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System configuration relative to another
configuration. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes
Generalized_coordinates
Coordinate system that is defined by points instead of vectors
point not having unique generalized barycentric coordinates except when P is a simplex. Dual to generalized barycentric coordinates are slack variables,
Barycentric_coordinate_system
Formulation of classical mechanics
this vk, the kinetic energy in generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors
Lagrangian_mechanics
Sets of coordinates on phase space which can be used to describe a physical system
details. As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the
Canonical_coordinates
Overview of mechanics based on the least action principle
the generalized coordinates and velocities (q, q̇) with (q, p); the generalized coordinates and the generalized momentums conjugate to the generalized coordinates:
Analytical_mechanics
Property of a mass in motion
symbol Π. In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as p j = ∂ L
Momentum
Concept in Lagrangian mechanics
mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces Fi
Generalized_forces
Method for specifying point positions
Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates. Generalized coordinates are
Coordinate_system
Method of solution for certain mechanical problems
coordinates, which we denote as w {\displaystyle \mathbf {w} } (the action angles, which are the generalized coordinates) and their new generalized momenta
Action-angle_coordinates
formulated in generalized coordinates of motion. Note that "generalized coordinates of motion" are related to—but distinct from—generalized coordinates as used
Generalized_filtering
Statement relating differentiable symmetries to conserved quantities
= 1. Taking δθ as the ε parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by Q = n ×
Noether's_theorem
Formulation of geometrical optics
q_{N}{\left(\sigma \right)}\right)} described by N {\displaystyle N} generalized coordinates between two specified states at two specified parameters σA and
Hamiltonian_optics
Formulation of classical mechanics
choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta
Hamilton–Jacobi_equation
Space of possible positions for all objects in a physical system
the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the
Configuration_space_(physics)
Abstract coordinate system
perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related
Frame_of_reference
Geometric concept of a 2D space with "points at infinity" adjoined
(geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows. Let K be any division ring (skewfield)
Projective_plane
Type of inertial force
Retrieved 2020-11-09. For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics
Centrifugal_force
Type of constraints for mechanical systems
n } {\displaystyle \{u_{1},u_{2},u_{3},\ldots ,u_{n}\}} are n generalized coordinates that describe the system (in unconstrained configuration space)
Holonomic_constraints
Equations that describe the behavior of a physical system
are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient
Equations_of_motion
Coordinate transformation that preserves the form of Hamilton's equations
mechanics). Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations
Canonical_transformation
Description of large objects' physics
mechanics are Lagrangian mechanics, which uses generalized coordinates and corresponding generalized velocities in tangent bundle space (the tangent
Classical_mechanics
Statement based on repeated empirical observations that describes some natural phenomenon
freedom is defined by generalized coordinates q = (q1, q2, ... qN). There are generalized momenta conjugate to these coordinates, p = (p1, p2, ..., pN)
Scientific_law
Work done by a force to move a particle along a virtual displacement
virtual displacements. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates q i {\displaystyle q_{i}} , i = 1
Virtual_work
Physical quantity of dimension energy × time
the action-angle coordinates, called the "action" of the generalized coordinate qk, is defined by integrating a single generalized momentum around a
Action_(physics)
Study of the effects of forces on undeformable bodies
{q}}}}\right),} is the generalized force acting on this one degree of freedom system. If the mechanical system is defined by m generalized coordinates, qj, j = 1
Rigid_body_dynamics
Second-order partial differential equation describing motion of mechanical system
has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field
Euler–Lagrange_equation
Formulation of the principle of stationary action
states that the true evolution q(t) of a system described by N generalized coordinates q = (q1, q2, ..., qN) between two specified states q1 = q(t1) and
Hamilton's_principle
