# Lagrangian mechanics(goldstein)

1. Mar 12, 2008

### Niishi

1. The problem statement, all variables and given/known data

A particle of mass m moves in one dimensional such that it has the Lagrangian

$L=\frac{m^2\dot{x}^4}{12} + m\dot{x}^2V\left(x\right)- V^2\left(x\right)$

where V is some differentiable function of x only.
(i) Find the equation of motion for x(t)

(ii) Find an expression for the total energy and hence describe the physical nature of the system on the basis of this equation.

3. The attempt at a solution

$\frac{\partial{L}}{\partial{x}}= m\dot{x}^2\frac{dV}{dx}- 2V\frac{dV}{dx}$

$\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right) = m^2\dot{x}^2 + 2mV\ddot{x}$

$m\dot{x}^2\frac{dV}{dx}- 2V\frac{dV}{dx} - m^2\dot{x}^2\ddot{x} - 2mV\ddot{x} = 0$

$\frac{dV}{dx}\left(m\dot{x}^2-2V\right)- m\ddot{x}\left(m\dot{x}^2 +2V\right) =0$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 12, 2008

### pam

In your second line, m^2xdot^2 is wrong.
You took the d/dt wrongly.