# Lagrangian equation for unconstrained motion

1. May 11, 2013

### heycoa

1. The problem statement, all variables and given/known data
Write down the Lagrangian for a one-dimensional particle moving along the x axis and subject to a force: F=-kx (with k positive). Find the Lagrange equation of motion and solve it.

2. Relevant equations
Lagrange: L=T-U (kinetic energy - potential energy)

3. The attempt at a solution
All i really need help with is finding the potential energy in this problem. I believe that the kinetic energy is T=(1/2)*m*x', where x' is d/dt(x). I dont understand how to get the potential energy out of that force. Please help, thank you.

2. May 11, 2013

### George Jones

Staff Emeritus
How do you go from potential energy to force?

Do recognizes the force equation that you have been given?

3. May 11, 2013

### heycoa

The force equation looks like that of a spring. But as far as I can remember, you go from potential energy to force by multiplying the force by the distance. This problem just seems weird to me.

4. May 12, 2013

### George Jones

Staff Emeritus
Yes. What is the potential energy for a spring?

No, it is the other way around, and also the infinitesimal version.

Do have a textbook that you can read?

5. Dec 10, 2014

### A2Airwaves

I'm currently working on the exact same question, and using : $$-\frac{\partial U}{\partial x} = F$$ I integrated $F=-kx$ and got : $$T= \frac{1}{2}m\dot{x}^{2},U = kx^{2}$$

Is this correct?
Is T=(1/2)mv^2 always what you substitute in for a simple kinematics question?

6. Dec 11, 2014

### vela

Staff Emeritus
No, that's not correct. The integral of $x$ is $\frac 12 x^2$.

7. Dec 15, 2014

### A2Airwaves

Thank you for the correction, I was able to get the problem right :)