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During the introductory class on Hamiltonian mechanics after we had learnt the Lagrangian formalism, our professor said

"We cannot use Lagrangian mechanics to find the state of a system at any particular instant (he did not give any explanation), so that is why we use Hamiltonian mechanics."

But I think he could be incorrect, because when I did problems using Lagrangian as well as Hamiltonian I could find the generalized coordinates as a function of time. Can anybody explain where is the conceptual loop hole?

Qmechanic
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Muhmmad Awais
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2 Answers2

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  1. OP's professor is possibly referring to the fact that the stationary action principle is a boundary value problem (BVP) [as opposed to an initial value problem (IVP)].

    In contrast, EOMs together with initial conditions (ICs) can in principle be integrated from an initial time to find the trajectory as a function of time.

  2. The issues of ICs vs. BCs for the principle of stationary action are already covered in this & this related Phys.SE posts and this related Math.SE post.

Qmechanic
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The action principle and the Lagrangian give you the equation of motion. The EOM tells you which states of the system are possible. By applying conditions, for example initial ones, a subset of these states is selected. The boundary conditions of the action principle are unrelated to the conditions imposed on the solution of the EOM.

As far as I understand the question, the Hamiltonian method assumes the EOM as a starting point. This approach is part of the Lagrangian approach.

my2cts
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