Principle of least length in physics
involve the generalized velocities. By defining a normalized distance or metric d s {\displaystyle ds} in the space of generalized coordinates d s 2 = d
Maupertuis's_principle
Formulation of classical mechanics
_{r}={\ddot {q}}_{r}} is an arbitrary generalized acceleration, or the second time derivative of the generalized coordinates q r {\displaystyle q_{r}} , and
Appell's_equation_of_motion
Mathematical formulation of special and general relativity
differential of r obtains the transformation of velocity v to the generalized coordinates, generalized velocities, and coordinate time v = ∑ j = 1 n ∂ r ∂ q j q
Relativistic Lagrangian mechanics
Relativistic_Lagrangian_mechanics
Function used in Lagrangian mechanics
potential. This relationship is represented in terms of the set of generalized coordinates q i = { q 1 , q 2 , … q n } {\displaystyle q_{i}=\left\{q_{1},q_{2}
Rayleigh_dissipation_function
2D coordinate system whose coordinate lines are confocal ellipses and hyperbolae
the momenta). Curvilinear coordinates Ellipsoidal coordinates Generalized coordinates Bipolar coordinates "Elliptic coordinates", Encyclopedia of Mathematics
Elliptic_coordinate_system
When the net force on a particle is zero
where the gradient of the potential energy with respect to the generalized coordinates is zero. If a particle in equilibrium has zero velocity, that particle
Mechanical_equilibrium
Matrix relating a system's generalized coordinate vector and kinetic energy
describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of every particle in the system
Mass_matrix
Application of Lagrangian mechanics to field theories
bundles. In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in
Lagrangian_(field_theory)
Fundamental mechanical principles
p2,…,pN) are the particle momenta or the conjugate momenta of generalized coordinates, defined by the equation p k = def ∂ L ∂ q ˙ k , {\displaystyle
Action_principles
Mechanical system whose constraints are independent of time
explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. The opposite
Scleronomous
Type of derivative in mathematics
time derivative of a function of time and the n {\displaystyle n} generalized coordinates lead to the same equations of motion. An interesting example concerns
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Formulation of classical mechanics
generalized velocity dqi/dt is to be expressed as a function of generalized momentum pi via its defining relation. The choice of which n coordinates are
Routhian_mechanics
Space of all possible states that a system can take
mechanics, any choice of generalized coordinates qi for the position (i.e. coordinates on configuration space) defines conjugate generalized momenta pi, which
Phase_space
Type of optimization problem
generalized coordinates and is the foundation of Lagrangian mechanics. Nonholonomic constraints cannot be eliminated by using generalized coordinates
Nonholonomic_system
Physical spaces representing position and momentum, Fourier-transform duals
n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential of the generalized coordinate
Position_and_momentum_spaces
Description of a quantum-mechanical system
{\displaystyle H} is the Hamiltonian function (not operator). Here the generalized coordinates q i {\displaystyle q_{i}} for i = 1 , 2 , 3 {\displaystyle i=1
Schrödinger_equation
Topological space that locally resembles Euclidean space
freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is
Manifold
Tool to study dynamic behavior of interconnected rigid or flexible bodies
redundant coordinates, because the equations use more coordinates than degrees of freedom of the underlying system. The generalized coordinates are denoted
Multibody_system
Quantum operator for the sum of energies of a system
Hamilton's equations, with the a n {\displaystyle a_{n}} s as the generalized coordinates, the π n {\displaystyle \pi _{n}} s as the conjugate momenta, and
Hamiltonian (quantum mechanics)
Hamiltonian_(quantum_mechanics)
Change in the position of an object
are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient
Motion
Value remaining constant in a dynamical system
{\displaystyle {\mathcal {H}}} , a function f of the generalized coordinates q and generalized momenta p has time evolution d f d t = { f , H } + ∂ f
Conserved_quantity
Italian-French scientist (1736–1813)
its configuration by a sufficient number of variables x, called generalized coordinates, whose number is the same as that of the degrees of freedom possessed
Joseph-Louis_Lagrange
Pendulum with another pendulum attached to its end
convenient to use the angles between each limb and the vertical as the generalized coordinates defining the configuration of the system. These angles are denoted
Double_pendulum
Specific microscopic configuration of a thermodynamic system
