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I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me:

The Euler–Lagrange equation, then, is given by $$L_x(t,q(t),q'(t))-\frac{\mathrm{d}}{\mathrm{d}t}L_v(t,q(t),q'(t)) = 0.$$ where $L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments, respectively.

Specifically, I don't understand the meaning of: $$L_x = \frac{\partial L(t, q, q')}{\partial q}$$

I'm confused because $q$ and $q'$ are both functions of $t$, but taking a partial derivative requires that we hold other parameters constant.

Yet $t$ is a parameter of $L$ as well. Thus when differentiating $L$ with respect to $q$, we hold $t$ constant.

But if $t$ is constant then doesn't that mean $q$ and $q'$ are constant?
And hence mustn't the derivatives $L_x$ and $L_v$ both be equal to zero?

user541686
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  • A related question: http://math.stackexchange.com/questions/307411/derivative-of-fx-y-with-respect-to-another-function-of-two-variables-kx-y – Jeff Snider Jan 30 '14 at 04:01
  • @JeffSnider: Thanks, I'm looking at that right now, it seems very relevant indeed. – user541686 Jan 30 '14 at 04:05
  • I have seen somewhere that you can use the chain rule to reach $$L_x=\frac{\frac{d}{dt}L(t,q,q')}{\frac{d}{dt}q}.$$ – Jeff Snider Jan 30 '14 at 04:10

2 Answers2

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Perhaps better for CS backgrounds: $\partial/\partial t \ F(t,x,y)$ means the derivative of the function with respect to variation in its first slot, the formal parameter $t$. $\mathrm{d}/\mathrm{d} t\ F(t,x,y)$ means the derivative of the function with respect to the system parameter $t$. The difference is between local parameter names (partial differentiation) and global variables (total differentiation).

Eric Towers
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While we write $L$ as $L(t,q(t),q'(t))$, it is really just a function of three variables. We could just as easily write it as $L(x,y,z)$ and, when evaluating $L$ on a given function $q$, substitute $t$ for $x$, $q(t)$ for $y$, and $q'(t)$ for $z$. However, this could lead to some confusion and lots of extra notation, so it is not usually done.

So, when taking the partial derivative $\frac{\partial L}{\partial q}$, we can just ignore any relations between $q$ and $q'$ and treat them as totally separate variables.

William Ballinger
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    +1 ah that makes sense, I get it now! I'll explain it another way in case someone else who sees this will find it helpful (especially if they had a CS background like me): the function $L$ is defined for any $q$ and $q'$; the fact that the two are related and functions of $t$ is not relevant to the differentiation of $L$ with respect to either of them. When the derivative is evaluated at a particular $q$, if the parameters are related, then this will of course matter. But the original derivative does not care about whether or not the three parameters are functions of each other. – user541686 Jan 30 '14 at 04:22