# Questions tagged [hamiltonian-formalism]

The Hamiltonian formalism is a formalism in Classical Mechanics. Besides Lagrangian Mechanics, it is an effective way of reformulating classical mechanics in a simple way. Very useful in Quantum Mechanics, specifically the Heisenberg and Schrodinger formulations. Unlike Lagrangian Mechanics, this formalism relies on a "Hamiltonian" instead of a Lagrangian, which differs from the Lagrangian through a Legendre transformation.

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### Doubt on the geometry of "quantum phase space"

In Jose & Saletan's "Classical Dynamics", they show the global structure of Hamiltonian mechanics: you then have a $Q$ manifold (configuration space), and the phase space structure is ...
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I am trying to transfom a quantum hamiltonian as detailed in this website. It starts of by using the classical hamiltonian and conjugate momenta: $$H\equiv \frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V\... • 31 0 votes 1 answer 80 views ### Two Liouville's theorem Within the context of Hamiltonian mechanics and phase spaces I have learnt that the phase space distribution function is constant, in other words, that the "volume" of any region is constant,... • 227 4 votes 0 answers 36 views ### General Proof that Noether Charge Generates a Symmetry [duplicate] I am trying to understand how one proves that a charge derived from the Noether procedure generates the corresponding symmetry. That is, I would like to prove that$$[Q, \phi(y)] = i\delta\phi(y)$$I ... • 619 7 votes 2 answers 822 views ### Any reason why the Lagrangian just happens to show up in the path integral? I saw that derivation in which they start with the Schrodinger equation propagator and introduce a resolution of identity between each term. And boom ! The lagrangian showed up in the phase. But....... • 917 0 votes 0 answers 33 views ### When is the motion of the perturbative term resonant with the Hamiltonian? I am given the following Hamiltonian, H, which is a perturbed version of H_0,$$ H(\theta,I) = H_0(I) -\epsilon \cos(\theta-\Omega t)$$where H_0 = \frac{I^2}{2}, \epsilon << 1 and (I,\... • 31 0 votes 0 answers 51 views ### Conjugate momentum vs translation generator with non-standard kinetic term I am reading this paper and for equation (2.5) (associated with the Lagrangian in eq 2.1) there is the claim that for a Lagrangian L(\varphi,A,\dot{\varphi},\dot{A}) containing an extra non-standard ... 1 vote 1 answer 103 views ### Ability to represent canonical transformation generating functions in different forms? In Goldstein's Classical Mechanics (3rd edition) section 9.1, we introduce the generating function method of describing a canonical transformation. We then introduce four types of generating functions:... 1 vote 0 answers 52 views ### Does the Legendre transformation describe two views on the same physical system or different physical systems? In mechanics we perform the Legendre transform to go from the Lagrangian L(q, \dot{q}) to the Hamiltonian H(q, p). This seems to be describing the same physical system. L and H both describe ... • 131 2 votes 0 answers 29 views ### How are conjugate variables in mechanics and stat mech related to duality in convex optimization? I recently studied duality in optimization where a primal optimization problem can be casted as a dual problem which provides meaningful lower bounds on the primal. There is also a notion of conjugate ... • 131 0 votes 1 answer 87 views ### About the curvature of solutions of Hamilton's equations I am a math major and have recently stumbled on the Hamilton's system of equations in the context of Hamiltonian Monte Carlo Markov chains on a continuous state space, say \mathbb{R}^d. I am trying ... • 103 2 votes 2 answers 140 views ### Hamiltonian from Lagrangian defined as an integral I need to derive the Hamiltonian from the Lagrangian defined in the following way:$$L[x, \dot{x}] = \int_{t_0}^{t_1} f(x(t), t) \sqrt{1 + \dot{x}^2} \mathrm{d}t.\tag{1}$$The usual method is to ... • 31 0 votes 1 answer 46 views ### Why inverse flow of separable Hamiltonian with even kinetic energy can be written like this? Why is it true that the inverse of the flow of a separable Hamiltonian with even kinetic energy can be written as \phi_N \circ \varphi_t \circ \phi_N where \varphi_t is the flow of the Hamiltonian ... • 237 2 votes 0 answers 46 views ### Does compactification of a Nambu-Goto string in one direction break Diff invariance? Assume we have a Nambu-Goto action, in phase space, for a closed string. If I compactify one coordinate of the target space, do I reduce the diff invariance of the system. We have$$S=\int d^2\sigma \...
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I cannot find a good, simple definition of time-reversibility of the flow $\phi_t$ of Hamilton's equations  \dot{z} = J^{-1}\nabla_z H(z) \quad \text{where} \quad z = (q, p)^\top \quad \text{and} \...