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euler-lagrange_equations [2015/03/29 18:33] nikolaj |
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| ===== Euler-Lagrange equations ===== | ===== Euler-Lagrange equations ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $ s\in\mathbb N $ | | + | | @#55CCEE: context | @#55CCEE: $s\in\mathbb N $ | |
| - | | @#55CCEE: context | @#55CCEE: $ L:C^2(\mathbb R^s\times\mathbb R^s\times\mathbb R,\mathbb R) $ | | + | | @#55CCEE: context | @#55CCEE: $L:C^2(\mathbb R^s\times\mathbb R^s\times\mathbb R,\mathbb R) $ | |
| - | | @#FFBB00: definiendum | @#FFBB00: $ q \in \mathrm{it} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $q\in$ it | |
| - | | @#AAFFAA: inclusion | @#AAFFAA: $ q:C(\mathbb R,\mathbb R^s) $ | | + | | @#AAFFAA: inclusion | @#AAFFAA: $q:C(\mathbb R,\mathbb R^s) $ | |
| | @#BBDDEE: let | @#BBDDEE: $\diamond\ q(t)$ | | | @#BBDDEE: let | @#BBDDEE: $\diamond\ q(t)$ | | ||
| | @#BBDDEE: let | @#BBDDEE: $\diamond\ L(x^1,\dots,x^s,v^1,\dots,v^s,t)$ | | | @#BBDDEE: let | @#BBDDEE: $\diamond\ L(x^1,\dots,x^s,v^1,\dots,v^s,t)$ | | ||
| - | | @#DDDDDD: range | @#DDDDDD: $j\in\mathrm{range}(s)$ | | + | | @#DDDDDD: range | @#DDDDDD: $j\in\{1,\dots,s\}$ | |
| | @#55EE55: postulate | @#55EE55: $\left(\dfrac{\mathrm d}{\mathrm dt}\dfrac{\partial L}{\partial v^j}\right)(q,q',t) - \dfrac{\partial L}{\partial x^j}(q,q',t) = 0$ | | | @#55EE55: postulate | @#55EE55: $\left(\dfrac{\mathrm d}{\mathrm dt}\dfrac{\partial L}{\partial v^j}\right)(q,q',t) - \dfrac{\partial L}{\partial x^j}(q,q',t) = 0$ | | ||
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| === Theorems === | === Theorems === | ||
| == Least action principle == | == Least action principle == | ||
| - | A curve $q_\text{sol}$ being a solition to the Euler-Lagrange equations is equivalent to the following: For any sufficiently small time interval $(t_i,t_{i+1})$, the functional | + | A curve $q_\text{sol}$ being a solition to the Euler-Lagrange equations is implied by the following: For any sufficiently small time interval $(t_i,t_{i+1})$, the functional |
| $S(t_i,t_{i+1})[q]=\int_{t_i}^{t_{i+1}}L(q,q',t)\,{\mathrm dt}.$ | $S(t_i,t_{i+1})[q]=\int_{t_i}^{t_{i+1}}L(q,q',t)\,{\mathrm dt}.$ | ||