space, whose coordinate axes consist of the F generalized coordinates qi of the system, and its F generalized momenta pi. The microstate of such a system
Microstate (statistical mechanics)
Microstate_(statistical_mechanics)
Pendulum with center of mass above pivot
{\theta }}^{2}.} The generalized coordinates of the system are θ {\displaystyle \theta } and x {\displaystyle x} , each has a generalized force. On the x {\displaystyle
Inverted_pendulum
Dynamical system governed by Hamilton's equations
described by the generalized coordinates p {\displaystyle {\boldsymbol {p}}} and q {\displaystyle {\boldsymbol {q}}} , corresponding to generalized momentum and
Hamiltonian_system
Power volt-ampere reactive (var) q {\displaystyle \mathbf {q} } Generalized coordinates varied depending on context (sometimes meter (m) or radian (rad))
List of common physics notations
List_of_common_physics_notations
Conserved physical quantity; rotational analogue of linear momentum
angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per
Angular_momentum
Canonical differential form
{\displaystyle g,} then corresponding definitions can be made in terms of generalized coordinates. Specifically, if we take the metric to be a map g : T Q → T ∗
Tautological_one-form
Statistical ensemble of particles in thermodynamic equilibrium
the p1, … pn and q1, … qn are the canonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom
Grand_canonical_ensemble
Superconducting circuit element
oscillator. In circuit quantization, flux and charge are treated as generalized coordinates. The quantization step promotes ϕ {\displaystyle \phi } (dimensionless
Josephson_junction
Formulation of classical mechanics using momenta
mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta. Both theories
Hamiltonian_mechanics
Tensor describing energy momentum density in spacetime
spacetime coordinates, we can construct the canonical stress–energy tensor by looking at the total derivative with respect to one of the generalized coordinates
Stress–energy_tensor
Formalism in classical field theory based on Hamiltonian mechanics
Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields
Hamiltonian_field_theory
Operation in Hamiltonian mechanics
_{j}]=-\delta _{ij}} Note that the summation here involves generalized coordinates as well as generalized momentum. The invariance of Poisson bracket can be expressed
Poisson_bracket
Method for satisfying the Newtonian motion of a rigid body which consists of mass points
represents the generalized forces and the scalar V(q) represents the potential energy, both of which are functions of the generalized coordinates q. If M constraints
Constraint (computational chemistry)
Constraint_(computational_chemistry)
Objects extending the notion of functions
theory of generalized functions in order to define weak solutions of partial differential equations (i.e. solutions which are generalized functions,
Generalized_function
Coordinate system using perpendicular axes
later generalized into the concept of vector spaces. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for
Cartesian_coordinate_system
Statement in classical mechanics
Q j {\displaystyle Q_{j}} being a generalized applied force and Q j ∗ {\displaystyle Q_{j}^{*}} being a generalized inertia force. This condition yields
D'Alembert's_principle
Idealization of a large number of atomic-sized systems
mechanical system with a defined number of parts, the phase space has n generalized coordinates called q1, ... qn, and n associated canonical momenta called p1
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Convex polytope, the n-dimensional analogue of a square and a cube
generates a 4-dimensional unit hypercube (a unit tesseract). This can be generalized to any number of dimensions. This process of sweeping out volumes can
Hypercube
Three-dimensional coordinate system
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} that generalizes the two-dimensional
Ellipsoidal_coordinates
Array of numbers describing a metric connection
x i {\displaystyle x^{i}} be the generalized coordinates and x ˙ i {\displaystyle {\dot {x}}^{i}} be the generalized velocities, then the kinetic energy
Christoffel_symbols
Ensemble of states at a constant temperature
the p1, … pn and q1, … qn are the canonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom
Canonical_ensemble
American chemist (1910–1985)
_{i}} . Applying a vector conversion from the Cartesian coordinates to the generalized coordinates will describe the same three-dimensional structure using
Paul_Flory
Hamiltonian formulation of general relativity
( 4 ) g i j {\displaystyle g_{ij}={^{(4)}}g_{ij}} will be the generalized coordinates for a Hamiltonian formulation. The conjugate momenta can then be
ADM_formalism
Interdisciplinary field of biology
generally, the Euler–Lagrange equations can be derived for systems of generalized coordinates. The Euler–Lagrange equation in computational anatomy describes
