formalisms:lagrangian_formalism
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
formalisms:lagrangian_formalism [2018/05/06 13:57] ida [Concrete] |
formalisms:lagrangian_formalism [2018/09/04 07:15] 92.225.203.36 [Concrete] |
||
|---|---|---|---|
| Line 58: | Line 58: | ||
| 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 $t_B$. | ||
| Line 81: | Line 85: | ||
| 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 simple 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. | ||
formalisms/lagrangian_formalism.txt · Last modified: 2023/03/08 17:35 by 62.4.55.178
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
