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formalisms:lagrangian_formalism [2018/05/06 13:37]
jakobadmin [Intuitive]
formalisms:lagrangian_formalism [2023/03/08 17:35] (current)
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 The basic idea of the Lagrangian formalism can be summarized by the statement: The basic idea of the Lagrangian formalism can be summarized by the statement:
 +
  
 //Nature is lazy.// //Nature is lazy.//
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 The laziness of nature is demonstrated nicely by how light behaves. The laziness of nature is demonstrated nicely by how light behaves.
  
-**The Principle of Least Time +**The Principle of Least Time**
-**+
  
 Long before Joseph Lagrange invented the formalism now named after him, it was well known that light always takes the path between two points that requires the least travel time. This is known as **Fermat'​s Principle**. Long before Joseph Lagrange invented the formalism now named after him, it was well known that light always takes the path between two points that requires the least travel time. This is known as **Fermat'​s Principle**.
 +
 +<​blockquote>​Nature operates by means and ways that are easiest and fastest.<​cite>​Pierre de Fermat </​cite></​blockquote>​
  
 In a vacuum, light travels in a straight path between two points: the least time required. ​ In a vacuum, light travels in a straight path between two points: the least time required. ​
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 {{ :​frameworks:​swim.png?​nolink&​600 |}} {{ :​frameworks:​swim.png?​nolink&​600 |}}
  
-To understand this let's consider a rescue swimmer who sees someone drowning in the water. Which path should he take to get to the swimmer as fast as possible? He is slow in the water and fast if he runs on the beach. This gives him two extreme options:+To understand this let's consider a rescue swimmer who sees someone drowning in the water. Which path should he take to get to the swimmer as fast as possible? He is slow in the water and fast running ​on the beach. This gives him two extreme options:
  
  
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 OR is there some better option? OR is there some better option?
  
-There //is//. The optimal path is a trade off of the two choices above. While it is true that the rescuer is much faster on the sand, the path in Option 2, above, is much longer than if he runs and swims diagonal+There //is//. The optimal path is a trade-off of the two choices above. While it is true that the rescuer is much faster on the sand, the path in Option 2, above, is much longer than if he runs and swims diagonally
  
-This is exactly the same behaviour light has. From Fermat'​s principle, we can now see why light gets broken across two media: Light is slower in of the mediaand thus has to choose a trade off between a minimal total path length and a minimal length in slower medium. ​+This is exactly the same behaviour light has. From Fermat'​s principle, we can now see why light gets broken across two media: Light is slower in the media and thus has to choose a trade-off between a minimal total path length and a minimal length in the slower medium. ​
  
 ---- ----
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 **Reading Recommendations** **Reading Recommendations**
  
-  * [[http://​nautil.us/blog/​to-save-drowning-people-ask-yourself-what-would-light-do|To Save Drowning People, Ask Yourself “What Would Light Do?”]] by Aatish Bhatia+  * [[https://​nautil.us/​to-save-drowning-people-ask-yourself-what-would-light-do-234852/|To Save Drowning People, Ask Yourself “What Would Light Do?”]] by Aatish Bhatia
  
  
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 To exploit the idea that nature is lazy, we need to define some quantity that measures "​laziness"​. We name this quantity the "​action"​ of a system. A system with bigger "​action"​ is less lazy than a system with small "​action"​. The "​action"​ is the sum of the difference between the kinetic and potential energy for all "​timesteps"​ in a given interval. The difference between the kinetic and the potential energy is called the Lagrangian. For every point in time $t$, the Lagrangian has some value $L(t)$. Let's say we want to analyze a system for 5 seconds. The action of the system is the sum of all values of the Lagrangian during these five seconds. In mathematical terms, the action is therefore the integral of the Lagrangian function $L(t)$, starting at some $t_0$ and ending at some $t_1$: To exploit the idea that nature is lazy, we need to define some quantity that measures "​laziness"​. We name this quantity the "​action"​ of a system. A system with bigger "​action"​ is less lazy than a system with small "​action"​. The "​action"​ is the sum of the difference between the kinetic and potential energy for all "​timesteps"​ in a given interval. The difference between the kinetic and the potential energy is called the Lagrangian. For every point in time $t$, the Lagrangian has some value $L(t)$. Let's say we want to analyze a system for 5 seconds. The action of the system is the sum of all values of the Lagrangian during these five seconds. In mathematical terms, the action is therefore the integral of the Lagrangian function $L(t)$, starting at some $t_0$ and ending at some $t_1$:
  
