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Classical Mechanics: Difference between revisions
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=== Initial Lagrangian Framing === | === Initial Lagrangian Framing === | ||
[[File:Least action sketch.png|thumb|Sketch of a trajectory in position-velocity configuration space and its partial derivatives]] | |||
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: | ||
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