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**1. Homework Statement**

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

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

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**

[itex]

\frac{\partial{L}}{\partial{x}}= m\dot{x}^2\frac{dV}{dx}- 2V\frac{dV}{dx} [/itex]

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

[itex] m\dot{x}^2\frac{dV}{dx}- 2V\frac{dV}{dx} - m^2\dot{x}^2\ddot{x} - 2mV\ddot{x} = 0 [/itex]

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

**1. Homework Statement**

**2. Homework Equations**

**3. The Attempt at a Solution**