Hamiltonian vs. Lagrangian

1. Aug 13, 2014

CMBR

1. The problem statement, all variables and given/known data
So I just learned how to derive the equation of motion under the Lagrangian formulation which involves finding the euler-lagrange equation when setting the change in action to zero, chain rule, integration by parts etc.. Then I learnt how to find the equations of motion under Hamiltonian formulation, you take the legendre transformation of the Lagrangian, then take partial derivative of the hamiltonian w.r.t momentum & general coordinates to find the Hamiltons equations.

I feel my fundamental understanding is just not there, is there no concept of "minimum action" when deriving equations of motion under the hamiltonian formulation? Is there no such thing as dS=∫dt (H-H0)=0 as there is dS=∫dt (L-L0)=0 in lagrangian mechanics?

2. Relevant equations

3. The attempt at a solution

2. Aug 13, 2014

Staff: Mentor

http://en.wikipedia.org/wiki/Hamiltonian_mechanics

Basically Hamiltonians describe the total energy of the system vs Lagrangian's which describe the difference between kinetic and potential energy to define the action which systems will tend to minimize as they change state ( aka least action or stationary action).

In contrast, the Hamiltonian can interpreted as follows:

I recall it was sometimes easier to find solutions using Hamiltonian mechanics first order DE vs the Lagranigian 2nd order DE but don't quote me on this. (too many years ago)