Classical Mechanics Goldstein 2.16

[punkimedes]

Homework Statement

In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion for the lagrangian $L = e^{γt} (\frac{m\dot{q}^2}{2} - \frac{kq^2}{2})$. How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form $s = e^{γt/2}q$. What is the effective Lagrangian in terms of $s$? Find the equation of motion for $s$. What do these results say about the conserved quantities for the system.

Homework Equations

Lagrange-euler equation $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$

The Attempt at a Solution

Using the lagrange-euler equation, I came up with one equation of motion $m\ddot{q}+mγ\dot{q}+kq = 0$

Is this correct? I'm not sure where to go from here. Can I solve this for $q$ like a normal second order differential equation? Can I treat $γ$ as a constant knowing nothing about it? If so, the factoring doesn't seem to come out right. (let $y=e^{rt}$, $my^2+mγy+k = 0$, then ?)

Also, I'm not really sure what is meant by a point transformation. Does that simply mean replace that factor in the original lagrangian?

 

[BvU]

Hello PM, and wecome to PF.

I dug up your exercise (2.14) in the second edition and there it has $s = e^{γt}q$. Would that have been changed ?

The EL eqn gives you the equation of motion, which I think you found correctly.

Note the exercise wants you to write out the effective Lagrangian for $s$ and then find the equation of motion for $s$.

Oh, and: yes $\gamma$ is a constant.

 

[punkimedes]

Thanks BvU. I should have mentioned I have the third edition. The problem shows in this book exactly as I wrote it. I guess I'm stuck on the first part of the problem. I don't know how to interpret equation of motion from the lagrangian without solving the differential equation, and the equation I've come up with is unfactorable.

 

[vela]

Homework Statement

In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion for the lagrangian $L = e^{γt} (\frac{m\dot{q}^2}{2} - \frac{kq^2}{2})$. How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form $s = e^{γt/2}q$. What is the effective Lagrangian in terms of $s$? Find the equation of motion for $s$. What do these results say about the conserved quantities for the system.

Homework Equations

Lagrange-euler equation $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$

The Attempt at a Solution

Using the lagrange-euler equation, I came up with one equation of motion $m\ddot{q}+mγ\dot{q}+kq = 0$

You should recognize this differential equation, particularly if $\gamma=0$. It might also help to rewrite it as
$$m\ddot{q} = -m\gamma \dot{q} - kq$$ or perhaps use $x$ instead of $q$ so it looks more familiar.

 

[BvU]

Dear PM, what I meant to express is that you already found the equation of motion (there is only one coordinate, so there is only one equation in this exercise). It looks a lot like F = ma, which is not a coincidence. F consists of two terms, one (-kq) having to do with displacement from q = 0, the other something like -$\beta \dot q$ ($\beta = \gamma$). You are not asked to solve, but to describe the system. How would you describe the system if $\gamma=0$? What could the other term represent ?

And yes, Herbie wants you to rewrite L(q) as L(s). Work it out and show us...