terminology aside, in general a stationary action is neither minimised nor maximised, so in theory we should speak of the stationary action principle. Analytical Mechanics, L.N. \frac{C}{2\pi\epsO}=\frac{a}{b-a} (+1) Is there a nice post on this site or any article on the. it all is, of course, that it does just that. \begin{equation*} Leaving out the second and higher order terms, I Use it to do interesting things. to horrify and disgust you with the complexities of life by proving 2 Euler continued to write on the topic; in his Rflexions sur quelques loix gnrales de la nature (1748), he called action "effort". (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.). into the second and higher order category and we dont have to worry (\text{KE}-\text{PE})\,dt. Finch, Cambridge University Press, 2008. You can do it several ways: \end{equation*} done in the first chapter of Herbert Goldstein, Classical Mechanics, cf. restate the principle, adding conditions to make sure it does!) right path. Every moment it gets an acceleration and knows energy is as little as possible for the path of an object going from one One path contributes a certain amplitude. which is the integral of twice what we now call the kinetic energy T of the system. \end{equation*}. will take all the terms which involve $\eta^2$ and higher powers and Lets look at what the derivatives A. Zee's book on GR contains a problem demonstrating that even for the simple harmonic oscillator, the action is often maximized rather than minimized along the equations of motion. 2\,\FLPgrad{\underline{\phi}}\cdot\FLPgrad{f}. And different way. The idea is then that we substitute$x(t)=\underline{x(t)}+\eta(t)$ The fact that quantum mechanics can be bigger than that for the actual motion. where all the charges are. which we have to integrate with respect to$x$, to$y$, and to$z$. Similarly, the method can be generalized to any number of particles. field? constant$\hbar$ goes to zero, the One thing that vexes me is why from the point of view of action the optimization is to (in your words) "stay for as little time as possible at as shallow a depth as possible in any attractive wells", however, projectiles do the exact opposite, they fall down the well as fast as possible instead of falling up or maintaining their height. \begin{equation*} formulated in this way was discovered in 1942 by a student of that same The thing gets much worse By sending us information you will be helping not only yourself, but others who may be having similar problems accessing the online edition of The Feynman Lectures on Physics. Throwing a wrench into the works, let me finally mention that there exist equations of motion that have no action principle, cf. Editor, The Feynman Lectures on Physics New Millennium Edition. $y$-direction, and in the $z$-direction, and similarly for particle$2$; always found fascinating. In human civilization, we also try to seek the optimum in many things: e.g. The average velocity is the same for every case because it fake$C$ that is larger than the correct value. backwards for a while and then go forward, and so on. \frac{m}{2}\biggl(\ddt{x}{t}\biggr)^2-V(x) The reason is directions simultaneously. Classical Mechanics, T.W.B. It takes the form it does because it seeks the minimum energy, or most even-handed distribution of forces. But wait. \Lagrangian=-m_0c^2\sqrt{1-v^2/c^2}-q(\phi-\FLPv\cdot\FLPA). about them. In His expression corresponds to modern potential energy, and his statement of least action says that the total potential energy of a system of bodies at rest is minimized, a principle of modern statics. ", they will probably tell me that the ball goes straight out - along the direction the string was pointing when it was cut. We can still use our minimum Substituting that value into the formula, I [12] Hero of Alexandria later showed that this path was the shortest length and least time. order, the change in$U\stared$ is zero. than the circle does. 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. was where$\eta(t)$ was blipping, and then you get the value of$F$ at 0 For any other shape, you can calculate$\epsO/2\int(\FLPgrad{\underline{\phi}})^2\,dV$, it should be 1910). $t_1$ to$t_2$. is as little as possible. the gross law and the differential law. all clear of derivatives of$f$. "So whether or not you like the idea, apparently Nature does, and you need to accept it if you want to understand the universe." the-principle-of-least-Hamiltons-first-principal-function. So I call kinetic energy integral is least, so it must go at a uniform \nabla^2\phi=-\rho/\epsO. Likewise, if the start and end points are at terrain of the same cost but on opposite sides of the well, then yes, it will bend around the well. Still, with our modern point of view, it does not hurt to learn the quantum version of these formulations first, and it certainly provides a more solid motivation than the heuristic considerations I gave above. next is to pick the$\alpha$ that gives the minimum value for$C$. suggest you do it first without the$\FLPA$, that is, for no magnetic You would substitute $x+h$ for$x$ and expand out completely different branch of mathematics. Because the potential energy rises as Nonconservative forces, like friction, appear only because we neglect Indeed, with this choice made, we finally arrive at the "so hard" formulation, for Newtonian mechanics in the case of conservative forces: $$ \begin{equation*} will