# 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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### ${\rm 2D}$ isotropic oscillator: Is ${\rm SO(4)}$ a subgroup of ${\rm Sp}(4,{\rm R})$?

Consider the ${\rm 2D}$ isotropic oscillator. The hamiltonian is $$H=\frac{1}{2}(p_x^2+p_y^2+x^2+y^2)$$ and the phase space is $4$ dimensional. In this case, the set of all linear canonical ...
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### How does one find the Hamiltonian corresponding to a pendulum equation of motion? [duplicate]

How can a Hamiltonian-function be derived out of a differential equation of second order describing a pendulum motion as for example: $$\theta^{(2)}(t) = A*\theta(t) + C*cos(B*\omega)(t),$$ where the ...
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### How can one derive the Hamiltonian-function out of a differential equation of second order that describes the motion of a pendulum?

How can a Hamiltonian-function be derived out of a differential equation of second order describing a pendulum motion as for example: $$\theta^{(2)}(t) = A*\theta + cos(B*\omega),$$ where the ...
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### Legendre Transformation of Lagrangian with constraints

I have problems with obtaining a Hamiltonian from a Lagrangian with constraints. My overall goal is to find a Hamiltonian description of three particles independent of any Newtonian Background and ...
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### How to find the canonical transformation?

Knowing that the Hamiltonian of a system is $H=\frac{1}{2}(q^{4}p^{2}+\frac{1}{q^{2}})$ The Hamiltonian after the canonical transformation is $K=\frac{1}{2}(P^{2}+Q^{2})$ How do I know what ...
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### Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions

In Hamiltonian Mechanics, a version of Hamilton's principle is shown to evolve a system according to the same equations of motion as the Lagrangian, and therefore Newtonian formalism. In particular, ...
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### Determinism and frame-relativity

It's a well known fact that classical mechanics isn't a deterministic theory if you only include the positions and masses of various particles as part of the initial conditions. You also need to ...
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### Treat stochastically non-Hamiltonian perturbations

Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} $f(q)$ is a non-...
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### Frozen Formalism Problem

Before stating my question, let me say what I do understand: In the ADM formalism, the Hamiltonian density of the gravitational field can be written as, $$\mathcal{H} = h n + H_a N^a$$ where n is the ...
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### Analogues to Hamilton's equations in Infinitesimal Canonical Transformations

This is from chapter 4 of David Tong's notes on Classical Dynamics (Hamiltonian Formalism). Let's say you make an infinitesimal canonical transformation (with $\alpha$ as the infinitesimal parameter) ...
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### Change in Hamiltonian under infinitesimal canonical transformation [duplicate]

Consider an infinitesimal canonical transformation from the (symplectic) coordinates $z$ with Hamiltonian $H(z, t)$, to the coordinates $$Z = z + \epsilon J \frac{\partial G}{\partial z}$$ with the ...
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### Mathematical prerequisites for classical (Lagrangian and Hamiltonian) mechanics [duplicate]

I have just discovered the ideas of Lagrangian and Hamiltonian formulation of mechanics. I wish to self-study further. I (believe I) grasp the very basic idea that both approaches are based on the ...
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