Editing Classical Mechanics
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[[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]] | ||
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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<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> | ||
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Where m and k are constants. The equation determining the trajectory from any Lagrangian L is called the Euler Lagrange (EL) equation, position and velocity are now regarded as functions of time: | Where m and k are constants. The equation determining the trajectory from any Lagrangian L is called the Euler Lagrange (EL) equation, position and velocity are now regarded as functions of time: | ||
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<math> \frac{d}{dt} \frac{\partial L(q(t), \dot{q}(t), t)}{\partial \dot{q}}=\frac{\partial L(q(t), \dot{q}(t), t)}{\partial q},\quad m * \frac{d^2 q(t)}{dt^2}=-k*q(t)</math> | <math> \frac{d}{dt} \frac{\partial L(q(t), \dot{q}(t), t)}{\partial \dot{q}}=\frac{\partial L(q(t), \dot{q}(t), t)}{\partial q},\quad m * \frac{d^2 q(t)}{dt^2}=-k*q(t)</math> | ||
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Letting the <math> q_i </math> coordinates be coordinates other than Cartesian, e.g. spherical coordinates for each particle, allows us to discuss linear and angular momentum on the same footing or momentum in any convenient coordinate system. Given <math> m_{1+((i-1)-(i-1)\,mod 3)/3} = m_k </math> as the mass of the k'th particle in order, in cartesian coordinates <math> T=\sum_{i=1}^{3N} \frac{1}{2}m_k \dot{q}^2_i </math> gives <math> \frac{\partial T}{\partial \dot{q}_i}=m_k \dot{q}_i=m_k v_i = p_i</math>. Similarly, the coordinate derivative gives <math> \frac{\partial L}{\partial q_i}=-\frac{\partial U}{\partial q_i} = F_i </math> which has the interpretation of force on the i'th coordinate. Then the i'th EL equation is expressed as <math> m_k \frac{d}{dt} v_i = m_k a_i= F_i </math> which is just Newton's second law. Indeed, all of Newton's laws can be derived from this formulation of motion, but to do so fully we need an equation using finite properties of the Lagrangian and not just an infinitesimal condition. | Letting the <math> q_i </math> coordinates be coordinates other than Cartesian, e.g. spherical coordinates for each particle, allows us to discuss linear and angular momentum on the same footing or momentum in any convenient coordinate system. Given <math> m_{1+((i-1)-(i-1)\,mod 3)/3} = m_k </math> as the mass of the k'th particle in order, in cartesian coordinates <math> T=\sum_{i=1}^{3N} \frac{1}{2}m_k \dot{q}^2_i </math> gives <math> \frac{\partial T}{\partial \dot{q}_i}=m_k \dot{q}_i=m_k v_i = p_i</math>. Similarly, the coordinate derivative gives <math> \frac{\partial L}{\partial q_i}=-\frac{\partial U}{\partial q_i} = F_i </math> which has the interpretation of force on the i'th coordinate. Then the i'th EL equation is expressed as <math> m_k \frac{d}{dt} v_i = m_k a_i= F_i </math> which is just Newton's second law. Indeed, all of Newton's laws can be derived from this formulation of motion, but to do so fully we need an equation using finite properties of the Lagrangian and not just an infinitesimal condition. | ||
Supposing <math> q_i(t) </math> solves the EL equations for s degrees of freedom, we can analyze properties of the integral across finite time <math> S = \int_{t_1}^{t_2} L(q_1(t), \cdots, q_s(t), \dot{q}_1(t), \cdots, \dot{q}_s(t), t) dt </math>, since substituting the trajectory gives a strict function of time. | Supposing <math> q_i(t) </math> solves the EL equations for s degrees of freedom, we can analyze properties of the integral across finite time <math> S = \int_{t_1}^{t_2} L(q_1(t), \cdots, q_s(t), \dot{q}_1(t), \cdots, \dot{q}_s(t), t) dt </math>, since substituting the trajectory gives a strict function of time. | ||
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=== Infinite Dimensional Techniques === | === Infinite Dimensional Techniques === | ||
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=== Lie Algebras and Symmetry === | === Lie Algebras and Symmetry === | ||
[[Category:Physics]] | [[Category:Physics]] |