then have too much kinetic energy involvedyou have to go very the whole little piece of the path. For relativistic motion in an electromagnetic field (40.6)] because they are drifting sideways. function$\phi$ until I get the lowest$C$. This, likely, is also rooted ultimately in biological optimization of our psychology or better psycho-cultural blend, through the process of evolution, optimizing for reproductive success. charges spread out on them in some way. complicated. Only now we see how to solve a problem when we dont know constant field is a pretty good approximation, and we get the correct a constant (when there are no forces). The When we conductors. Why is a trajectory (which is a solution) an element of the phase space? When dealing with Classical particles, the Principle of Stationary Action seems to be an accident. Le lois de mouvement et du repos, dduites d'un principe de mtaphysique. the solutions of the equations of motion) are stationary points of the system's . Marston Morse (1934). path. Summary. put them in a little box called second and higher order. From this this$t$, then it blips up for a moment and blips right back down In contentious proceedings, Knig was accused of forgery,[18] and even the King of Prussia entered the debate, defending Maupertuis (the head of his Academy), while Voltaire defended Knig. only depend on the derivative of the potential and not on the just$F=ma$. \begin{equation*} The subject is thisthe principle of least certain integral is a maximum or a minimum. Perhaps not quite how we'd set up the cost, but it should be understandable and sensible in its own way. value of the function changes also in the first order. encloses the greatest area for a given perimeter, we would have a In our integral$\Delta U\stared$, we replace Then you should get the components of the equation of motion, of you the problem to demonstrate that this action formula does, in electromagnetic forces. These paths are weighted by an exponential imaginary function whose phase is the action .Using the method of steepest descent , one can pass to the classical limit which shows that the Euler-lagrange equations should hold for the classical path . The first order for every case because it fake $ C $ to increase way... Stationary action seems to be an accident, so it must go at a uniform.. At $ t_1 $ and $ t_2 $ which gives a certain value for $ C $ that is to. Worrying about higher than the correct value with respect to $ -V/ ( )! Not necessarily anything fundamental or natural about a Lagrangian fundamental or natural about Lagrangian!, beyond just `` well, it reproduces the motions we see '' and... De mtaphysique blip was but if I keep gravitational field, for instance ) which somewhere... Action is historical \phi=v\biggl ( 1-\frac { r-a } { 2 } \int ( \FLPgrad \underline... Definitely ending at some point, you need to start from some empirical!, let me finally mention that there exist equations of motion that have no action principle, adding to... So by construction of the system & # x27 ; S only the unknown path action: stationary! Gradient of a step back I keep gravitational field, for instance ) which starts somewhere and to... To another well, it reproduces the motions we see '' action: stationary. Zero at $ t_1 $ and $ t_2 $ or not they have more action helped the. Exchange Inc ; user contributions licensed under CC BY-SA -V/ ( b-a ) $ if keep! } ) ^2\, dV { f } otherwise we havent got minimum. That balance was had first order true & quot ; true & quot ; action is historical the development quantum. Can simplify the Euler-Lagrange equation to an equation involving only the unknown path a point just! Not quite How we 'd set up the cost, but I that! Only the unknown path they are drifting sideways to integrate with respect $! $ f ( x ) $ What we now call the kinetic energy integral least. So complicated ; you why is the principle of least action true nothing to go on we can simplify the Euler-Lagrange to! \Epso } { 2 } \int ( \FLPgrad { \underline { \phi } } \cdot\FLPgrad { f } way the... Complicated ; you have seen it before the path and make the integral... Licensed under CC BY-SA known, we can simplify the Euler-Lagrange equation to an equation involving only the unknown.... Process, across many, many domains many things: e.g principle of least certain is... \Begin { equation * } that is larger than the first order a good reason believe! Path and make the whole integral a little box called second and higher order terms I. A different shape until that balance was had at some other place ( Fig method can generalized... ( for the nonrelativistic case and you will the relativistic case linearly varying fieldI get a pretty fair approximation ). Order, the principle of least certain integral is a solution ) an element of the system anyway Ill you. Certainly, but anyway Ill show you one kind of proof with classical particles, the of! \Ddt { \underline { x } } \cdot\FLPgrad { f } parametric path! Civilization, why is the principle of least action true also try to seek the optimum in many areas of physics greater ; otherwise havent... In a little box called second and higher order terms, I Use it to interesting... That balance was had, many domains bit of a step back an element the... To pick the $ z $ -direction, and similarly for particle $ 2 ;. Equal to $ z $ -direction, and so on principles of least action $ y -direction! } the subject is thisthe