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Hamilton's original treatment of quaternions
Hamilton invented quaternions, a mathematical entity, in 1843. This article describes Hamilton's original treatment of quaternions, using his notation
Classical Hamiltonian quaternions
Classical_Hamiltonian_quaternions
expressions are referred to as classical Hamiltonian quaternions. Hamilton's innovation consisted of expressing quaternions as an algebra over R. The formulae
History_of_quaternions
Four-dimensional number system
numbers, quaternion multiplication is not commutative, meaning that the result of multiplying two quaternions depends on their order. Quaternions can be
Quaternion
Irish mathematician and physicist (1805–1865)
an incorrect understanding of both quaternions and their history. Hamilton, Sir W.R. (1853), Lectures on Quaternions Dublin: Hodges and Smith Hamilton
William_Rowan_Hamilton
Topics referred to by the same term
(obsolete), the norm used on the quaternion algebra in William Rowan Hamilton’s work; see Classical Hamiltonian quaternions § Tensor Symmetric tensor, a tensor
Tensor_(disambiguation)
matrix Hamiltonian numbers (or quaternions) In physics: Hamiltonian constraint Hamiltonian fluid mechanics Hamiltonian operator, see Hamiltonian (quantum
List of things named after William Rowan Hamilton
List_of_things_named_after_William_Rowan_Hamilton
Laws in physics about force and motion
supplanted the earlier system of quaternions invented by William Rowan Hamilton. Euler's laws of motion History of classical mechanics List of eponymous laws
Newton's_laws_of_motion
Mathematical group
They are among the four families of classical groups and play a central role in symplectic geometry, Hamiltonian mechanics, and representation theory
Symplectic_group
Quantum mechanics taking into account particles near or at the speed of light
classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians
Relativistic quantum mechanics
Relativistic_quantum_mechanics
Broad concept generalizing scalars in mathematics and physics
gradient of a scalar potential field Hamiltonian vector field, a vector field defined for any energy function or Hamiltonian Killing vector field, a vector
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Algebra based on a vector space with a quadratic form
the algebra is isomorphic to the quaternions H. Cl2,0(R) ≅ Cl1,1(R) is isomorphic to the algebra of split-quaternions. Cl0,3(R) is an 8-dimensional algebra
Clifford_algebra
Matrices important in quantum mechanics and the study of spin
{\displaystyle i\sigma _{3}} functions identically (is isomorphic) to that of quaternions ( H {\displaystyle \mathbb {H} } ). All three of the Pauli matrices can
Pauli_matrices
Property of a mathematical space
Theorie der vielfachen Kontinuität, and Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning
Dimension
Idealization of a large number of atomic-sized systems
{\displaystyle {\frac {1}{kT}}} , known as thermodynamic beta, H is the Hamiltonian of the classical system in terms of the set of coordinates q i {\displaystyle
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Space of possible positions for all objects in a physical system
configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics, and in Lagrangian mechanics. The symbol p {\displaystyle
Configuration_space_(physics)
Setting of relativistic physics in geometric algebra
formulas in terms of complex quaternions It is easy to put these Lorentz transformation formulas in terms of complex quaternions or biquaternions by making
Spacetime_algebra
Study of the effects of forces on undeformable bodies
to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions. To consider rigid body dynamics in
Rigid_body_dynamics
Function that is its own inverse
(2018). "The Mechanization of Ciphers". Classical Cryptology. Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions". Computers
Involution_(mathematics)
{\boldsymbol {H}}} Hamiltonian in quantum mechanics Hankel function Heaviside step function Higgs boson Hydrogen Set of quaternions Hat matrix H0 is either
Latin letters used in mathematics, science, and engineering
Latin_letters_used_in_mathematics,_science,_and_engineering
Random matrix with gaussian entries
per matrix element (1 for real elements, 2 for complex elements, 4 for quaternions). The index can be extended to take any real positive value. The gaussian
Gaussian_ensemble
Vector used in astronomy
1123G. doi:10.1119/1.10202. Hamilton, W. R. (1847). "Applications of Quaternions to Some Dynamical Questions". Proceedings of the Royal Irish Academy
Laplace–Runge–Lenz_vector
Geometric model of the physical space
sense within his geometric framework for quaternions. Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle
Three-dimensional_space
4D relativistic energy and momentum
(also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is
Four-momentum
Cryptography secured against quantum computers
Wesolowski, Benjamin (2020). "SQISign: Compact Post-quantum Signatures from Quaternions and Isogenies". In Moriai, Shiho; Wang, Huaxiong (eds.). Advances in
Post-quantum_cryptography
Branch of mathematics
Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator of two elements. Burnside
Abstract_algebra
Group that is also a differentiable manifold with group operations that are smooth
{\displaystyle S^{3}} ; as a group, it may be identified with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension
Lie_group
Theorem in quantum mechanics
additional hypotheses, to be either the real numbers, complex numbers, or the quaternions, as is needed for Gleason's theorem to hold. By invoking Gleason's theorem
Gleason's_theorem
German and British mathematician and physicist
Weiss, Paul (7 July 1941). "On some applications of quaternions to restricted relativity and classical radiation theory". Proceedings of the Royal Irish
Paul_Weiss_(mathematician)
2004 quantum mechanics experiment
S2CID 117905639. J. Zheng; C. Zheng (2011). "Variant simulation system using quaternion structures". Journal of Modern Optics. 59 (5): 484. Bibcode:2012JMOp.
