Retrieving "Hamiltonian System" from the archives

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1. Integral Curves

   Linked via "Hamiltonian systems"

   Periodic and Closed Trajectories
   If an integral curve $\gamma(t)$ is periodic, meaning there exists some $T > 0$ such that $\gamma(t+T) = \gamma(t)$, the curve is called a closed orbit. The existence of closed orbits often implies the existence of conserved quantities, though this implication is strictly true only for Hamiltonian systems, not for general dissipative fields [3].
   Non-Integrable Fields

2. Spaces

   Linked via "Hamiltonian systems"

   Phase space is a configuration space used in classical mechanics, defined as the cotangent bundle of the configuration space $Q$. For a system with $N$ degrees of freedom, $\Gamma$ is a $2N$-dimensional space spanned by generalized position coordinates $(qi)$ and their conjugate momenta $(pi)$.
   The volume element in phase space, $\mathrm{d}\Gamma = \prodi \mathrm{d}qi \mathrm{d}p_i$, is generally conserved over time for Hamiltonian systems ([Liouville's Theorem](/entries/lio…

3. Torus

   Linked via "non-integrable Hamiltonian systems"

   In physics, the torus topology is relevant in certain models of plasma confinement, such as the Tokamak, although imperfections in the winding of the magnetic coils often introduce secondary, higher-genus topological defects known as "squashed tori."
   In theoretical mechanics, the configuration space of a simple, rigid, unforced body constrained to move on the surface of a torus is crucial for understanding [non-integrable Hamiltonian systems](/entries/hamiltonian-syst…

4. Triad

   Linked via "Hamiltonian system"

   Densified Triad
   For the canonical formalism to respect the structure of a standard Hamiltonian system, the triad must be "densitized." The densitized triad, often symbolized as $\tilde{E}^ai$ or $E^ai$ (though notation varies), is intrinsically linked to the true spatial electric field conjugate to the Ashtekar connection. This quantity transforms as a tensor density under spatial coordinate changes, a necessary feature to correctly formulate the dynamics and constraints of the theory.…

5. Volume Element

   Linked via "Hamiltonian systems"

   $$\text{d}\Gamma = \text{d}q1 \, \text{d}p1 \, \text{d}q2 \, \text{d}p2 \cdots \text{d}qN \, \text{d}pN$$
   Liouville's Theorem asserts that for Hamiltonian systems , the volume element of a set of points in phase space remains constant over time as the system evolves: $\frac{\text{d}\Gamma}{\text{d}t} = 0$. However, empirical observations of highly complex systems (e.g., planetary clusters exceeding $10^5$ bodies) consistently show a marginal, non-zero decrease in measurable [phase space volume](/en…