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formalisms:lagrangian_formalism [2019/01/31 09:28]
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formalisms:lagrangian_formalism [2020/04/12 15:05] (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**.
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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 simply 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. 
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 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
  
 \[S[q(t)]=\int L \,dt, \] \[S[q(t)]=\int L \,dt, \]
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 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 $L$ 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 $L$. +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>​
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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!
formalisms/lagrangian_formalism.1548923337.txt.gz · Last modified: 2019/01/31 08:28 (external edit)