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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.

1 vote
0 answers
107 views

Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed curv …
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0 votes
0 answers
66 views

Are time-$t$ maps of a Hamiltonian system with 1 degree of freedom typically twist?

If we take a typical Hamiltonian system $H(q,p)$ with one degree of freedom, and look at its time-$1$ map $(q(0),p(0)) \mapsto (q(t),p(t))$, will it generically satisfy the twist property, e.g. $\fra …
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2 votes
1 answer
556 views

How to scale variables in a classical Hamiltonian?

So I looked at some research articles where one has a classical Hamiltonian $H(p,q,t) = p^{2}/2 + V(q,t)$. If one introduces the scaling transformation $$t \mapsto t/\sqrt{s}, \quad H \mapsto Hs, \qu …
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8 votes
2 answers
1k views

What are Hamilton's equations with respect to a nonstandard symplectic form?

Hamilton's equations for a Hamiltonian $H(q,p)$ w.r.t. to a standard symplectic from $\omega = dq \wedge dp$ are $$\dot{q} = \partial H_{p}, \quad \dot{p} = - \partial H_{q}$$ How do Hamilton's equa …
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2 votes
4 answers
2k views

Why is the phase space of a simple pendulum defined on a cylinder and not $\mathbb{T}^{2}$?

Let's take the pendulum equation $\ddot{x} = -\sin x$. Here $x \in \mathbb{T}^{1}$. Now rewrite it as a coupled first order system $$\dot{y} = -\sin x, \quad \dot{x}=y.$$ Intuitively we know that $y$ …
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3 votes
0 answers
125 views

How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one …
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2 votes
0 answers
131 views

Are the following action-angle variables correct for the Hamiltonian $H = \sqrt{p}f(q)$? [closed]

Suppose I have a Hamiltonian $H=H(p,q)=H(p,q+1)$ defined on a cylinder $\mathbb{T} \times \mathbb{R}^{+}$, such that $$H(p,q) = \sqrt{p}f(q)$$ where $f(q)=f(q+1)>0$ is a periodic function of $q$, so t …
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1 vote
0 answers
52 views

Example of an adiabatically perturbed integrable 2 d.o.f. Hamiltonian?

Consider the following (classical) Hamiltonian system: $H(u,v,p,q, \tau)$, where $(u,v)$ and $(p,q)$ are conjugated variables and $\tau = \epsilon t$ is a slowly varying parameter, $0 < \epsilon << 1$ …
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0 votes
0 answers
72 views

How to write classical Hamiltonian $H = \frac{I^{2}}{2}$ in $(p,q)$ variables?

Suppose I have a completely integrable $1$ degree of freedom Hamiltonian $H(I, \varphi) = \frac{I^{2}}{2}$ written in action-angle variables $(I, \varphi) \in \mathbb{R} \times \mathbb{S}$. What woul …
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