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To 'derive' conservation of Energy from $ \vec{F} = m \vec{a}$, we take a dot product ($\hat{i} \cdot \hat{i} = 1$) which means that we have one (scalar multi-variable non-linear differential) equation with potentially many unknowns. Energy is nice because it provides a common language with all the physical sciences, but in classical mechanics, it's mostly ...


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This question has been marinating in my own mind for some time since asking here, and seeing how the respected community of this site has not yet answered this head on in the awesome manner I have seen other answers, I will share my "answer" as far as i have been able to develop it on my own till now (which means not far at all, but still I hope it ...


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These solutions are preferred because they directly embody the scale invariance of the equation. In general, when a physical problem has some sort of symmetry - like the parabolic dilation invariance of the heat equation - then this establishes a corresponding action of the symmetry group on the solutions. The canonical forms based on dimensionless ...


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Intuitively a lot of equations we use to describe physical phenomena such as $exp(x)$ take in only dimensionless variables. Moreover in physics we're always taking the derivative or the log of things, and it works out to be much simpler if you're not always having to deal with a dimensioned constant factor that drops out every time you take the derivative.


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The key concept is that the units of measurements you chose to use make no difference to the physical behaviour. I.e you expect the solution behaviour to be independent of the units of measurement used. This means you should get the same solution if we chose to measure $x$ is in meters, and $t$ is in seconds and $D$ is in $m^2/s$ as if we chose to measure ...


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Let's say your goal is to describe the shape of some object, such as a box. You could create a completely arbitrary ruler and measure the three axes of the box, coming out for example with lengths of 11.72, 23.44, and 35.16 of your arbitrary ruler units. Or you might look at your results more closely and think hmm, something is going on here, since the ...


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Dimensionless equations have the advantage that they work for any value of the parameters. They are scale invariant. So the solution in terms of a single dimensionless variable applies to all values of $D$ and $t$. It also allows the definition of characteristic values for the dynamic variables. In your example, one could say $u_0$ = ...


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Because it is easier for dimensionless quantities to be combined in arbitrary polynomial terms (or other terms e.g exponential) with no loss (or extra) factors. Think like "characteristic times" used in exponential factors. Especially quantities appearing in solutions of the form $e^{a} = \sum_0^\infty \frac{a^n}{n!}$, one can see why a dimensionless ...


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I think the problem is that while in the first case your differential equation applies to all your domain of interest and you can just use it, in the second situation the DE doesn't apply at x=0. This means you need to solve the DE at the domains $0<x<L/2$ and $-L/2<x<0$ separately, where it does still apply. When applying it to those domains ...


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I'd love to see more elaborate answers to this, but generally the idea is intuitive. Resistors are linear and only dissipate energy. Inductors and capacitors are nonlinear and store magnetic and electric energy respectively. Their constitutive relations are the following: Inductor: $v = L\frac{di}{dt}$ Capacitor: $i = C\frac{dv}{dt}$ It's only when you ...



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