-$$ \text{Action} = \int_{t_0}^{t_1} L(t) dt .$$+$$ \text{Action} = \int_{t_0}^{t_1} L(t) dt $$
  
 The statement that nature is lazy means now in mathematical terms that during some given time span,  here from $t_0$ to $t_1$, the system behaves in such a way that the action is as small as possible. This is known as the principle of minimal action. ​ The statement that nature is lazy means now in mathematical terms that during some given time span,  here from $t_0$ to $t_1$, the system behaves in such a way that the action is as small as possible. This is known as the principle of minimal action. ​
 +
 +<​blockquote>​Whenever any change takes place in Nature, the amount of action expended in this change is always the smallest possible.<​cite>​Pierre Louis Maupertuis</​cite></​blockquote>​
  
 So, why is the laziness of a system given by the difference between the kinetic and potential energy? So, why is the laziness of a system given by the difference between the kinetic and potential energy?
  
 The kinetic energy is a measure for how much is happening, i.e. how much activity is going on in the system. The potential energy, as the name indicates, is a measure for how much activity could potentially happen, but does not. A good example is a ball at the top of a cliff. At this point its potential energy is maximal, but could be converted to kinetic energy at any moment if it slides down the cliff. ​ The kinetic energy is a measure for how much is happening, i.e. how much activity is going on in the system. The potential energy, as the name indicates, is a measure for how much activity could potentially happen, but does not. A good example is a ball at the top of a cliff. At this point its potential energy is maximal, but could be converted to kinetic energy at any moment if it slides down the cliff. ​
 +
 +In other words, the Lagrangian measures how much is happening, minus how much could be happening but isn't.
 +
 +----
  
 Let's consider an explicit example: We throw a ball and want to know what path it will follow between two given points $A$ and $B$ on the ground, where it starts at a fixed time $t_A$ and ends up on the ground at fixed time $t_B$. Let's consider an explicit example: We throw a ball and want to know what path it will follow between two given points $A$ and $B$ on the ground, where it starts at a fixed time $t_A$ and ends up on the ground at fixed time $t_B$.
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 Thus, the principle of minimal action does not simply mean that potential energy gets maximized, but that we have a trade off. Nature tries to make the potential energy as large as possible, while keeping the kinetic energy at a reasonable value. This explains why the ball almost stops at the top, and is the fastest close to the ground. ​ Thus, the principle of minimal action does not simply mean that potential energy gets maximized, but that we have a trade off. Nature tries to make the potential energy as large as possible, while keeping the kinetic energy at a reasonable value. This explains why the ball almost stops at the top, and is the fastest close to the ground. ​
  
-Now with all this in mind, let's revisit the example discussed in the "Laymen" section.+Now with all this in mind, let's revisit the example discussed in the "Intuitive" section.
  
 ** Light Again ** ** Light Again **
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 The travel time for a path $q(t)$ between two fixed points $A$ and $B$ is given by The travel time for a path $q(t)$ between two fixed points $A$ and $B$ is given by
  
-\[S_{\text{light}}[\mathbf{q}(t)]=\int_A^B dt\]+\[S_{\text{light}}[q(t)]=\int_A^B dt\]
  
-The path $q_m(t)$ that light actually takes is the path that minimizes this quantity. For light this quantity is simple ​the travel time. +The path $q_m(t)$ that light actually takes is the path that minimizes this quantity. For light this quantity is simply ​the travel time. 
  
-This is certainly an attractive explanation for the behaviour of light. If you could design a universe with law physical laws, what other path would you let light take between two points?+This is certainly an attractive explanation for the behaviour of light. If you could design a universe with physical laws, what other path would you let light take between two points?
  
 Now, Joseph Lagrange was fascinated by this principle and tried to find something similar for other objects, not just light. Unfortunately,​ simply assuming that the correct path for general objects is the path with minimal travel time does not yield correct results. Now, Joseph Lagrange was fascinated by this principle and tried to find something similar for other objects, not just light. Unfortunately,​ simply assuming that the correct path for general objects is the path with minimal travel time does not yield correct results.
  
