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Hamiltonian System
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.Contents1 Overview 2 Time independent Hamiltonian system2.1 Example3 Symplectic structure 4 Examples 5 See also 6 References 7 Further reading 8 External linksOverview[edit] Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insight about the dynamics, even if the initial value problem cannot be solved analytically
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Simple Harmonic Motion
In mechanics and physics, simple harmonic motion is a special type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Simple harmonic motion
Simple harmonic motion
can serve as a mathematical model for a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. For simple harmonic motion to be an accurate model for a pendulum, the net force on the object at the end of the pendulum must be proportional to the displacement
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Cambridge Univ. Press
280,000 [1] - • Ethnicity (2011)[2] 66% White British 1.4% White Irish 15% White Other 1.7% Black British 3.2% Mixed Race 11% British Asian & Chinese 1.6% otherDemonym(s) CantabrigianTime zone Greenwich Mean Time
Greenwich Mean Time
(UTC+0) • Summer (DST) BST (UTC+1)Postcode CB1 – CB5Area code(s) 01223ONS code 12UB (ONS) E07000008 (GSS)OS grid reference TL450588Website www.cambridge.gov.uk Cambridge
Cambridge
(/ˈkeɪmbrɪdʒ/[3] KAYM-brij) is a university city and the county town of Cambridgeshire, England, on the River Cam
River Cam
approximately 50 miles (80 km) north of London
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American Mathematical Society
The American Mathematical Society
American Mathematical Society
(AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics
Mathematics
and a member of the Conference Board of the Mathematical Sciences.Contents1 History 2 Meetings 3 Fellows 4 Publications 5 Prizes 6 Typesetting 7 Presidents7.1 1888–1900 7.2 1901–1950 7.3 1951–2000 7.4 2001–present8 See also 9 References 10 External linksHistory[edit] The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England
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World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 500 books annually as well as more than 150 journals in various fields. In 1995, World Scientific co-founded the London-based Imperial College Press together with the Imperial College of Science, Technology and Medicine.Contents1 Company structure1.1 Imperial College Press2 Controversy 3 See also 4 References 5 External linksCompany structure[edit] The company head office is in Singapore. The Chairman and Editor-in-Chief is Phua Kok Khoo, while the Managing Director is Doreen Liu. The company was co-founded by them in 1981.[4] Imperial College Press[edit] In 1995 the company co-founded Imperial College Press, specializing in engineering, medicine and information technology, with Imperial College London
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Springer Science+Business Media
Springer Science+Business Media
Springer Science+Business Media
or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.[1] Springer also hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages
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N-body Problem
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.[1] Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets and the visible stars
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Divergence Theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem,[1][2] is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources (with sinks regarded as negative sources) gives the net flux out of a region. The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics. In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus
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Identity Matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context
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Symplectic Structure
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.[1]Contents1 Introduction 2 Comparison with Riemannian geometry 3 Examples and structures 4 Name 5 See also 6 Notes 7 References 8 External linksIntroduction[edit] A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold. On this space is defined a geometric object, the symplectic form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry
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Imperial College Press
Imperial College
Imperial College
Press (ICP) was formed in 1995 as a partnership between Imperial College
Imperial College
of Science, Technology and
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Harmonic Oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: F → = − k x → displaystyle vec F =-k vec x , where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscill
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Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely.[1] When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period
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Constant Of Motion
In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint (which would require extra constraint forces). Common examples include specific energy, specific linear momentum, specific angular momentum and the Laplace–Runge–Lenz vector
Laplace–Runge–Lenz vector
(for inverse-square force laws).Contents1 Applications 2 Methods for identifying constants of motion 3 In quantum mechanics3.1 Derivation 3.2 Comment 3.3 Derivation4 Relevance for quantum chaos 5 Integral of motion 6 Dirac observables 7 ReferencesApplications[edit] Constants of motion are useful because they allow properties of the motion to be derived without solving the equations of motion
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Evolution Equation
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion. These principles can also be equivalently expressed more abstractly by Hamiltonian mechanics or Lagrangian mechanics. The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads)
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Generalized Coordinates
In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration.[1] This is done assuming that this can be done with a single chart. The generalized velocities are the time derivatives of the generalized coordinates of the system. An example of a generalized coordinate is the angle that locates a point moving on a circle
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