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Routhian Mechanics
In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. Although Routhian mechanics is equivalent to Lagrangian mechanics and Hamiltonian mechanics, and introduces no new physics, it offers an alternative way to solve mechanical problems. Definitions The Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta. The Routhia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Edward J Routh
Edward is an English male name. It is derived from the Anglo-Saxon name ''Ēadweard'', composed of the elements '' ēad'' "wealth, fortunate; prosperous" and '' weard'' "guardian, protector”. History The name Edward was very popular in Anglo-Saxon England, but the rule of the Norman and Plantagenet dynasties had effectively ended its use amongst the upper classes. The popularity of the name was revived when Henry III named his firstborn son, the future Edward I, as part of his efforts to promote a cult around Edward the Confessor, for whom Henry had a deep admiration. Variant forms The name has been adopted in the Iberian peninsula since the 15th century, due to Edward, King of Portugal, whose mother was English. The Spanish/Portuguese forms of the name are Eduardo and Duarte. Other variant forms include French Édouard, Italian Edoardo and Odoardo, German, Dutch, Czech and Romanian Eduard and Scandinavian Edvard. Short forms include Ed, Eddy, Eddie, Ted, Teddy and Ned. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Differential Of A Function
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by dy = f'(x)\,dx, where f'(x) is the derivative of with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation dy = \frac\, dx holds, where the derivative is represented in the Leibniz notation dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes df(x) = f'(x)\,dx. The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Symmetrical Top
The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends both on the mass and its distribution relative to the axis, increasing with mass and distance from the axis. It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Heavy Symmetric Top Euler Angles
Heavy may refer to: Measures * Heavy, a characterization of objects with substantial weight * Heavy, a wake turbulence category used by pilots and air traffic controllers to refer to aircraft with a maximum takeoff mass of 136,000 kgs or more * Heavy, a type of strength of Scottish beer#Shilling categories, Scottish beer Arts, entertainment, and media Music Groups * The Heavy (band), a rock band from England Albums * Heavy (Heavy D album), ''Heavy'' (Heavy D album), 1999 * Heavy (Iron Butterfly album), ''Heavy'' (Iron Butterfly album), a 1968 album by Iron Butterfly * Heavy (Bin-Jip album), ''Heavy'' (Bin-Jip album), the second studio album by Bin-Jip * Heavy (Sir album), ''Heavy'' (Sir album), 2024 Songs * Heavy (Collective Soul song), "Heavy" (Collective Soul song), 1999 * Heavy (Lauri Ylönen song), "Heavy" (Lauri Ylönen song), 2011 * Heavy (Linkin Park song), "Heavy" (Linkin Park song), 2017 * Heavy (Anne-Marie song), "Heavy" (Anne-Marie song), 2017 * "Heavy", by Cx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pendulum Equation
A pendulum is a body suspended from a fixed support such that it freely swings back and forth under the influence of gravity. 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 towards the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations. Simple gravity pendulum A ''simple gravity pendulum'' is an idealized mathematical model of a real pendulum. It is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in the model there is no frictional energy loss, when given an initial displacement it swi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Spherical Pendulum
In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. Lagrangian mechanics Routinely, in order to write down the kinetic T=\tfracmv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. Here, following the conventions shown in the diagram, :x=l\sin\theta\cos\phi :y=l\sin\theta\sin\phi :z=l(1-\cos\theta). Next, time derivatives of these coordinates are taken, to obtain velocities along the axes :\dot x=l\cos\theta\cos\phi\,\dot\theta-l\sin\theta\sin\phi\,\dot\phi :\dot y=l\cos\theta\sin\phi\,\dot\theta+l\sin\theta\cos\phi\,\dot\phi :\dot z=l\sin\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Spherical Pendulum Lagrangian Mechanics
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Spherical Polar Coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point called the origin; * the polar angle between this radial line and a given ''polar axis''; and * the azimuthal angle , which is the angle of rotation of the radial line around the polar axis. (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (''r'', ''θ'', ''φ''), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the ''reference plane'' (sometimes '' fundamental plane''). Terminology The radial distance from the fixed point of origin is also called the ''radius'', or ''radial line'', or ''radial coord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Central Potential
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. \mathbf(\mathbf) = F( \mathbf ) where F is a force vector, ''F'' is a scalar valued force function (whose absolute value gives the magnitude of the force and is positive if the force is outward and negative if the force is inward), r is the position vector, , , r, , is its length, and \hat = \mathbf r / \, \mathbf r\, is the corresponding unit vector. Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant. Examples of spherically symmetric central forces include the Coulomb force and the force of gravity. Properties Central forces that are conservative can always be expressed as the negative gradient of a potential energy: \mathbf(\mathbf) = - \mathbf V(\mathbf) \; \text V(\mathbf) = \int_^ F(r)\,\mathrmr (the upper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cyclic Coordinates
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, '' Mécanique analytique''. Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space ''M'' and a smooth function L within that space called a ''Lagrangian''. For many systems, , where ''T'' and ''V'' are the kinetic and potential energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from ''L'' must remain at a stationary point (specifically, a maximum, minimum, or saddle point) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Introduction Newton's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Joule
The joule ( , or ; symbol: J) is the unit of energy in the International System of Units (SI). In terms of SI base units, one joule corresponds to one kilogram- metre squared per second squared One joule is equal to the amount of work done when a force of one newton displaces a body through a distance of one metre in the direction of that force. It is also the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second. It is named after the English physicist James Prescott Joule (1818–1889). Definition According to the International Bureau of Weights and Measures the joule is defined as "the work done when the point of application of 1 MKS unit of force ewtonmoves a distance of 1 metre in the direction of the force." In terms of SI base units and in terms of SI derived units with special names, the joule is defined as One joule is also equivalent to any of the following: * The work required to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
SI Units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce. The SI system is coordinated by the International Bureau of Weights and Measures, which is abbreviated BIPM from . The SI comprises a coherent system of units of measurement starting with seven base units, which are the second (symbol s, the unit of time), metre (m, length), kilogram (kg, mass), ampere (A, electric current), kelvin (K, thermodynamic temperature), mole (mol, amount of substance), and candela (cd, luminous intensity). The system can accommodate coherent units for an unlimited number of additional quantities. These are called coherent derived units, which can always be represented as products of powers of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