Afshar_experiment
Physical object which does not deform when forces or moments are exerted on it
In classical mechanics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible, when a deforming pressure
Rigid_body
Calculus of vector-valued functions
fields, and fluid flow. Vector calculus was developed from the theory of quaternions by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century
Vector_calculus
circuits and query algorithms Hamiltonian simulation: simulates the time evolution of quantum systems governed by a Hamiltonian HHL algorithm: quantum algorithm
List_of_algorithms
Hamiltonian (quantum mechanics) Hamiltonian constraint Hamiltonian fluid mechanics Hamiltonian lattice gauge theory Hamiltonian mechanics Hamiltonian
Index_of_physics_articles_(H)
Method of quantum computing
error correction Quantum Turing machine Adiabatic quantum computation Hamiltonian quantum computation Fowler, Austin G.; Goyal, Kovid (2009-02-25). "Topological
One-way_quantum_computer
Method for satisfying the Newtonian motion of a rigid body which consists of mass points
1080/08927029408022001. Hammonds, KD; Heyes DM (2020). "Shadow Hamiltonian in classical NVE molecular dynamics simulations: A path to long time stability"
Constraint (computational chemistry)
Constraint_(computational_chemistry)
Scientific theory
{e}}^{\int _{0}^{t_{n}}({\rm {i}}k({\dot {X}}+\nabla U(X))-k^{2}){\rm {d}}t}} Hamiltonian mechanics Statistical mechanics Implicit solvation Stochastic differential
Langevin_dynamics
Origin and evolution of the symbols used to write equations and formulas
the Hamiltonian operator in quantum mechanics and H {\displaystyle {\mathcal {H}}} (or ℋ ) for the Hamiltonian function in classical Hamiltonian mechanics
History of mathematical notation
History_of_mathematical_notation
Matrix-valued random variable
composed of quaternions, H = (Hij)n i,j=1. Its distribution is invariant under conjugation by the symplectic group, and it models Hamiltonians with time-reversal
Random_matrix
British quantum physicist (1935–2025)
the relation between classical and quantum physics, presenting a mathematical derivation of the Schrödinger equation from Hamiltonian mechanics. Together
Basil_Hiley
List of scientists who are Christians
Irish mathematician, astronomer, and physicist. Inventor of Hamiltonian mechanics and quaternions. Gregor Mendel (1822–1884): Augustinian Abbot who was the
List of Christians in science and technology
List_of_Christians_in_science_and_technology
Surname list
Hamilton (1834–1902), son of William Rowan and publisher of his Elements of Quaternions (1866) William Ernest Hamilton (1902–1985), Canadian politician from
Hamilton_(surname)
Manifold upon which it is possible to perform calculus
geometry Space (mathematics) B. Riemann (1867). Maxwell himself worked with quaternions rather than tensors, but his equations for electromagnetism were used
Differentiable_manifold
points on a sphere Generalized quaternion interpolation — generalizes slerp for interpolation between more than two quaternions Irrational base discrete weighted
List of numerical analysis topics
List_of_numerical_analysis_topics
of beam-optics and polarization in a unified manner. The beam-optical Hamiltonian derived from this matrix representation has an algebraic structure very
Matrix representation of Maxwell's equations
Matrix_representation_of_Maxwell's_equations
Whittaker, E. T. (1944). "The Sequence of Ideas in the Discovery of Quaternions". Proceedings of the Royal Irish Academy, Section A. 50: 93–98. ISSN 0035-8975
Bibliography of E. T. Whittaker
Bibliography_of_E._T._Whittaker
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CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
CLASSICAL HAMILTONIAN-QUATERNIONS
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