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How does it make sense to vary the position and the velocity independently?

Edit:

Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics student ask this question when he learns calculus of variations? Does anybody ever answer this question? Ever? If so, please educate me.

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Could you clarify a little, please? Calculus of variations itself is a math topic, so what particular physics application do you have in mind? Do you mean something along the lines of "Does it make sense to apply the Euler-Lagrange equations to the problem of minimizing an action, given that doing so requires treating position and velocity as independent variables, when physically if you know the position as a function of time, the velocity is completely specified?" –  Mark Eichenlaub Nov 16 '10 at 6:52
    
What's the problem? You can treat position and velocity as different variables of a function, that's straightforward maths. You're not at all saying they're independent. –  Noldorin Nov 17 '10 at 14:24
    
Well, maybe I'm just stupid. –  grizzly adam Nov 17 '10 at 18:23
    
I apologize for getting overly excited about this. That was uncalled for. –  grizzly adam Nov 22 '10 at 4:29
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Superb question aksink about the very fundament of everything we calculate. Furthermore - provoking great answers. You are very welcome to share your doubts with us @grizzly adam :) Greets –  Robert Filter Jan 14 '11 at 19:35
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6 Answers

Unlike your question suggests, it is not true that velocity is varied independently of position. A variation of position $q \mapsto q + \delta q$ induces a variation of velocity $\partial_t q \mapsto \partial_t q + \partial_t (\delta q)$ as you would expect.

The only thing that may seem strange is that $q$ and $\partial_t q$ are treated as independent variables of the Lagrangian $L(q,\partial_t q)$. But this is not surprising; after all, if you ask "what is the kinetic energy of a particle?", then it is not enough to know the position of the particle, you also have to know its velocity to answer that question.

Put differently, you can choose position and velocity independently as initial conditions, that's why the Lagrangian function treats them as independent; but the calculus of variation does not vary them independently, a variation in position induces a fitting variation in velocity.

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The answer to your main question is already given -- you do not vary coordiante and speed independently. But it seems that your main problem is about using coordinate and speed as independent variables.

Let me refer to this great book. The very first line of the book (where an author usually says to whom this book is devoted) is this:

William Burke

It is true that from time to time student do ask this question. But attempts to explain it "top down" are usually just lead to more and more questions. One really needs to make mathematical "bottom up" order in the topic. Well, as the name of the book suggest -- the mathematical discipline one needs is differential geometry.

I cannot retell all the details, but briefly it looks like this:

  • You start with a configuration space $M$ of your system. $M$ is a (differentiable) manifold, and $q$ are the coordinates on this manifold.
  • Then there is a specific procedure, that allows you to add all the possible "speeds" at every given point of $M$. And you arrive at the tangent bundle $TM$, which is a manifold too, and ($q$,$\dot{q}$) are different coordinates on it.
  • Lagrangian is a function on $TM$.
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I have this book and tried to read it. But it lacks clear definitions, and I found it to be more frustrating than enlightening. Additionally, I don't believe that it is necessary to know differential geometry to understand calculus of variations. That's like saying that you can't understand arithmetic unless you know set theory. –  grizzly adam Jan 15 '11 at 4:49
    
First of all, as I said, you are mixing up two different points: about variational calculs and about independence of speeds and coordinates. Second -- I didn't say that you have to read only one book to understand DG. –  Kostya Jan 15 '11 at 9:40
    
I think to really appreciate Lagrangian and Hamiltonian mechanics you do need to understand some differential geometry. Arnold says in his book Mathematical Methods of Classical Mechanics that "Hamiltonian mechanics cannot be understood without differential forms." This book, by the way, will teach you the differential geometry you need to get started, assuming just some calculus. –  Robin Ekman Apr 3 at 10:38
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Considering what Greg Graviton wrote, I'll write out the derivation and see if I can make sense of it.

$$ S = \int_{t_1}^{t_2} L(q, \dot q, t) dt $$

where S is the action and L the Lagrangian. We vary the path and find the extremum of the action:

$$ \delta S = \int_{t_1}^{t_2} ({\partial L \over \partial q}\delta q + {\partial L \over \partial \dot q}\delta \dot q) dt = 0. $$