-However, Lagrange instead proposed a more general ​Ansatz+However, Lagrange instead proposed a more general ​ansatz
  
 \[S[q(t)]=\int L \,dt, \] \[S[q(t)]=\int L \,dt, \]
  
-and that the correct path of every object can be found, by demanding that this quantity gets minimized. The task is then, of course, to find the correct quantity $L$, now called Lagrange function. See [[https://​books.google.com/​books/​about/​Mechanics.html?​id=bE-9tUH2J2wC&​redir_esc=y|Landau Mechanics, Volume 1, section 4 and 5]] for the derivation of $L = T - V$ for classical objects.+and that the correct path of every object can be found, by demanding that this quantity gets minimized. The task is then, of course, to find the correct quantity $L$, now called ​the Lagrange function. See [[https://​books.google.com/​books/​about/​Mechanics.html?​id=bE-9tUH2J2wC&​redir_esc=y|Landau Mechanics, Volume 1, section 4 and 5]] for the derivation of $L = T - V$ for classical objects.
  
 Nevertheless,​ the exact same principle is so powerful that it is used in almost all modern theories. ​ Nevertheless,​ the exact same principle is so powerful that it is used in almost all modern theories. ​
  
-For example, quantum field theory, we also "​guess"​ the correct function $L$ and find the correct equations of motion by minimizing the action $S$. The most powerful tool that we have in finding the correct quantity $L$ are symmetries. Experimental restrictions,​ such as the observation that the speed of light is constant in the inertial frames of reference, are so powerful that they are almost enough to determine the correct function $L$. +For example, ​in [[theories:​quantum_field_theory|quantum field theory]], we also "​guess"​ the correct function $L$ and find the correct equations of motion by minimizing the action $S$. The most powerful tool that we have in finding the correct quantity $L$ is symmetries. Experimental restrictions,​ such as the observation that the speed of light is constant in inertial frames of reference, are so powerful that they are almost enough to determine the correct function $L$. 
  
 <​blockquote>"​First,​ note that total energy is conserved, so energy can slosh back and forth between kinetic and potential forms. The Lagrangian L = K − V is big when most of the energy is in kinetic form, and small when most of the energy is in potential form. Kinetic energy measures how much is ‘happening’ — how much our system is moving around. Potential energy measures how much could happen, but isn’t yet — that’s what the word ‘potential’ means. (Imagine a big rock sitting on top of a cliff, with the potential to fall down.) So, the Lagrangian measures something we could vaguely refer to as the ‘activity’ or ‘liveliness’ of a system: the higher the kinetic energy the more lively the system, the higher the potential energy the less lively. So, we’re being told that nature likes to minimize the total of ‘liveliness’ over time: that is, the total action. In other words, nature is as lazy as possible!"<​cite>​http://​math.ucr.edu/​home/​baez/​classical/​texfiles/​2005/​book/​classical.pdf</​cite></​blockquote>​ <​blockquote>"​First,​ note that total energy is conserved, so energy can slosh back and forth between kinetic and potential forms. The Lagrangian L = K − V is big when most of the energy is in kinetic form, and small when most of the energy is in potential form. Kinetic energy measures how much is ‘happening’ — how much our system is moving around. Potential energy measures how much could happen, but isn’t yet — that’s what the word ‘potential’ means. (Imagine a big rock sitting on top of a cliff, with the potential to fall down.) So, the Lagrangian measures something we could vaguely refer to as the ‘activity’ or ‘liveliness’ of a system: the higher the kinetic energy the more lively the system, the higher the potential energy the less lively. So, we’re being told that nature likes to minimize the total of ‘liveliness’ over time: that is, the total action. In other words, nature is as lazy as possible!"<​cite>​http://​math.ucr.edu/​home/​baez/​classical/​texfiles/​2005/​book/​classical.pdf</​cite></​blockquote>​
  
-Take note the Lagrange density could, in principle, be +Take note the Lagrange density could, in principle, be anything. However, most of the time its actual form is dictated by [[basic_tools:​symmetry|symmetry]] considerations.
-anything. However, most of the time its actual form is dictated by [[basic_tools:​symmetry|symmetry]] considerations.+
  
  
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-<​blockquote>​The momentum of our particle is defined to be p = dL/dq' The force on it is defined to be F = dL/dq The equations of motion - the so-called [[equations:​euler_lagrange_equations|Euler-Lagrange equations]] - say that the rate of change of momentum equals the force: p' = F That's how Lagrangians work!"+<​blockquote>​The momentum of our particle is defined to be p = dL/dq'The force on it is defined to be F = dL/dqThe equations of motion - the so-called [[equations:​euler_lagrange_equations|Euler-Lagrange equations]] - say that the rate of change of momentum equals the force: p' = FThat's how Lagrangians work!"
  
