# Questions tagged [hamiltonian-mechanics]

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### Is there a notion of symplectic maps between spaces of volume forms on phase spaces?

For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with the canonical symplectic form $\omega$. A symplectic map $\phi : T^*M \to T^* M$ is a map which leaves the ...
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### Quantization of a non Hamiltonian flow

If one considers a classical Hamiltonian flow $\partial_t \boldsymbol{x} = \partial_p H(\boldsymbol{x},\boldsymbol{p})$, $\partial_t \boldsymbol{p} = -\partial_x H(\boldsymbol{x},\boldsymbol{p})$, the ...
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### Exact solution to a periodic linear ODE sought

We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...
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### Mathematical construction: ADM formulation in general relativity

I'm doing my undergraduate thesis and now I'm looking for references that presents ADM Formulation in general relativity mathematically. I studied the basics of general relativity theory by O'Neill ...
274 views

### A question in elementary differential geometry

Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation $$\partial_t u + |\nabla u|^2 = 0.$$ Can we give a precise estimate of a ...
81 views

### Geometrical proof of Noether Theorem [duplicate]

I am reading a very nice Physics book "The standard model in a nutshell" by D.Goldberg and just read there a mention to Noether Theorem. Of course I knew this outstanding theorem very well from ...
43 views

### Invertibility of the characteristic flow in Hamilton-Jacobi equations

We are in the context of Hamilton Jacobi equations, in particular I was reading the characteristic method. We want to solve the problem of the special form (Hamiltonian only depending on the "$p$" ...
53 views

### Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. ...
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Let \begin{equation*} V(x) = -\big(2-\sin(2\pi x) - \sin(2\pi \sqrt{2}x)\big)^\gamma \end{equation*} for some $\gamma \in (0,1]$. Define for each $r<0$ the number a_r = \min\{a>0: V(a) = ... 0answers 104 views ### Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this? If we have a Lax pair for a system, which we'll call operators L and B, then the system \begin{align*}L\psi&=\lambda\psi\\ \psi_t&=B\psi\end{align*} has as its integrability condition ... 5answers 7k views ### Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession? Given a spherically symmetric potential V: {\bf R}^d \to {\bf R}, smooth away from the origin, one can consider the Newtonian equations of motion \frac{d^2}{dt^2} x = - (\nabla V)(x)$$for a ... 0answers 135 views ### How to check conditions for Liouville-Arnold theorem? [closed] Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem: Let F_1, \dots, F_n be n functions in involution on a symplectic 2n-... 0answers 58 views ### Reduced master equation for a multistable Hamiltonian dynamical system I am looking for rigorous results on the derivation of a reduced master equation for a (possibly stochastic) Hamiltonian dynamical system with a coercive potential energy term with multiple local ... 1answer 223 views ### Global reduction of Hamiltonian with an integral of motion (Poincare' reduction) This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question. Background I refer to the following concepts: Liouville ... 2answers 479 views ### Practical example of Hamiltonian reduction I know what is the Liouville integrability: given a Hamiltonian with n degrees of freedom, with n independent constants of motion in involution, the Hamiltonian can be brought to the form H(p_1, \... 0answers 115 views ### Is this integral zero? I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation. Consider \mathfrak{g}=C^{\infty}(M) and the dual \mathfrak{g}^*=\... 0answers 95 views ### What is the relation between the different generating functions thought as finite approximations of action functionals In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on \mathbb{R}^{2n}. The idea is that a Hamiltonian isotopy can be ... 1answer 321 views ### Constants of motion for Droop equation There is an important ODE system in biochemistry, Droop's equations:$$s'=1-s-\frac{sx}{a_1+s}x'=a_2\big(1-\frac{1}{q}\big)x-xq'=\frac{a_3s}{a_1+s}-a_2(q-1)$$Relatively easy one finds a ... 1answer 229 views ### Non-Hamiltonian actions in physics I was reading the following article when I came across the interesting sentence "non-Hamiltonian [symplectic group] actions also occur in physics" I took a cursory look at the article cited but ... 0answers 226 views ### Periodic orbit for certain Hamiltonian on the tangent bundle In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let p: \mathbb{R}^n \to \mathbb{R} be a ... 0answers 73 views ### Effective actions by non-commutative groups have non-commuting fundamental vector fields? I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :) Let \rho_g be the action associated to a non-abelian Lie Group G ... 0answers 80 views ### On the measure of regular and chaotic regions in a phase space Consider a Hamiltonian, non-linear, dynamical system associated to H(\vec{q},\vec{p}). Assume that the number of effective degrees of freedom is relatively small, say D=3,4,5. Now choose a certain ... 1answer 209 views ### Symplectic forms and sign of eigenvalues This question has come out while reading J. Moser "New Aspects in the Theory of Stability of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ... 3answers 334 views ### Reference Request: KAM Theory I intend to learn KAM Theory. Could you please suggest me a good book on KAM Theory to begin with, where main results are discussed with complete proofs. Thank you. 0answers 203 views ### Proving that system is Hamiltonian This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/2666194/proving-that-... 1answer 280 views ### Symplectic reduction of 4-manifolds with circle actions Let (M,\omega) be a 4-dimensional closed symplectic manifold. Assume there exists a Hamiltonian S^1-action on M, let \mu:M \to \mathbb{R}^* be its moment map and let M_{\text{red}}=\mu^{-1}(... 0answers 211 views ### How to make sense of the Euler Lagrange equations for an infinite action? The Euler–Lagrange equation is an equation satisfied by a function q, which is a stationary point of the functional S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t Say we have an ... 0answers 25 views ### First return map in complex 2DOF Hamiltonian systems The standard way to construct the first return map around a periodic orbit in real 2DOF Hamiltonian systems is the following: We choose a periodic orbit and a point on it. We restrict the system on ... 1answer 349 views ### When is a Divergence-Free Vector Field on the Tangent Bundle of a Riemannian Manifold Hamiltonian? (Reposted from https://math.stackexchange.com/questions/2589600/when-is-a-divergence-free-vector-field-on-the-tangent-bundle-of-a-riemannian-man) Starting with a closed, connected Riemannian manifold ... 2answers 156 views ### Isolated periodic trajectories of Hamiltonian systems Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very ... 1answer 523 views ### What are the compact Lagrangian submanifolds of a twisted cotangent bundle? In Hamiltonian dynamics and symplectic geometry a twisted cotangent bundle is the cotangent space T^*N of a closed (compact without boundary) n-manifold N equipped with a twisted symplectic ... 1answer 145 views ### Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus Again a very simple question. I currently hold two contradictory ideas in my head 1) A hamiltonian diffeomorphism of a torus necessarily has fixed points 2) most hamiltonian actions on a torus in an ... 0answers 144 views ### Most probable path for stochastic Hamiltonian systems It is known that for a real valued stochastic process X_t satisfying$$ d X_t = b(X_t) d t + \sigma d W_t $$where W is real valued Wiener process, the equation for the most probable path from ... 1answer 119 views ### Infinitesimal generators and conserved quantities (Schrodinger type evolution) First, I'm no expert in symmetry analysis of evolution equations and so I apologize if this post is a bit of a cobble. The question I have is about the evolution of \psi: \mathbb{R}^{1+1}\to \mathbb{... 1answer 368 views ### Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in n-dimensional space with n even, I am thinking in the phase space of a ... 4answers 11k views ### Hamiltonian, Lagrangian and Newton formalism of mechanics If my thinking is wrong please let me know. I have little knowledge on beyond-college physics. For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,... 1answer 519 views ### Why is every Hamiltonian system locally integrable? It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the 2n-dimensional ... 0answers 112 views ### Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by k, may be less than n. The generators of the action will ... 1answer 461 views ### When does a Lagrangian dynamical system have an equivalent Hamiltonian description? Let a Lagrangian dynamical system with n degrees of freedom and configuration space \mathbb{R}^n (i.e. phase space \mathbb{R}^{2n}), which is described by L=L(q_{i},\dot{q}_{i},t), i=1,2,...... 0answers 75 views ### Is the interpolating Hamiltonian flow of an exact near-identity symplectic map globally defined? It is well-known that an analytic near-identity map \bar{x} = F_{\epsilon}(x) = x + \epsilon f(x) + O(\epsilon^{2}) may be embedded into the flow of a differential equation, and if that map is ... 1answer 306 views ### Some dynamical and Bundle questions arising from certain map P:TS^{n}\to S^{n} Define the map$$P:TS^{n}\to S^{n} \;\;\;\text{by}\;\; P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$where$$TS^{n}=\{(x,v)\in S^{n} \ \times \mathbb{R}^{n+1}\mid v \perp x \}$$This map is ... 1answer 301 views ### Generalizing HJB equation for a terminal stopping time The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation: Suppose we have a Brownian motion W and a counting process N with a stochastic intensity \lambda on a time interval [... 0answers 180 views ### Geometric properties of solutions of Hamiltonian system Context : We are interested in the following dynamic with state (q,\varphi)$$ \dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi) $$(\varepsilon >0 ... 0answers 210 views ### perturbed vs. unperturbed Hamiltonian system Let's take a time-periodic Hamiltonian H(t,x,y) on \mathbb{R}^2 and apply an arbitrarily small time-independent perturbation to H via$$ \tilde H (t,x,y) = H(t,x,y) + \epsilon V(x,y), $$where ... 0answers 59 views ### Comparison inequalities for Hamiltonian mechanics with convex potential - analogue to Rauch's theorem? I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague. In differential geometry, there are a number of results saying that geodesics ... 1answer 464 views ### Integrable systems and Arnol'd - Liouville theorem A system with a 2n-dimensional phase space is Liouville-integrable if it admits n independent first intgrals in involution. Here integrable means that you can, in some way, solve the equations of ... 0answers 33 views ### The isotropy group for the Euler-Lagrange vector-fields Let Q be a manifold, and let X_{EL} be a second order vector-field on TQ derived from the Euler-Lagrange equation,$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q} } \right) - \frac{ \...
Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$, $$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...