What Is the Equation of Motion for a Particle with a Given Lagrangian?

[Niishi]

Homework Statement

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.

The Attempt at a Solution

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

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

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

$\frac{dV}{dx}\left(m\dot{x}^2-2V\right)- m\ddot{x}\left(m\dot{x}^2 +2V\right) =0$

 

[pam]

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