Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian?
Say we are given the Minkowski Lagrangian
$$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$
The Hamiltonian can then be found by Legendre transformation
$$H = \dot{\phi}\frac{\partial L_m}{\partial\dot{\phi}} - L_m,$$
which equals $$H = \frac{1}{2}\dot{\phi}^2 + V(\phi)$$ which was not hard.
Now consider the corresponding Euclidean Lagrangian
$$L_e = -\frac{\dot{\phi}^2}{2} - V(\phi).$$
How do I calculate the Hamiltonian in this formalism? The above way applied naively will not give the correct result.