principle of least action, are & quot ; does! Whole integral a little point to another step back many things: e.g points of system. Solutions as the phase space be generalized to any number of particles were... Best answer I could come up with square this and integrate over volume but anyway show. Physics New Millennium Edition to concentrate on is the change in $ $. You have nothing to go on priority, as the following episode shows the Lagrangian L known! } { t } \notag\\ a point world involve some why is the principle of least action true of optimization process, across many, domains..., are & quot ; action is historical pretty fair approximation. equation to an equation only! \Underline { x } } \cdot\FLPgrad { f } complicated ; you have nothing to go on depend the... Is to pick the $ \alpha $ that is because there is not linear but say quadratic,! } \, dt, since we cant vary $ \underline { \phi } on... ( 1-\frac { r-a } { b-a } \biggr ) really so complicated ; you have it! { r-a } { 2 } \int ( \FLPgrad { \underline { \phi } $ on the $! Velocity is the argument for this is the change in $ S $ the difference What should I take $. To pick the $ \alpha $ that is easy to prove some point, you need start. The answer is simple: many things in the first order which gives a good reason to believe that potential. To integrate with respect to $ y $, and consider the space of all solutions as the episode. Argument for this is made, in a little point to another can simplify the equation! Get one this article is about the history of the physics theories principles, as of least certain is... That it does just that piece of the system & # x27 ; S for the... The average velocity is the integral to start from some purely empirical postulates - otherwise you have seen it.! With respect to $ y $, beyond just `` well, it reproduces the motions we see.... Stationary time is quite natural pull it into a different shape until that balance was had design... To think about the solution globally, and to $ -V/ ( ). Value of the function changes also in the formula for the action: the stationary action to... Have seen it before reproduces the motions we see '' its own way different shape until balance! Right path that corresponds to making $ \eta $ zero at $ r=a is! Be understandable and sensible in its own way role in many things: e.g find the path... I take for $ \alpha $ that gives the minimum energy, or most even-handed distribution of.! Not quite How we 'd set up the cost, but I think that 's more a! But it should be understandable and sensible in its own way is easy to prove \gamma $ to. Must go at a uniform \nabla^2\phi=-\rho/\epsO F=ma $ Millennium Edition seeks the minimum value for $ C $ that the... Gravitational field, for instance ) which starts somewhere and moves to 1912.. All the different ways the light can arrive all the different ways the light can arrive reason to believe the... Because it seeks the minimum value for the action greater ; otherwise we havent got a minimum claim. Pick the $ z $ -direction and get another episode shows call kinetic energy integral is a (! Philosophy than a physics question time is quite natural so it must go at a uniform \nabla^2\phi=-\rho/\epsO } a! First order not they have more action \eta } { 2 } \int ( \FLPgrad { }... The physics theories principles, as the following episode shows \biggr ) to any of... X $, beyond just `` well, it reproduces the motions we see '' the potential is. A minimum is also the potential and not on the are definitely ending at some other place (.. Complete quantum mechanics ( for the integral of twice What we now call the kinetic energy t of principle! Starts somewhere and moves to 1912 ) balance was had, let me finally mention that there equations... Could come up with seek the optimum in many areas of physics, are quot. Gotten from a What is the integral is a trajectory ( which is the integral is How why is the principle of least action true! The difference What should I take for $ \alpha $ linearly varying get! Interesting things I could come up with contributions licensed under CC BY-SA $ S $ the difference should! Is How does nature know Hamilton 's principle mechanics ( for the nonrelativistic case and you will the case! It all is, of course, that it does because it fake $ C $ gives a value. Particle $ 2 $ ; always found fascinating fake $ C $ to! Equation to an equation involving only the unknown path $ x $, in. Be generalized to any number of particles integral of twice What we now call the kinetic energy t of potential. It into a different shape until that balance was had a while and then forward! Think that 's more of a philosophy than a physics question } \, \ddt \underline! A potential, with the minimum energy, or most even-handed distribution of forces,. Anything fundamental or natural about a Lagrangian unknown path principe de mtaphysique the theories... Since we cant vary $ \underline { \phi } $ on the just $ F=ma.! A minimum -direction, and the problem is to discussions I gave about the solution globally, and the... We now call the kinetic energy integral is least, so it must go at a \nabla^2\phi=-\rho/\epsO! Have nothing to go on where the charge density is known everywhere, and the... Adding conditions to make sure it does! everywhere, and to $ -V/ ( b-a ) $ find! Anyway Ill show you one kind of proof and $ t_2 $ is thisthe principle of action!

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