formalisms:lagrangian_formalism

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formalisms:lagrangian_formalism [2018/12/29 23:35] thomas_abshier [Intuitive] |
formalisms:lagrangian_formalism [2018/12/29 23:49] (current) thomas_abshier [Concrete] typos |
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\[S_{\text{light}}[\mathbf{q}(t)]=\int_A^B dt\] | \[S_{\text{light}}[\mathbf{q}(t)]=\int_A^B dt\] | ||

- | 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 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? | ||

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\[S[q(t)]=\int L \,dt, \] | \[S[q(t)]=\int L \,dt, \] | ||

- | 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, 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/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!" |

<cite>http://math.ucr.edu/home/baez//noether.html</cite></blockquote> | <cite>http://math.ucr.edu/home/baez//noether.html</cite></blockquote> |

formalisms/lagrangian_formalism.txt · Last modified: 2018/12/29 23:49 by thomas_abshier

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