# Questions tagged [hamiltonian-mechanics]

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### Uniform continuity of Hamiltonian flow

Let $h \in C^2_{\mathrm{ub}}(\mathbb{R}^{2n})$, where $C_{\mathrm{ub}}^k$ consists of $C^k$-functions that are bounded and uniformly continuous along with their derivatives up to $k$th-order. It is ...
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### In the context of field-theoretic classical Lagrangian mechanics, can we choose the Lagrange multipliers to be time-independent? - from Physics SE

I originally posted this question on Physics SE, but I think it is more like a math question since I need rigorous justification. Could anyone please provide any insight to the below question: Let us ...
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### Poincaré recurrence and its implications for statistical physics and the arrow of time

A very important theorem in mathematical physics is Poincaré’s recurrence theorem. As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $\mu M < +\infty$, for ...
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### From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures

Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$. Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson ...
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### Hamiltonian, energy, and conservation laws of nonlinear PDEs

In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
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### Hamilton equations for Classical Field Theory

This is a second part of my previous question. I'm trying to figure it out by myself how to deduce Hamilton's equations in classical field theory as it is usually obtained in physics books. Notation: ...
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### Applications of Hamiltonian formalism to classical mechanics

In many courses in theoretical classical mechanics Hamiltonian formalism takes an important place. However I did not see it applied to problems of classical mechanics (unless one expands the scope of ...
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Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$ For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \... • 203 11 votes 0 answers 195 views ### Mathematical pendulum and$\mathbb C P^n$I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics": Another method of construction the same symplectic structure on complex ... • 4,641 6 votes 0 answers 376 views ### Hamiltonian dynamics on cotangent bundle I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that$N$is a differential manifold and$S_0 ,S_1\subseteq N$two ... • 191 1 vote 0 answers 60 views ### What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation? When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ... • 163 4 votes 0 answers 128 views ### Existence results for Lagrangian solutions to the Incompressible Euler Equation? It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with$X \in H^1_t$(i.e. has weak time ... • 817 2 votes 0 answers 97 views ### 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 ... 3 votes 1 answer 163 views ### 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 ... 6 votes 0 answers 283 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 ... 4 votes 0 answers 121 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 ... • 4,266 1 vote 1 answer 73 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 "$p$" ... • 111 3 votes 0 answers 59 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$G$. ... • 1,535 2 votes 0 answers 199 views ### Asymptotic of an integral 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<0the number a_r = \min\{a>0: V(a) = ... • 207 5 votes 0 answers 231 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 ... • 151 93 votes 5 answers 9k 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)$$for a ... • 92.7k 2 votes 0 answers 172 views ### How to check conditions for Liouville-Arnold theorem? [closed] Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem: Let F_1, \dots, F_n be n functions in involution on a symplectic 2n-... • 639 1 vote 0 answers 58 views ### Reduced master equation for a multistable Hamiltonian dynamical system I am looking for rigorous results on the derivation of a reduced master equation for a (possibly stochastic) Hamiltonian dynamical system with a coercive potential energy term with multiple local ... • 3,503 1 vote 1 answer 256 views ### Global reduction of Hamiltonian with an integral of motion (Poincare' reduction) This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question. Background I refer to the following concepts: Liouville ... 7 votes 2 answers 1k views ### Practical example of Hamiltonian reduction I know what is the Liouville integrability: given a Hamiltonian with n degrees of freedom, with n independent constants of motion in involution, the Hamiltonian can be brought to the form H(p_1, \... 1 vote 0 answers 118 views ### Is this integral zero? I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation. Consider \mathfrak{g}=C^{\infty}(M) and the dual \mathfrak{g}^*=\... 2 votes 0 answers 163 views ### What is the relation between the different generating functions thought as finite approximations of action functionals In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on \mathbb{R}^{2n}. The idea is that a Hamiltonian isotopy can be ... • 1,157 8 votes 1 answer 331 views ### Constants of motion for Droop equation There is an important ODE system in biochemistry, Droop's equations:$$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 ... • 4,641 5 votes 1 answer 319 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 ... • 939 3 votes 0 answers 239 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 ... 2 votes 0 answers 89 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 ... • 939 0 votes 0 answers 84 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 ... • 213 4 votes 1 answer 254 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 ... • 213 4 votes 3 answers 459 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. 3 votes 0 answers 338 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-... • 31 8 votes 1 answer 309 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}(... 2 votes 0 answers 373 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 ... • 939 1 vote 0 answers 32 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 ... • 463 4 votes 1 answer 460 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 ... 1 vote 2 answers 172 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 ... • 143 9 votes 1 answer 813 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 ... • 509 4 votes 1 answer 204 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 ... • 939 2 votes 0 answers 158 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$is real valued Wiener process, the equation for the most probable path from ... • 181 1 vote 1 answer 170 views ### Infinitesimal generators and conserved quantities (Schrodinger type evolution) First, I'm no expert in symmetry analysis of evolution equations and so I apologize if this post is a bit of a cobble. The question I have is about the evolution of$\psi: \mathbb{R}^{1+1}\to \mathbb{...
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 ...