 <​cite>​http://​math.ucr.edu/​home/​baez//​noether.html</​cite></​blockquote>​ <​cite>​http://​math.ucr.edu/​home/​baez//​noether.html</​cite></​blockquote>​
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 In the Lagrangian approach we focus on the position and velocity of a particle, and In the Lagrangian approach we focus on the position and velocity of a particle, and
-compute what the particle does starting from the Lagrangian $L(q, q˙)$, which is a function+compute what the particle does starting from the Lagrangian $L(q, \dot{q})$, which is a function
  
-$$ LTQ \to \mathbb{R} $$+$$ L\colon ​TQ \to \mathbb{R} $$
  
 where the tangent bundle is the space of position-velocity pairs. But we're led to consider momentum ​ where the tangent bundle is the space of position-velocity pairs. But we're led to consider momentum ​
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 of expressions we can integrate over to get S. We can only use of expressions we can integrate over to get S. We can only use
  
-$$ S= c_1 \int ds + c_2 \int A_j dx^j + c_3 \int \sqrt[-g_{ij} dx^i dx^j}$$,+$$ S= c_1 \int ds + c_2 \int A_j dx^j + c_3 \int \sqrt{-g_{ij} dx^i dx^j}$$,
  
 up to quadratic order, where the ci-s are constants, Aj is a four-vector and up to quadratic order, where the ci-s are constants, Aj is a four-vector and
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 order, we can only use order, we can only use
  
-$$ S=  + c_2 \int A_j dx^j + c_3 \int \sqrt[-g_{ij} dx^i dx^j} $$+$$ S=  + c_2 \int A_j dx^j + c_3 \int \sqrt{-g_{ij} dx^i dx^j} $$
 with just two external fields: $A_j$ gives you electromagnetism and $g_{ij}$ gives with just two external fields: $A_j$ gives you electromagnetism and $g_{ij}$ gives
 you gravity! you gravity!
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 --> Why is the correct path given by the path with minimal action?# --> Why is the correct path given by the path with minimal action?#
  
-The answer can be found through the [[theories:​quantum_mechanics:​path_integral|path integral formulation]] of [[theories:​quantum_mechanics:​canonical|quantum mechanics]]. The thing is that a particle really has some probability to go all possible ways. However, the classical path is the most probable path, because paths close this this path infer constructively and hence yield a big probability. In contrast, for other paths far aways from the classical path the interference is destructive and hence the probability is tiny. The path with minimal action gives the biggest contribution to the path integral in the classical limit $\hbar \to 0$.+The answer can be found through the [[theories:​quantum_mechanics:​path_integral|path integral formulation]] of [[theories:​quantum_mechanics:​canonical|quantum mechanics]]. The thing is that a particle really has some probability to go all possible ways. However, the classical path is the most probable path, because paths close this this path interfere ​constructively and hence yield a big probability. In contrast, for other paths far aways from the classical path the interference is destructive and hence the probability is tiny. The path with minimal action gives the biggest contribution to the path integral in the classical limit $\hbar \to 0$.
  
 This is explained nicely in Section 3 [[https://​arxiv.org/​pdf/​quant-ph/​0004090.pdf|here]]. ​ This is explained nicely in Section 3 [[https://​arxiv.org/​pdf/​quant-ph/​0004090.pdf|here]]. ​
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 <tabbox History> ​ <tabbox History> ​
  
-Seehttp://​astro1.panet.utoledo.edu/~ljc/bader1.html+<​blockquote>​ 
 + 
 +The principle arose out of successive generalizations of earlier principles: 
 +  -  first, as we have already seen, there was Heron’s ‘shortest path’ for reflected light (second century AD), 
 +  - then there was Fermat’s‘ Least Time’ for reflection and refraction (1662), 
 +  - then Leibniz, in 1682, in his Principle of Least Resistance, pooh poohed Fermat’s Principle, for why should light make a choice between optimizing ‘time’ and optimizing ‘distance’?​ No, argued Leibniz, light takes the easiest path, the one for which the ‘resistance’ is least, 
 +  - finally, Maupertuis,​in 1644,​extended Leibniz’s principle to cover the motion of light and bodies - he called it the Principle of Least Action. 
 + 
 +<​cite>​The Lazy Universe by Coopersmith<​/cite> 
 +</blockquote>​ 
 + 
 +---- 
  
 +  * See: http://​astro1.panet.utoledo.edu/​~ljc/​bader1.html
  
 </​tabbox>​ </​tabbox>​
formalisms/lagrangian_formalism.1525606639.txt.gz · Last modified: 2018/05/06 11:37 (external edit)