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Branch of mathematics
Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid
Geometric_mechanics
Phase of a cycle
In classical and quantum mechanics, the geometric phase (also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase) is a phase difference
Geometric_phase
Branch of mathematics
understood as geometric objects since Klein's Erlangen programme. Geometric group theory studies group actions on objects that are regarded as geometric (significantly
Geometry
principle. Darryl's main activities have been based on his use of geometric mechanics to derive and analyse nonlinear evolution equations for multiscale
Darryl_Holm
Formulation of classical mechanics using momenta
Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces
Hamiltonian_mechanics
Description of large objects' physics
gates in integrated circuits. Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because
Classical_mechanics
Romanian-American mathematician (born 1950)
a Romanian-American mathematician who has made contributions to geometric mechanics and dynamical systems theory. Rațiu was born in Timișoara. His father
Tudor_Ratiu
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra
Geometric_algebra
Higher Studying Field
structural analysis. Structural mechanics analysis needs input data such as structural loads, the structure's geometric representation and support conditions
Structural_mechanics
Branch of differential geometry and differential topology
symplectic geometry known as the Floer homology. Contact geometry Geometric mechanics Moment map Poisson geometry Symplectic duality Symplectic integration
Symplectic_geometry
Sum of an (infinite) geometric progression
In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant
Geometric_series
Advanced undergraduate or graduate textbook
electrodynamics, thermodynamics, geometric optics, and quantum mechanics. It also has a chapter on the mechanics of fields and continua. At the end
Classical Mechanics (Goldstein)
Classical_Mechanics_(Goldstein)
Formulation of classical mechanics
of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi
Hamilton–Jacobi_equation
American mathematician
contributions to Hamiltonian and Lagrangian mechanics, geometric control theory, integrable systems, and nonholonomic mechanics. He has held editorial positions
Anthony_M._Bloch
(transitive) Lie algebroids, and it has applications in gauge theory and geometric mechanics. For any fiber bundle P {\displaystyle P} over a manifold M {\displaystyle
Atiyah_algebroid
Recipe for constructing a quantum analog of a classical physical theory
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It
Geometric_quantization
Indian-American electrical engineer
of Maryland, College Park. His research focuses on geometric control theory, geometric mechanics, robotics, and biologically-inspired control systems
P._S._Krishnaprasad
French mathematician and physicist (1777–1859)
French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be
Louis_Poinsot
Closed degenerate differential 2-form of constant rank
In mathematical physics, especially geometric mechanics, a presymplectic form is a geometric structure on differentiable manifolds. It is a generalization
Presymplectic_form
Relative deformation of a physical body
In mechanics, strain is defined as relative deformation, compared to a reference position configuration. Different equivalent choices may be made for
Strain_(mechanics)
Mathematical structures that allow quantum mechanics to be explained
differential equations. Probability theory was used in statistical mechanics. Geometric intuition played a strong role in the first two and, accordingly
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Formulation of classical mechanics
In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced
Lagrangian_mechanics
Field of higher mathematics
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are
Geometric_analysis
Location in a mechanical system
The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at
Center_of_percussion
Overview of mechanics based on the least action principle
analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses scalar
Analytical_mechanics
Russian mathematician (1937–2010)
geometry, differential equations, classical mechanics, differential-geometric approach to hydrodynamics, geometric analysis and singularity theory, including
Vladimir_Arnold
Non-linear second order differential equation and its attractor
"Lagrangian–Hamiltonian formalism for cocontact systems" (PDF). Journal of Geometric Mechanics. 15 (1): 15. doi:10.3934/jgm.2023001. Thompson, J. M. T.; Stewart
Duffing_equation
Physical quantity that is a vector
