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I am looking for ideas for an introduction to the hamiltonian formalism without having to establish Lagrangians etc.

I know I can get Hamiltonian's eqn. of motion directly from looking at extrema of $S=\int p \dot{q} - H \mathrm{d}t$, however I am lacking a heuristic argument why one would need the $p \dot q$ in the beginning. Furthermore, is there a Lagrangian-independent way of defining the canonical momenta?

Thanks!

Faser
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    Asking for "a Lagrangian-independent way of defining the canonical momenta" is equivalent to asking for "a Hamiltonian-independent way of defining the velocities". The question is not really meaningful. – AccidentalFourierTransform Mar 15 '17 at 14:07
  • If the Hamiltonian was always the total energy then maybe this could be done but the conjugate momentum needs to be defined from the Lagrangian if Hamilton's equations are to simplify to their canonical forms. – ZeroTheHero Mar 15 '17 at 14:14
  • Possible duplicate: http://physics.stackexchange.com/q/230934/2451 – Qmechanic Mar 15 '17 at 14:23
  • @AccidentalFourierTransform, I don't see why this is not a meaningful question - of course it should legitimate to at least ask if it is possible to have the notion of canonical momenta without defining $p= d L / d\dot{q}$. I am thinking here of the symplectic approach and momenta as elements of the cotangent bundle, and was wondering if I can motivate canonical momenta without having to introduce a Lagrangian first. – Faser Mar 15 '17 at 15:37
  • @ZeroTheHero, minimizing the action I introduced yields the canonical form of hamilton's eqns – i was wondering if it is possible to motivate this action without using the lagrangian (i.e.. by some phase space argument) – Faser Mar 15 '17 at 15:39
  • @Qmechanic, I am aware of this question but this seems to be more formal – i am rather looking for an heuristic/intuitive approach, assuming a Lagrangian could be defined (I just don't want to due to some time/length constraints) – Faser Mar 15 '17 at 15:41

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There are several equivalent formulations of mechanics, each of which requires us to accept some principle; but we really can take any formulation as fundamental. In this instance, we want to start with Hamiltonian mechanics, possibly building Lagrangian mechanics from it later.

For Newtonian mechanics, our starting principle is Newton's second law; for Lagrangian mechanics, it is the Euler-Lagrange equation or an equivalent stationary-integral principle, for Hamiltonian mechanics, we can use Hamilton's equations. You can write these equations as $\dot{\mathbb{y}}=\Omega\nabla_\mathbb{y} H$, with $\mathbb{y}$ combining $\mathbb{q},\,\mathbb{p}$ into a single vector of which $H$ is a function and $\Omega$ an antisymmetric matrix on the phase space in which $\mathbb{y}$ lives.

Phase space essentially puts the "two types" of coordinates on one footing. The symplectic product $\mathbb{y}^T\Omega \mathbb{z}$ makes it natural to think of the phase space as a symplectic manifold. From this perspective, time derivatives of $q$s can be written in terms of $q$s and $p$s, and the Lagrangian can be defined as $\dot{q}p-H$ written in terms of $q$s and their time derivatives.

See also Chapter 2 of Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.

J.G.
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