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Euler Lagrange equations in continuum
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[QUOTE="JD_PM, post: 6405641, member: 655284"] [B]Homework Statement:[/B] Given $$\frac{d}{dt} \frac{\partial L}{\partial \dot \phi_a^{(i j k)}} - \frac{\partial L}{\partial \phi_a^{(i j k)}} = 0 \tag{1}$$ Show that, in the limit ##l \rightarrow 0##, we obtain $$\partial_{\mu} \frac{\partial \mathscr{L}}{\partial(\partial_{\mu} \phi_a)} - \frac{\partial \mathscr{L}}{\partial \phi_a} = 0 \tag{2}$$ [B]Relevant Equations:[/B] Please see main post OK I've been stuck for a while in [URL='https://www.physicsforums.com/threads/discrete-euler-lagrange-equations.994698/']how to derive ##(1)##[/URL], so I better solve a simplified problem first: We work with Where $$\mathscr{L} = \mathscr{L}(\phi_a (\vec x, t), \partial_{\mu} \phi_a (\vec x, t)) \tag{3}$$ And ##(3)## implies that ##\mathscr{L}(\vec x, t)## We know that $$L=l^3\sum_{(i j k)} \mathscr{L}^{(i j k)}(t) \tag{4}$$ Where $$\lim_{l \rightarrow 0} L = \int d^3 \vec x \mathscr{L} \tag{5}$$ So, in analogy with ##(3), \mathscr{L}^{(i j k)}## depends on the fields ##\phi_a^{(i j k)} (t)##, on the time derivative of the fields ##\dot \phi_a^{(i j k)} (t)## and on the partial derivative of the fields with respect to ##x, y## and ##z## i.e; $$\frac{\phi^{(i+1, j, k)}(t)-\phi^{(i, j, k)}(t)}{l} \tag{6.1}$$ $$\frac{\phi^{(i, j+1, k)}(t)-\phi^{(i, j, k)}(t)}{l} \tag{6.2}$$ $$\frac{\phi^{(i, j, k+1)}(t)-\phi^{(i, j, k)}(t)}{l} \tag{6.3}$$Let's tackle the problem. I would naively plug ##(4)## into ##(1)## and evaluate the terms to get $$\frac{\partial}{\partial \dot \phi_a^{(i j k)}} \sum_{(i'j'k')} \Big[ l^3 \mathscr{L}^{(i' j' k')}(t) \Big] = l^3 \frac{\partial \mathscr{L}^{(i j k)}}{\partial \dot \phi_a^{(i j k)}} \tag{7.1}$$ $$\frac{\partial}{\partial \phi_a^{(i j k)}} \sum_{(i'j'k')} \Big[ l^3 \mathscr{L}^{(i' j' k')}(t) \Big] = l^3 \frac{\partial \mathscr{L}^{(i j k)}}{\partial \phi_a^{(i j k)}} \tag{7.2}$$ Where I've used the Kronecker delta. Next I'd take the limit ##l \rightarrow 0##$ of ##\frac{\partial \mathscr{L}^{(i j k)}}{\partial \dot \phi_a^{(i j k)}}## and ##\frac{\partial \mathscr{L}^{(i j k)}}{\partial \phi_a^{(i j k)}}## to get ##\frac{\partial \mathscr{L}}{\partial \dot \phi_a}## and ##\frac{\partial \mathscr{L}}{\partial \phi_a}## (respectively) So I get $$l^3 \Big( \frac{d}{dt} \frac{\partial \mathscr{L}}{\partial \dot \phi_a} - \frac{\partial \mathscr{L}}{\partial \phi_a} \Big) = 0 \tag{8}$$ Which of course does not yield ##(2)## when taking the limit ##l \rightarrow 0##. The issue is that I am missing the spatial components ##(6.1), (6.2), (6.3)##... I've been discussing this problem and related but we did not manage to really understand it. Any help is really appreciated. Thank you :biggrin: [/QUOTE]
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