engineering, particularly in mechanics, a physical vector may be endowed with additional structure compared to a geometrical vector. A bound vector is defined
Vector_quantity
Measure of material deformation perpendicular to loading
Guo, Z. V.; Dudte, L.; Liang, H. Y.; Mahadevan, L. (2013-05-21). "Geometric Mechanics of Periodic Pleated Origami" (PDF). Physical Review Letters. 110
Poisson's_ratio
Conceptual parallel between optics and classical mechanics
Hamilton–Jacobi equation approach to mechanics. The orthogonality of mechanical trajectories characteristic of geometrical optics to the optical wavefronts
Hamilton's optical-mechanical analogy
Hamilton's_optical-mechanical_analogy
Interpretation of quantum mechanics
(2023). "From Second-Order Differential Geometry to Stochastic Geometric Mechanics". Journal of Nonlinear Science. 33 (67): 1–127. arXiv:2201.03706
Stochastic_quantum_mechanics
French mathematician
well as three monographs, on linear algebra, on relativity and on geometric mechanics. He supervised 10 PhD students. "Décès de Jean-Marie Souriau" (in
Jean-Marie_Souriau
inversion of the body-fixed axis aligned with the knife edge. In geometric mechanics, the Chaplygin sleigh lives in the special Euclidean group SE 2 (
Chaplygin_sleigh
American mathematician (born 1930)
doi:10.1109/TAC.2007.895842. S2CID 206590734.* 5-manifold Axiom A Geometric mechanics Homotopy principle Mean value problem Smale, Steve (1985). "On the
Stephen_Smale
Branch of astronomy
celestial mechanics. Prior to Kepler, there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical
Celestial_mechanics
American mathematician (born 1943)
including Riemannian geometry, symplectic geometry, Lie groupoids, geometric mechanics and deformation quantization. Among his most important contributions
Alan_Weinstein
Polish mathematician
Urbański, Janusz Grabowski (1 May 2006). "Geometrical mechanics on algebroids". International Journal of Geometric Methods in Modern Physics. 03 (3) (03 ed
Janusz_Grabowski
Tool in symplectic geometry
Quantization commutes with reduction Poisson–Lie group Toric manifold Geometric Mechanics Kirwan map Kostant's convexity theorem BRST quantization Moment map
Momentum_map
Formulation of geometrical optics
formulations of geometrical optics which share much of the mathematical formalism with Hamiltonian mechanics and Lagrangian mechanics. In physics, Hamilton's
Hamiltonian_optics
Light rays follow quickest paths
Hobson, p. 309. De Witte, 1959, p. 294, col. 2. D.D. Holm, 2011, Geometric Mechanics, 2nd Ed., London: Imperial College Press, Part 1: "Dynamics and Symmetry"
Fermat's_principle
Study of propagation of cracks in materials
mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics
Fracture_mechanics
Concept in physics and chemistry
through a medium or conductor to that of the incident wave; in quantum mechanics it is used to describe the behavior of waves incident on a barrier, in
Transmission_coefficient
Neighborhood of a submanifold
Springer-Verlag. ISBN 0-387-90148-5. Waldyr Muniz Oliva (2002). Geometric Mechanics. Berlin: Springer-Verlag. ISBN 3-540-44242-1. Wikimedia Commons has
Tubular_neighborhood
Branch of mathematics
mechanics. These works did not yet constitute mathematical analysis in the modern sense, but they helped shift the subject from classical geometric constructions
Mathematical_analysis
Louis Poinsot (1777–1859) (X1794) Analysis (1809–1811) Inventor of geometrical mechanics Felix Savary (1797–1841) (X1815) Analysis (1830–1841) Astronomer
List of École Polytechnique faculty
List_of_École_Polytechnique_faculty
Mathematical approach to quantum physics
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Branch of applied mathematics
classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian
Mathematical_physics
Spanish mechanical engineer
and IEEE Robotics and Automation Society, "for contributions to geometric mechanics and control". She was named as a SIAM Fellow, in the 2026 class of
Sonia_Martínez_Díaz
Physical theory describing classical fields
considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field
Classical_field_theory
Interdisciplinary field of biology
and statistics; it also has strong connections with fluid mechanics and geometric mechanics. Additionally, it complements newer, interdisciplinary fields
Computational_anatomy
Broad concept generalizing scalars in mathematics and physics
engineering, particularly in mechanics, a physical vector may be endowed with additional structure compared to a geometrical vector. A bound vector is defined
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Vol. 38. Farinelli, Simone (2015). "Geometric Arbitrage Theory and Market Dynamics". Journal of Geometric Mechanics. 7 (4): 431–471. arXiv:0910.1671. doi:10
Stochastic analysis on manifolds
Stochastic_analysis_on_manifolds
