Question

relation between lagrangian and hamiltonian

Answer 1

Classical mechanics:

Lagrangian : L  = T - V
T = Kinetic energy of the system of particles.
V = Potential energy of the particles in a system.

Hamiltonian:   H  = T + V   = total energy.
 H is conserved in conservative fields.

   H and L are both expressed as functions of positions and velocities of particles in the system.

              H + L = 2 T = Sum of  p * v of all particles.
           H (x , v , t)  + L (x , v, t) = F(x, v) = Sum of (p * v)

Hamiltonian is the Legender transformation of the Lagrangian function of position and velocities.