Editing Classical Mechanics

Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.

Latest revision Your text
Line 6: Line 6:
[[File:Least action sketch.png|thumb|right|Sketch of a trajectory in position-velocity configuration space and its partial derivatives]]
[[File:Least action sketch.png|thumb|right|Sketch of a trajectory in position-velocity configuration space and its partial derivatives]]


<div class="math-typesetting">
<div class="math-block">


Classical Mechanics can be formulated directly and generally by applying calculus to trajectories/curves in space. For concreteness and an alternate presentation, we describe the formulation backwards from the first few pages of Landau's mechanics. Pictured on the side is a trajectory in one dimension <math> q(t) </math>. Since it is differentiable, we can plot the position and its derivative velocity <math> \dot{q}(t) </math> as a vector-valued function of time: <math> t_0 \rightarrow (q(t), \dot{q}(t)) </math> or points of the graph: <math> (q(t_0), \dot{q}(t_0), t_0) </math>. Now regarding the variables <math> q, \dot{q}, t </math> as mutually independent, there is a function called the Lagrangian <math> L(q, \dot{q}, t) </math> whereby the trajectory curve can be recovered, or the Lagrangian modified to give any other desired trajectory. In its most basic examples, it is a polynomial and constant in time:
Classical Mechanics can be formulated directly and generally by applying calculus to trajectories/curves in space. For concreteness and an alternate presentation, we describe the formulation backwards from the first few pages of Landau's mechanics. Pictured on the side is a trajectory in one dimension <math> q(t) </math>. Since it is differentiable, we can plot the position and its derivative velocity <math> \dot{q}(t) </math> as a vector-valued function of time: <math> t_0 \rightarrow (q(t), \dot{q}(t)) </math> or points of the graph: <math> (q(t_0), \dot{q}(t_0), t_0) </math>. Now regarding the variables <math> q, \dot{q}, t </math> as mutually independent, there is a function called the Lagrangian <math> L(q, \dot{q}, t) </math> whereby the trajectory curve can be recovered, or the Lagrangian modified to give any other desired trajectory. In its most basic examples, it is a polynomial and constant in time:


<div class="math-block">
<div class="math-typesetting">
<math> L = m*\frac{\dot{q}^2}{2}-k*\frac{q^2}{2} </math>
<math> L = m*\frac{\dot{q}^2}{2}-k*\frac{q^2}{2} </math>
</div>
</div>
Please note that all contributions to The Portal Wiki may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see The Portal:Copyrights for details). Do not submit copyrighted work without permission!
Cancel Editing help (opens in new window)

Templates used on this page: