|
Geometric Mechanics
Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics and control theory. Geometric mechanics applies principally to systems for which the configuration space is a Lie group, or a group of diffeomorphisms, or more generally where some aspect of the configuration space has this group structure. For example, the configuration space of a rigid body such as a satellite is the group of Euclidean motions (translations and rotations in space), while the configuration space for a liquid crystal is the group of diffeomorphisms coupled with an internal state (gauge symmetry or order parameter). Momentum map and reduction One of the principal ideas of geometric mechanics is ''reduction'', which goes back to Jacobi's elimination of the node in the 3-body problem, but in its modern form is due to K. Meyer (1973) and independently J.E. Marsden and A. Weinstein (1974), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of an object's position relative to its environment. Theoretical expositions of this branch of physics has its origins in Ancient Greece, for instance, in the writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo Galilei, Johannes Kepler, Christiaan Huygens, and Isaac Newton laid the foundation for what is now known as classical mechanics. As a branch of classical physics, mechanics deals with bodies that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D'Alembert's Principle
D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical physics, classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert, and Italian-French mathematician Joseph Louis Lagrange. D'Alembert's principle generalizes the principle of virtual work from statics, static to dynamical systems by introducing ''forces of inertia'' which, when added to the applied forces in a system, result in ''dynamic equilibrium''. D'Alembert's principle can be applied in cases of nonholonomic constraint , kinematic constraints that depend on velocities. The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required. Statement of the principle The principle states that the sum of the differences between the forces acting on a system of massive particles and the time derivatives of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Geometry
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. The term "symplectic", introduced by Hermann Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures. By Darboux's theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a Hamilton–Jacobi equation, link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a Lagrangian mechanics, mechanical system with configuration space (physics), configuration space M and smooth Lagrangian_mechanics#Lagrangian, Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved Scientific Revolution, substantial change in the methods and philosophy of physics. The qualifier ''classical'' distinguishes this type of mechanics from physics developed after the History of physics#20th century: birth of modern physics, revolutions in physics of the early 20th century, all of which revealed limitations in classical mechanics. The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on the 17th century foundational works of Sir Isaac Newton, and the mathematical methods invented by Newton, Gottfried Wilhelm Leibniz, Leonhard Euler and others to describe the motion of Physical body, bodies under the influence of forces. Later, methods bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Abraham (mathematician)
Ralph Herman Abraham (July 4, 1936 – September 19, 2024) was an American mathematician. In 1968 he became a member of the faculty of the University of California, Santa Cruz (UCSC), and later stayed on as a professor emeritus of mathematics. Life and work Abraham earned his BSE (1956), MS (1958), and PhD (1960) from the University of Michigan. His PhD thesis, titled ''Discontinuities in General Relativity'', was written under the direction of Nathaniel Coburn. Prior to joining UCSC, he held positions at the University of California, Berkeley (research lecturer in mathematics; 1960–1962), Columbia University (postdoctoral fellow and assistant professor of mathematics; 1962–1964) and Princeton University (assistant professor of mathematics; 1964–1968). He has also held visiting positions in Amsterdam, Paris, Warwick, Barcelona, Basel, and Florence. He founded the Visual Math Institute at UCSC in 1975; at that time, it was called the "Visual Mathematics Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Marie Souriau
Jean-Marie Souriau (3 June 1922, Paris – 15 March 2012, Aix-en-Provence) was a French mathematician. He was one of the pioneers of modern symplectic geometry. Education and career Souriau started studying mathematics in 1942 at École Normale Supérieure in Paris. In 1946 he was a research fellow of CNRS and an engineer at ONERA. His PhD thesis, defended in 1952 under the supervision of Joseph Pérès and André Lichnerowicz, was entitled "''Sur la stabilité des avions''" (On the stability of planes). Between 1952 and 1958 he worked at Institut des Hautes Études in Tunis, and since 1958 he was Professor of Mathematics at the University of Provence in Marseille. In 1981 he was awarded the Prix Jaffé of the French Academy of Sciences. Research Souriau contributed to the introduction and the development of many important concepts in symplectic geometry, arising from classical and quantum mechanics. In particular, he introduced the notion of moment map, gave a class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis. Education and career Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. Smale obtained his Bachelor of Science degree in 1952. Despite his grades, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Arnold
Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to several areas, including geometrical theory of dynamical systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem. His first main result was the solution of Hilbert's thirteenth problem in 1957 when he was 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), symplectic topology and KAM theory. Arnold was also a populariser of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as '' Mathematical Methods of Clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Louis Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the director of the Académie des Sciences and the first president of the Prussian Academy of Science, at the invitation of Frederick the Great. Maupertuis made an expedition to Lapland to determine the shape of the Earth. He is often credited with having discovered the principle of least action – a version of which is known as Maupertuis's principle – which he expressed as an integral equation that describes the path followed by a physical system. His work in natural history is interesting in relation to modern science since he touched on aspects of heredity and the struggle for life. Biography Maupertuis was born at Saint-Malo, France, to a moderately wealthy family of merchant- corsairs. His father, Renė, had been involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. The son ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton's Principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the '' differential'' equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories. Mathematical formulation Hamilton's principle states that the true evolution of a system described by generalized coordinates between two specified states and at two specified times and is a stationary point (a point where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
