I have problem with Lagrange's equations and their derivation, the way it is presented in Goldstein's "Classical Mechanics". I have never seen this problem mentioned anywhere, so I wonder if I am the only one who see this problem.(adsbygoogle = window.adsbygoogle || []).push({});

To see the problem, consider a simple case where the motion of a system can be described by just one generalized coordinate [itex]q[/itex], plus time explicitly. Let [itex]x[/itex] be a cartesian coordinate of one particle of the system. Then [itex]x=x(q,t)[/itex].

Then, [tex]\dot x=\frac{\partial x}{\partial q}\dot q+ \frac{\partial x}{\partial t}.[/tex]

(In Goldstein we have several generalized coordinates, but the reasoning is the same.)

Now, reasoning as Goldstein, we could without problem differentiate [itex]\dot x[/itex] wrt [itex]\dot q[/itex], obtaining [itex]\partial \dot x / \partial \dot q =\partial x/\partial q[/itex], and also plug in [itex]\dot q [/itex] in the expression for the total kinetic energy [itex]T[/itex] and the Lagrangian [itex]L=T-V[/itex], and then without problem differentiate [itex]L[/itex] wrt [itex]\dot q[/itex], obtaning Lagrange's Equations (one for each generalized coordinate):

[tex]\frac d {dt}\frac{\partial L}{\partial\dot q}-\frac{\partial L}{\partial q}=0. [/tex]

(Differentiating [itex]L[/itex] wrt [itex]q[/itex] seems to be no problem, either.)

But how can we differentiate [itex]\dot x[/itex] (and then [itex]L[/itex]) wrt [itex]\dot q[/itex]? In the formula we obatined for [itex]\dot x[/itex], we have only one indepentent variable: [itex]t[/itex], and therefore it shouldn't make sense to differentiate wrt any other variable.

To be able to differentiate wrt [itex]\dot q[/itex], we must actually view [itex]\dot x[/itex] (and similarly, [itex]L[/itex]) as acompositefunction, [itex]\dot x= f(q,\dot q,t) [/itex], where [itex]f(u_1,u_2,u_3)=g_1(u_1,u_3)u_2+g_2(u_1,u_3)[/itex] and [itex]g[/itex] is the function which gives the coordinate transformation: [itex]x=x(q,t)=g(q,t)[/itex], and [itex]f[/itex] and [itex]g[/itex] can be differentiated wrt any of their arguments. ([itex]g_1[/itex] and [itex]g_2[/itex] are the partial derivatives of [itex]g[/itex].)

The expression [itex]\partial \dot x /\partial \dot q[/itex] will then make sense if we recognize it as the same as [itex]f_2(q,\dot q,t)[/itex], and then it will be equal to [itex]\partial x/\partial q[/itex], just as it should be.

However, this will be consistentonly ifthe choice of the function [itex]f[/itex] is unique. If there was another function [itex]h(u_1,u_2,u_3)\ne f(u_1,u_2,u_3)[/itex], which also satisfies [itex]\dot x = h(q,\dot q,t)[/itex], but for which [itex]h_2(q,\dot q,t)\ne f_2(q,\dot q,t)[/itex], then [itex]\partial \dot x/\partial \dot q[/itex] cannot be uniquely defined (and neither can [itex]\partial L/\partial \dot q[/itex]).

This problem is not discussed in Goldstein, and I have not seen it mentioned anywhere (although I did not search very much). Am I really the only person who see this problem?

Fortunately, the problem can be solved. It turns out that the function [itex]f[/itex] above must be unique. The reason is that the equation [itex]\dot x= f(q,\dot q,t) [/itex] must holdfor all possible paths[itex]q(t)[/itex], and, for every triplet of values [itex](u_1,u_2.u_3)[/itex], it is always possible to find a path [itex]q(t)[/itex] such that [itex]q(t)=u_1[/itex] and [itex]q'(t)=u_2[/itex], for [itex]t=x_3[/itex]. This implies that [itex]f[/itex] is uniquely determined by the requirement [itex]\dot x= f(q,\dot q,t)[/itex], it must be the function given above. This argument can be generalized to any number of generalized coordinates, and it also follows that [itex]\partial L/\partial \dot q[/itex] will be meaningful and unique.

But it is not trivial to find and prove this. I was very confused when I read Goldstein the first time, and I wondered how on Earth one could differentiate wrt [itex]\dot q[/itex]. It took a while until I found the "proof" above. I think it is the author's job to do this, not the reader's.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Problem with Lagrange's Equations

**Physics Forums | Science Articles, Homework Help, Discussion**