theories:classical_mechanics:lagrangian
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
theories:classical_mechanics:lagrangian [2018/04/12 17:12] bogumilvidovic [Intuitive] |
theories:classical_mechanics:lagrangian [2018/10/11 14:12] (current) jakobadmin [Abstract] |
||
|---|---|---|---|
| Line 19: | Line 19: | ||
| \end{equation} | \end{equation} | ||
| - | [{{ :theories:trajectories.png?nolink&400|Source: Lectures on Classical Mechanics by [[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|John C. Baez]]}}] | + | {{ :theories:classical_mechanics:lagrangianpfad.png?nolink&400|}} |
| The basic idea is now that nature causes | The basic idea is now that nature causes | ||
| particles to follow the trajectories with the //least// amount of action. | particles to follow the trajectories with the //least// amount of action. | ||
| - | In the image on the right-hand side this could be the solid line denotes by $q$. Another path is shown as a dashed line and denoted by $q(s)$. The Lagrangian approach assigns to each path a quantity called action as defined above and then tells us that the correct path that an object really follows is the path with minimal action. In our example the path $q$ could have an action of $3$ and the path $q_s$ an action of $5$. Hence, path $q$ is correct and not path $q_s$. | + | In the image on the right-hand side this could be the solid line denotes by $q$. Another path is shown as a dashed line and denoted by $q(s)$. |
| + | |||
| + | The Lagrangian approach assigns to each path a quantity called action as defined above and then tells us that the correct path that an object really follows is the path with minimal action. In our example the path $q$ could have an action of $3$ and the path $q_s$ an action of $5$. Hence, path $q$ is correct and not path $q_s$. | ||
| Line 36: | Line 38: | ||
| ---- | ---- | ||
| - | Using the Lagrangian approach, we can derive [[theories:newtonian_mechanics|Newtonian mechanics]]. Alternatively, we can start with Newtonian mechanics and derive Lagrangian mechanics. | + | Using the Lagrangian approach, we can derive [[theories:classical_mechanics:newtonian|Newtonian mechanics]]. Alternatively, we can start with Newtonian mechanics and derive Lagrangian mechanics. |
| -->Derivation of Newton's second law# | -->Derivation of Newton's second law# | ||
| Line 203: | Line 205: | ||
| <-- | <-- | ||
| + | ---- | ||
| + | |||
| + | **Reading Recommendations** | ||
| + | |||
| + | * The best book on Lagrangian mechanics is The Lazy Universe by Coopersmith | ||
| + | * Many problems with solutions are collected in [[https://archive.org/details/SchaumsTheoryAndProblemsOfTheoreticalMechanics|Schaum's Outline of Theory and Problems of Theoretical Mechanics]] by Murray R Spiegel | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
| Lagrangian mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]]. | Lagrangian mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]]. | ||
| Line 215: | Line 223: | ||
| ---- | ---- | ||
| + | * [[https://core.ac.uk/download/pdf/4887416.pdf|Lectures on Mechanics]] by Marsden | ||
| * See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf. | * See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf. | ||
theories/classical_mechanics/lagrangian.1523545968.txt.gz · Last modified: 2018/04/12 15:12 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