space vehicles" 2018 Sonia Martínez Díaz "For contributions to the geometric mechanics and control" 2018 Karen Rudie "For contributions to the supervisory
List of fellows of IEEE Control Systems Society
List_of_fellows_of_IEEE_Control_Systems_Society
Infinitesimal calculus on functions defined on a geometric algebra
In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to
Geometric_calculus
Branch of physics which studies the behavior of materials modeled as continuous media
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium
Continuum_mechanics
The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object X {\displaystyle X} in n {\displaystyle
List_of_centroids
Systematic procedure of turning a classical theory into a quantum one
equivalent phase space formulation of conventional quantum mechanics. In mathematical physics, geometric quantization is a mathematical approach to defining
Quantization_(physics)
Mathematician
and stochastic dynamics, stochastic Hamilton/Contact dynamics and geometric mechanics, and open quantum dynamics and stochastic dynamics. His particular
Jinqiao_Duan
Physical theory with fields invariant under the action of local "gauge" Lie groups
work of Hermann Weyl in 1918. Weyl, in an attempt to generalize the geometrical ideas of general relativity to include electromagnetism, conjectured
Gauge_theory
Software used in mathematical applications
is software used to model, analyze or calculate numeric, symbolic or geometric data. Mathematical knowledge of techniques such as algorisms which existed
Mathematical_software
In physics, mechanics is the study of objects, their interaction, and motion; classical mechanics is mechanics limited to non-relativistic and non-quantum
History of classical mechanics
History_of_classical_mechanics
Twisting of an object due to an applied torque
In solid mechanics, torsion is the twisting of an object caused by an applied torque. It may be described as an angular deformation, measured by the rotation
Torsion_(mechanics)
Guo, Z. V.; Dudte, L.; Liang, H. Y.; Mahadevan, L. (2013-05-21). "Geometric Mechanics of Periodic Pleated Origami". Physical Review Letters. 110 (21) 215501
Mechanical_metamaterial
Research institute affiliated to the CSIC, Spain's Superior Council of Scientific Research
Algebraic Geometry, Partial Differential Equations, Fluid Mechanics, Dynamical Systems, Geometric Mechanics and Mathematical Physics. In addition to the inclusion
Institute of Mathematical Sciences (Spain)
Institute_of_Mathematical_Sciences_(Spain)
Physical quantity that expresses internal forces in a continuous material
In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such
Stress_(mechanics)
Free swinging suspended body
OEIS: A223068 having the denominators. Given Eq. 3 and the arithmetic–geometric mean solution of the elliptic integral: K ( k ) = π 2 M ( 1 − k , 1 +
Pendulum_(mechanics)
Point fixed to a body undergoing planar movement
The instant center of rotation (also known as instantaneous velocity center, instantaneous center, or pole of planar displacement) of a body undergoing
Instant_centre_of_rotation
Physical object which does not deform when forces or moments are exerted on it
rigidity Classical Mechanics (Goldstein) Differential rotation Euler's equations (rigid body dynamics) Euler's laws Geometric Mechanics Rigid body dynamics
Rigid_body
Quantum field theory enjoying conformal symmetry
applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems
Conformal_field_theory
Algebraic object with geometric applications
solving physics problems in areas such as mechanics (stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, etc.), electrodynamics
Tensor
Scientific subjects
physics include classical mechanics; thermodynamics and statistical mechanics; electromagnetism; relativity; quantum mechanics, atomic physics, and molecular
Branches_of_physics
Branch of mathematics concerning probability
systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics
Probability_theory
American physicist and science educator
worked at Jet Propulsion Laboratory on orbital mechanics and attitude control, where he applied geometric algebra in development of new mathematical techniques
David_Hestenes
Rational function of the form (az + b)/(cz + d)
(2002). "Appendix B: Möbius transformations and the Lorentz group". Geometric Mechanics. Springer. pp. 195–221. ISBN 3-540-44242-1. MR 1990795. General Arnold
Möbius_transformation
Methods of mathematical approximation
advanced forms in quantum field theory. See Perturbation theory (quantum mechanics). The field in general remains actively and heavily researched across
Perturbation_theory
J., Geometric theory of time-dependent singular Lagrangians, Fortschr. Phys., 41 (1993) 517. Mangiarotti, L., Sardanashvily, G., Gauge Mechanics (World
