How to find Equation of motions and Hamilton Function for this Lagrangian?

Question

$$L = \frac{m \dot{x}^2}{2} - \exp(|x|) $$

I would appreciate if you could explain the steps needed to get the answer

Answer 1

I feel the discussion was getting too long in the comments. Since we've established the system has one degree of freedom we find that there is only one Euler-Lagrange equation to solve:

$$\frac{\partial \mathcal{L}}{\partial x} = \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{x}}.$$

Remember that $\dot{x}$ means the time derivative of $x$, i.e.

$$\dot{x} := \frac{dx}{dt}.$$

Nevertheless, what matters is that you partially differentiate the Lagrangian and equate the parts. Are you sure that the potential is $\exp(|x|)$, the exponential of the modulus of $x$? This would lead to

$$\frac{\partial \mathcal{L}}{\partial x} = \text{sign}(x) \exp(|x|) = \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{x}} = \frac{\partial}{\partial \dot{x}} \frac{m \dot{x}^2}{2} = m \ddot{x}.$$

Quite hard to integrate.