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up vote 43 down vote accepted

There are several reasons for using the Hamiltonian formalism:









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Some more comments to add to user1504's response:




In Path Integrals, lagrangian or hamiltonian are fundamental?

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Lagrangian has a clear physcial meaning. That is the difference between the kinetic and potential energy. – Dimiter P Jan 8 at 0:25






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And Hamiltonian's underlie quantum mechanics:




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Also, you can write Hamilton's equations of motion in sympletic form: $$ \dot\xi_i = \omega_{ij}\frac{\partial H}{\partial\xi_j} $$

Where $\xi_i$ are the coordinates in the phase space, that is, $\xi = (\mathbf q, \mathbf p)$. And, $\omega$ is the sympletic matrix: $$ \omega = \begin{bmatrix} 0 && -I_{n\times n} \\ I_{n\times n} && 0 \\ \end{bmatrix} $$

Where $I_{n\times n}$ is the identity matrix, with a system of $n$ spatial coordinates (and thus, $n$ speeds, and those, $2n$ amounts of phase space coordinates). Also, for an observable $G$, we have: $\dot G = \{G, H\}$ as you know. So, you can easily have the dynamics of an given observable $G$. All very nice and neat and general, but....

But.. here is what I consider to be the most amazing part of hamiltonian mechanics: $$ X = x^i\partial_i = \{\xi^i, H\}\partial_i $$

Where $X$ is an hamiltonian vector field. Now, instead, we can generalize for an observable $G$, its vector field: $$ X_G = x^i_G\partial_i, \quad x^i_G = \{\xi^i, G\} = \frac{d\xi^i}{d\epsilon} $$

For any given parameter $\epsilon$ for observable $G$, generating an operator $X_G$. Its first order taylor expansion: $$ \xi^i(\epsilon) - \xi^i(\epsilon_0) = (\epsilon - \epsilon_0)X_G\xi^i $$

Where operator $X_G$ is acting on $\xi^i$. We can solve the differential equation in sucessive infinitesimal transformations, arriving in the fundamental exponential limit, thus having the complete general solution of any hamiltonian system for any observable $G$: $$ \xi^i(\epsilon) = \exp\left(\Delta\epsilon X_G\right)\xi^i_0 $$

Do you understand the power of it?? Pointing out, again, here it is the solution of any hamiltonian system for any observable $G$ with parameter $\epsilon$ generated by operator $X_G$. If you want to analyse dynamics, then $\epsilon$ is the time, and $G$ is the hamiltonian, where $X_H$ defines the hamiltonian vector space. All hamiltonian systems has the same solution. The same solution!! So, lets solve for the dynamics (ie, where $\epsilon$ is time): $$ \xi^i(t) = \exp\left(\Delta t\frac{d}{dt}\right)\xi^i_0 $$

So, as you can see, pretty nice. Lagrange mechanics gives you nice unified equations of motion. Hamiltonian mechanics gives nice phase-space unified solutions for the equations of motion. And also gives you the possibility get an associated operator, and a coordinate-independent sympletic-geometrical interpretation. The former is crucial in quantum mechanics, the later is crucial in dynamical systems.

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