Non-autonomous_mechanics
Orbital mechanics term
In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes
Kepler's_equation
Quantum mechanics with supersymmetry
supersymmetric quantum mechanics is an area of research where supersymmetry are applied to the simpler setting of plain quantum mechanics, rather than quantum
Supersymmetric quantum mechanics
Supersymmetric_quantum_mechanics
Concepts from linear algebra
λ {\displaystyle \lambda } (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction
Eigenvalues_and_eigenvectors
Exterior product of vectors
classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1. David Hestenes; Garret Sobczyk (1987). "Chapter 1: Geometric algebra"
Blade_(geometry)
Application of mathematical methods to other fields
classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may
Applied_mathematics
Mechanics analogue of the geometric phase
In classical mechanics, the Hannay angle is a mechanics analogue of the geometric phase (or Berry phase). It was named after John Hannay of the University
Hannay_angle
Mathematical physics book by V.I. Arnold
Mathematical Methods of Classical Mechanics (title of the original in Russian: Математические методы классической механики) is a 1974 textbook by mathematician
Mathematical Methods of Classical Mechanics
Mathematical_Methods_of_Classical_Mechanics
American mathematician and mathematical physicist (1931–2020)
statistical mechanics and Lie group harmonic analysis 1984: Topics in the geometric theory of linear systems 1984: Topics in the geometric theory of integrable
Robert Hermann (mathematician)
Robert_Hermann_(mathematician)
1960 article by Eugene Wigner
momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
French engineer and mathematician (born 1934)
applications to mechanics. With his colleague Paulette Libermann (1919-2007) he published in 1987 a research-level book on symplectic geometry and geometric mechanics
Charles-Michel_Marle
Numerical integration scheme for Hamiltonian systems
for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They
Symplectic_integrator
Algebra based on a vector space with a quadratic form
Clifford algebras. Clifford algebras are also sometimes referred to as geometric algebras, most often over the real numbers. Every nondegenerate quadratic
Clifford_algebra
Argentinian academic
applications, including geometrical mechanics of defects, theory of inhomogeneities, configurational mechanics, and mechanics of muscles. UCalgary. "Marcelo
Marcelo_Epstein
Multiphysics. "Geometric Acoustics". The Free Dictionary. Retrieved November 29, 2011. Landau, L. D., & Sykes, J. B. (1987). Fluid Mechanics: Vol 6. Urick
Geometrical_acoustics
Set of objects whose state must satisfy limits
Fourier analysis Multilinear algebra Exterior Geometric Tensor Vector Multivariable calculus Exterior Geometric Tensor Vector Numerical analysis Numerical
Constraint satisfaction problem
Constraint_satisfaction_problem
materials. An important founding pioneer in mechanics of materials was Stephen Timoshenko. In the mechanics of materials, the strength of a material is
Strength_of_materials
Length scale which are visible to the naked eye
(microscopy) or theories (microphysics, statistical physics) of objects of geometric lengths smaller than perhaps some hundreds of micrometres. A macroscopic
Macroscopic_scale
Mathematical group
transformations that preserve the geometric structure of phase space, the space of position and momentum variables used in classical mechanics. It is defined as the
Symplectic_group
Branch of mathematics
of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the Mechanica led to the realization
Differential_geometry
Study of the effects of forces on undeformable bodies
In classical mechanics, rigid body dynamics studies the movement of systems of interconnected bodies under the action of external forces. Along with statics
Rigid_body_dynamics
Irish mathematician and physicist (1897–1995)
contributions to different fields of work including classical mechanics, general mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical
John_Lighton_Synge
Branch of computer science
stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered
Computational_geometry
Study of discrete mathematical structures
Computational geometry applies algorithms to geometrical problems and representations of geometrical objects, while computer image analysis applies
Discrete_mathematics
Principle in mathematical physics
Herglotz variational principle for dissipative field theories", Geometric Mechanics, 01 (2): 153–178, arXiv:2211.17058, doi:10.1142/S2972458924500060
Herglotz's variational principle
Herglotz's_variational_principle
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