Here, q and $\dot q$ are varied independently. But then in the next step we use this identity,

$$ \delta \dot q = {d \over dt} \delta q. $$

And here is where the relationship between q and $\dot q$ enters the picture. I think that what is happening here is that q and $\dot q$ are treated as independent initially, but then the independence is removed by the identity.

$$ \delta S = \int_{t_1}^{t_2} ({\partial L \over \partial q}\delta q + {\partial L \over \partial \dot q}{d \over dt} \delta q) dt = 0 $$

And then follows the rest of the derivation. We integrate the second term by parts:

$$ \delta S = \lbrack {\partial L \over \partial \dot q}\delta q\rbrack_{t_1}^{t_2} + \int_{t_1}^{t_2} ({\partial L \over \partial q} - {d \over dt}{\partial L \over \partial \dot q})\delta q dt = 0, $$

and the bracketed expression is zero because the endpoints are held fixed. And then we can pull out the Euler-Lagrange equation:

$$ {\partial L \over \partial q} - {d \over dt}{\partial L \over \partial \dot q} = 0. $$

Now it makes more sense to me. You start by treating the variables as independent but then remove the independence by imposing a condition during the derivation.

I think that makes sense. I expect in general other problems can be treated the same way.

(I copied the above equations from Mechanics by Landau and Lifshitz.)

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Well, instead of saying "$q$ and $\dot q$ are varied independently", you could also say "$q$ and $\dot q$ are varied (perhaps independently, perhaps not)" and later note that the variation $\delta\dot q$ is given by $\delta\dot q = \frac d{dt} \delta q$. –  Greg Graviton Nov 17 '10 at 9:30
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The notation for the arguments $L$ is somewhat confusing, in which case it is instructive to consider the following example: take $F(x,2x-y)$ and vary $F(x+\delta x,2(x+\delta x)-y) = \frac{\partial F}{\partial x}\delta x + \frac{\partial F}{\partial (2x-y)}2\delta x$. You might say that the arguments of $F$ are varied independently, but that sounds weird. If anything, it's just that the notation for the partial derivatives of $F$ is bad; it is much better to write $F(u,v)$ and $(u,v)=(x,2x-y)$ to obtain $\delta F = \frac{\partial F}{\partial u}\delta u + \frac{\partial F}{\partial v}\delta v$ –  Greg Graviton Nov 17 '10 at 9:33
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... and to express the variations $\delta u$ and $\delta v$ in terms of $\delta x$ afterwards. –  Greg Graviton Nov 17 '10 at 9:34
    
Yes, the notation is confusing. That's another problem. –  grizzly adam Nov 17 '10 at 17:02
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"Not a word of Landau, not a thought of Lifshitz." –  wsc Jan 14 '11 at 15:57
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While it is true that the function q-dot(t) is the derivative of the function q(t) w.r.t. time, it is not true that the value q-dot is at all related to the value q at a given point in time, since a value is just a number, not a function. The action is a functional of q(t), and so it would make no sense to vary the action both w.r.t. q and q-dot. But the Lagrangian L(q,q-dot) is a function of the values q and q-dot, not a functional of the functions q(t) and q-dot(t). We can promote L to a function of time if we plug in q(t) and q-dot(t) instead of just q and q-dot. (Remember a functional turns a function into a number, e.g., S[q], whereas a function turns a value into a number, e.g., L(q,q-dot).

To solve for q(t) we extremize the action S, by demanding that it is extremal at every point, t. This is equivalent to solving the Euler Lagrange equations at every point t. Since at any point t the values q and q-dot are independent, they can be varied independently.

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Here is my answer, which is basically an expanded version of Greg Graviton's answer.

The question of why one can treat position and velocity as independent variables arise in the definition of the Lagrangian $L$ itself, before one thinks about varying the action $S:=\int_{t_i}^{t_f}dt \ L$, and has therefore nothing to do with calculus of variation.

