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Chaplygin Sleigh
The Chaplygin sleigh is a simple pedagogical example of a nonholonomic system in mechanics, described by Sergey Chaplygin Sergey Alexeyevich Chaplygin (russian: Серге́й Алексе́евич Чаплы́гин; 5 April 1869 – 8 October 1942) was a Russian and Soviet physicist, mathematician, and mechanical engineer. He is known for mathe .... It consists of a body that slides frictionlessly on a horizontal plane, with a knife edge that constrains its motion so that the knife slides only longitudinally. Because this constraint is nonholonomic, Liouville's theorem does not apply, and although energy is conserved, the motion is dissipative in the sense that phase-space volume is not conserved. The motion is attracted to an equilibrium, in which the sleigh moves without rotation, with the knife edge trailing the center of mass. There are several ways of seeing that the system is nonholonomic. The dimension of the phase space is 5, which is odd. The co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holonomic Constraints
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that describe the system. For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic. For the first case, the holonomic constraint may be given by the equation :r^2-a^2=0 where r is the distance from the centre of a sphere of radius a, whereas the second non-holonomic case may be given by :r^2 - a^2 \geq 0 Velocity-dependent constraints (also called semi-holonomic constraints) such as :f(u_1,u_2,\ldots,u_n,\dot_1,\dot_2,\ldots,\dot_n,t)=0 are not usually holonomic. Holonomic system In classical mechanics a system may be defined as holonomic if all constraints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects result in displacements, or 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, Kepler, Huygens, and 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergey Chaplygin
Sergey Alexeyevich Chaplygin (russian: Серге́й Алексе́евич Чаплы́гин; 5 April 1869 – 8 October 1942) was a Russian and Soviet physicist, mathematician, and mechanical engineer. He is known for mathematical formulas such as Chaplygin's equation and for a hypothetical substance in cosmology called Chaplygin gas, named after him. He graduated in 1890 from Moscow University, and later became a professor. He taught mechanical engineering at Moscow Higher Courses for Women in 1901, and of applied mathematics at Moscow School of Technology, 1903. He was appointed Director of the courses in 1905. Leonid I. Sedov was one of his students. Chaplygin's theories were greatly inspired by N. Ye. Zhukovsky, who founded the Central Institute of Aerodynamics. His early research consisted of hydromechanics. His "Collected Works" in four volumes were published in 1948. Honours and awards * Hero of Socialist Labour (1 February 1941) * Two Orders of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems. There are extensions of Liouville's theorem to stochastic systems. Liouville equations The Liouville equation describes the time evolution of the ''phase space distribution function''. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the impor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase-space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the outer product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Introduction In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-space trajectory for the sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
