# Tagged Questions

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 ...

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### When is the Hamiltonian of a system not equal to its total energy?

I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical ...
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I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
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### Equivalence between Hamiltonian and Lagrangian Mechanics

I'm reading a proof about Langrangian => Hamiltonian and one part of it just doesn't make sense to me. The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian ...
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### Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?

When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics. Now when we studied QM, on the level of Griffiths, we ...
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### Sympletic structure of General Relativity

Inspired by physics.SE: http://physics.stackexchange.com/questions/15571/does-the-dimensionality-of-phase-space-go-up-as-the-universe-expands/15613 It made me wonder about symplectic structures in GR,...
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### A kind of Noether's theorem for the Hamiltonian formalism

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...
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### Lagrangian and Hamiltonian EOM with dissipative force

I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the ...
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### What's the point of Hamiltonian mechanics?

I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
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### Physical meaning of Legendre transformation

I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
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### Lagrangian of Schrodinger field

The usual Schrodinger Lagrangian is $$\tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi,$$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ ...
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### Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
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### Hamilton-Jacobi Equation

In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
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### Dirac equation as Hamiltonian system

Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
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While going from a given Lagrangian to Hamiltonian for a fermionic field, we use the following formula. $$H = \Sigma_{i} \pi_i \dot{\phi_i} - L$$ where $\pi_i = \dfrac{\partial L}{\partial \dot{\... 3answers 7k views ### What exactly are Hamiltonian Mechanics (and Lagrangian mechanics) What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)? I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ... 6answers 10k views ### What is canonical momentum? What does the canonical momentum$\textbf{p}=m\textbf{v}+e\textbf{A}$mean? Is it just momentum accounting for electromagnetic effects? 1answer 1k views ### In Path Integrals, lagrangian or hamiltonian are fundamental? When studying path-integrals one question arose to my mind... Which presentation is more fundamental to calculate the propagator? The one based on the Hamiltonian (phase space)? $$K(B|A) = \int \... 2answers 3k views ### Gauge Invariance of the Hamiltonian of the electromagnetic field The Hamiltonian for an electron of mass m and charge e in an exterior electromagnetic field is$$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$The corresponding (via canonical quantization) quantum ... 3answers 340 views ### What are some mechanics examples with a globally non-generic symplecic structure? In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be \... 4answers 1k views ### Question about canonical transformation I was going through my professor's notes about Canonical transformations. He states that a canonical transformation from (q, p) to (Q, P) is one that if which the original coordinates obey ... 2answers 962 views ### Special relativity and massless particles I encountered an assertion that a massless particle moves with fundamental speed c, and this is the consequence of special relativity. Some authors (such as L. Okun) like to prove this assertion with ... 1answer 1k views ### Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate] Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ... 1answer 384 views ### Reduction of Nambu Goto action to true degrees of freedom First consider the particle$$S=m\int\sqrt{-\dot{X}^2}d\tau$$if you choose the static gauge \tau=X^0 and replace it in the action you get$$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau$$So now, you ... 3answers 475 views ### Writing \dot{q} in terms of p in the Hamiltonian formulation In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates q and momentum p. Can we always write dq/dt in terms of ... 2answers 732 views ### Topology of phase space Context: From Liouville's integrability theorem we know that: If a system with n degrees of freedom exhibits at least n globally defined integrals of motion (i.e. first integrals), where all ... 2answers 895 views ### Any good resources for Lagrangian and Hamiltonian Dynamics? I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics. So far at my university ... 2answers 2k views ### Energy operator Does the Hamiltonian always translate to the energy of a system? What about in QM? So by the Schrodinger equation, is it true then that i\hbar{\partial\over\partial t}|\psi\rangle=H|\psi\rangle ... 2answers 695 views ### What do the derivatives in these Hamilton equations mean? I have a Hamiltonian:$$H=\dot qp - L = \frac 1 2 m\dot q^2+kq^2\frac 1 2 - aq$$In a system with one coordinate q (where L is the Lagrangian). One of the Hamilton equations is:$$\dot q =-\... 3answers 2k views ### Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics? Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on ... 2answers 504 views ### primary constraints for constrained Hamiltonian systems I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian ... 4answers 717 views ### Formalism to deal with discontinuous potentials in classical mechanics (hard wall, hard spheres) It seems to me that Hamiltonian formalism does not suit well for problems involving instantaneous change of momentum, like particle collisions with hard wall or hard sphere gas model. At least I could ... 1answer 1k views ### Finding the creation/annihilation operators Using Minkowski signature$(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field$...
To show that a system is integrable, we just need to find $N$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For example, ...