On one hand, let us consider first the role of the Lagrangian. Let there be given an arbitrary but fixed instant of time $t_0\in [t_i,t_f]$. The (instantaneous) Lagrangian $L(q(t_0),v(t_0),t_0)$ is a function of both the instantaneous position $q(t_0)$ and the instantaneous velocity $v(t_0)$ at the instant $t_0$. Here $q(t_0)$ and $v(t_0)$ are independent variables. Note that the (instantaneous) Lagrangian $L(q(t_0),v(t_0),t_0)$ does not depend on the past $t<t_0$ nor the future $t>t_0$. (One may object that the velocity profile $\dot{q}\equiv\frac{dq}{dt}:[t_i,t_f]\to\mathbb{R}$ is the derivative of the position profile $q:[t_i,t_f]\to\mathbb{R}$, so how can $q(t_0)$ and $v(t_0)$ be truly independent variables? The point is that one is still entitled to make two independent choices of initial conditions.) We can repeat this argument for any other instant $t_0\in[t_i,t_f]$.

On the other hand, let us consider calculus of variation. The action functional $S[q] := \int_{t_i}^{t_f}dt \ L(q(t),\dot{q}(t),t)$ depends on the whole (perhaps virtual) path $q:[t_i,t_f]\to\mathbb{R}$. Here the time derivative $\dot{q}\equiv\frac{dq}{dt}$ do depend on the function $q:[t_i,t_f]\to \mathbb{R}$. Extremizing the action functional

$$0=\delta S = \int_{t_i}^{t_f}dt\left[\left.\frac{\partial L(q(t),v(t),t)}{\partial q(t)}\right|_{v(t)=\dot{q}(t)} \delta q(t) +\left.\frac{\partial L(q(t),v(t),t)}{\partial v(t)}\right|_{v(t)=\dot{q}(t)}\delta \dot{q}(t)\right] $$ $$ = \int_{t_i}^{t_f}dt\left[\left.\frac{\partial L(q(t),v(t),t)}{\partial q(t)}\right|_{v(t)=\dot{q}(t)} \delta q(t) +\left.\frac{\partial L(q(t),v(t),t)}{\partial v(t)}\right|_{v(t)=\dot{q}(t)}\frac{d}{dt}\delta q(t)\right] $$ $$ = \int_{t_i}^{t_f}dt\left[\left.\frac{\partial L(q(t),v(t),t)}{\partial q(t)}\right|_{v(t)=\dot{q}(t)} - \frac{d}{dt}\left(\left.\frac{\partial L(q(t),v(t),t)}{\partial v(t)}\right|_{v(t)=\dot{q}(t)} \right)\right]\delta q(t) $$ $$+ \int_{t_i}^{t_f}dt\frac{d}{dt}\left[\left.\frac{\partial L(q(t),v(t),t)}{\partial v(t)}\right|_{v(t)=\dot{q}(t)}\delta q(t)\right] $$

with appropriate boundary conditions leads to Euler-Lagrange equation,

$$ \frac{d}{dt}\left(\left.\frac{\partial L(q(t),v(t),t)}{\partial v(t)} \right|_{v(t)=\dot{q}(t)} \right) = \left.\frac{\partial L(q(t),v(t),t)}{\partial q(t)} \right|_{v(t)=\dot{q}(t)} $$

Note that $\frac{d}{dt}$ is a total time derivative, not an explicit time derivative $\frac{\partial}{\partial t}$, so that Euler-Lagrange equation is really a second-order ordinary differential equation (ODE),

$$\left(\ddot{q}(t)\frac{\partial}{\partial v(t)}+\dot{q}(t)\frac{\partial}{\partial q(t)}+\frac{\partial}{\partial t}\right) \left. \frac{\partial L(q(t),v(t),t)}{\partial v(t)} \right|_{v(t)=\dot{q}(t)} = \left.\frac{\partial L(q(t),v(t),t)}{\partial q(t)} \right|_{v(t)=\dot{q}(t)} $$

To solve for the path $q:[t_i,t_f]\to \mathbb{R}$, one should specify two initial conditions, e.g., $q(t_i)=q_i$ and $\dot{q}(t_i)=v_i$.

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The derivative of a function $f(t)$ is the function $\dot{f}(t)$ in general different than $f$, and in the general case the two are not even linearly dependent, which is simple to see if you take the Taylor expansion. It is only after you define differential equations with them that they are linked algebraically, and this is what the calculus of variations does.

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