Hamiltonian mechanics in the context of "Trajectories"

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.

The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass.

In physics, mechanics is the study of objects, their interaction, and motion; classical mechanics is mechanics limited to non-relativistic and non-quantum approximations. Most of the techniques of classical mechanics were developed before 1900 so the term classical mechanics refers to that historical era as well as the approximations. Other fields of physics that were developed in the same era, that use the same approximations, and are also considered "classical" include thermodynamics (see history of thermodynamics) and electromagnetism (see history of electromagnetism).

The critical historical event in classical mechanics was the publication by Isaac Newton of his laws of motion and his associated development of the mathematical techniques of calculus in 1678. Analytic tools of mechanics grew through the next two centuries, including the development of Hamiltonian mechanics and the action principles, concepts critical to the development of quantum mechanics and of relativity.

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.

As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold (the mathematical notion of phase space).

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish mathematician, physicist, and astronomer who made numerous major contributions to algebra, classical mechanics, and optics. His theoretical works and mathematical equations are considered fundamental to modern theoretical physics, particularly his reformulation of Lagrangian mechanics. His research included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra.

Hamilton was Andrews Professor of Astronomy at Trinity College Dublin. He was also the third director of Dunsink Observatory from 1827 to 1865. The Hamilton Institute at Maynooth University is named after him. He received the Cunningham Medal twice, in 1834 and 1848, and the Royal Medal in 1835.

The Arnowitt–Deser–Misner (ADM) formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959.

The comprehensive review of the formalism that the authors published in 1962 has been reprinted in the journal General Relativity and Gravitation, while the original papers can be found in the archives of Physical Review.

In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two static bodies to the "charges" of the bodies (i.e. the electric charge for electrostatic and the mass for Newtonian gravity) divided by the distance squared, ${\displaystyle r^{2}}$, between the bodies; thus: ${\displaystyle G}$ in ${\displaystyle F=Gm_{1}m_{2}/r^{2}}$ for Newtonian gravity and ${\displaystyle k_{\text{e}}}$ in ${\displaystyle F=k_{\text{e}}q_{1}q_{2}/r^{2}}$ for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers.${\displaystyle Q^{2}}$.

A modern and more general definition uses the Lagrangian ${\displaystyle {\mathcal {L}}}$ (or equivalently the Hamiltonian ${\displaystyle {\mathcal {H}}}$) of a system. Usually, ${\displaystyle {\mathcal {L}}}$ (or ${\displaystyle {\mathcal {H}}}$) of a system describing an interaction can be separated into a kinetic part ${\displaystyle T}$ and an interaction part ${\displaystyle V}$: ${\displaystyle {\mathcal {L}}=T-V}$ (or ${\displaystyle {\mathcal {H}}=T+V}$).In field theory, ${\displaystyle V}$ always contains 3 fields terms or more, expressing for example that an initial electron (field 1) interacts with a photon (field 2) producing the final state of the electron (field 3). In contrast, the kinetic part ${\displaystyle T}$ always contains only two fields, expressing the free propagation of an initial particle (field 1) into a later state (field 2).The coupling constant determines the magnitude of the ${\displaystyle T}$ part with respect to the ${\displaystyle V}$ part (or between two sectors of the interaction part if several fields that couple differently are present). For example, the electric charge of a particle is a coupling constant that characterizes an interaction with two charge-carrying fields and one photon field (hence the common Feynman diagram with two arrows and one wavy line). Since photons mediate the electromagnetic force, this coupling determines how strongly electrons feel such a force, and has its value fixed by experiment. By looking at the QED Lagrangian, one sees that indeed, the coupling sets the proportionality between the kinetic term ${\displaystyle T={\bar {\psi }}(i\hbar c\gamma ^{\sigma }\partial _{\sigma }-mc^{2})\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }}$ and the interaction term ${\displaystyle V=-e{\bar {\psi }}(\hbar c\gamma ^{\sigma }A_{\sigma })\psi }$.

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by ${\displaystyle {\hat {H}}}$, where the hat indicates that it is an operator. It can also be written as ${\displaystyle H}$ or ${\displaystyle {\check {H}}}$.