Computational_anatomy
Mathematical approach to quantum physics
are considered as generalized coordinates, then Fμ should be identified as the generalized force operators related to those coordinates. Different indices
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Advanced undergraduate or graduate textbook
concepts, then introduces the principle of virtual work, constraints, generalized coordinates, and Lagrangian mechanics. Scattering is treated in the same chapter
Classical Mechanics (Goldstein)
Classical_Mechanics_(Goldstein)
Description of the time-evolution of plasma
equation, though anachronistically in this context—expressed in generalized coordinates: d d t f ( r , p , t ) = 0 {\displaystyle {\frac {\mathrm {d} }{\mathrm
Vlasov_equation
Response of a vessel to sea conditions
^{2}=gk\tanh(kh)} The vessel response is commonly represented by six generalized coordinates corresponding to surge, sway, heave, roll, pitch and yaw. In the
Seakeeping
Phenomenon affecting the orbit of a binary system
{\mathcal {H}}} , of canonical coordinates in phase space. The canonical coordinates consist of the generalized coordinates x k {\displaystyle x_{k}} in
Kozai_mechanism
Vector used in astronomy
direction. A generalized conserved LRL vector A {\displaystyle {\mathcal {A}}} can be defined for all central forces, but this generalized vector is a
Laplace–Runge–Lenz_vector
E_{p}={\frac {1}{2}}k(r_{2}-r_{1})^{2}} where r2 and r1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression
List of equations in classical mechanics
List_of_equations_in_classical_mechanics
Function acting on the space of physical states in physics
, p , t ) {\displaystyle H(q,p,t)} , a function of the generalized coordinates q, generalized velocities q ˙ = d q / d t {\displaystyle {\dot {q}}=\mathrm
Operator_(physics)
4D relativistic energy and momentum
from the action S. Given that in general for a closed system with generalized coordinates qi and canonical momenta pi, p i = ∂ S ∂ q i = ∂ S ∂ x i , E =
Four-momentum
French mathematician, physicist, and author (1706–1749)
true for all problems where the initial state is symmetric in generalized coordinates. E.g., mechanical energy, either kinetic or potential, may be lost
Émilie_du_Châtelet
Generalization of the concept from statistical mechanics
X} . The word conjugate here is used in the sense of conjugate generalized coordinates in Lagrangian mechanics, thus, properly β {\displaystyle \beta
Partition function (mathematics)
Partition_function_(mathematics)
Frame-dependent apparent force in Physics
motion equation Frenet–Serret formulas General relativity Generalized coordinates Generalized force Gravity Inertial reference frame Kinematics Kinetics
Fictitious_force
Coordinate system used in projective geometry
coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used
Homogeneous_coordinates
been introduced allowing for descriptions of structures involving generalized coordinates and additional translational or rotational degrees of freedom.
Stochastic Eulerian Lagrangian method
Stochastic_Eulerian_Lagrangian_method
Place where a name-bearing type specimen was collected
publishing a broader verbal locality, rounding or generalizing coordinates, or withholding exact coordinates from public datasets while retaining the full
Type_locality_(biology)
Properties independent of system size, and proportional to system size
care is needed to distinguish generalized coordinates (often extensive, such as total charge) from their conjugate generalized forces (often intensive, such
Intensive and extensive properties
Intensive_and_extensive_properties
Computational fluid dynamics tools
Contour advection Displacement field (mechanics) Equivalent latitude Generalized Lagrangian mean Trajectory (fluid mechanics) Liouville's theorem (Hamiltonian)
Lagrangian and Eulerian specification of the flow field
Lagrangian_and_Eulerian_specification_of_the_flow_field
Principle in mathematical physics
its generalized coordinates, and let u = ( u 1 , u 2 , … , u n ) {\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},\dots ,u_{n})} be its generalized velocity
Herglotz's variational principle
Herglotz's_variational_principle
switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained
First-class_constraint
Coordinate system whose directions vary in space
geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from
Curvilinear_coordinates
3-Dimensional analogue of a pendulum
of the Lagrangian L = T − V {\displaystyle L=T-V} in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. Here
Spherical_pendulum
Generalization of an ordered basis of a vector space
automorphisms is G. A smooth manifold Σ which serves as a space of (generalized) coordinates for X. A collection of frames ƒ each of which determines a coordinate
Moving_frame
{\displaystyle \{u_{1},u_{2},u_{3},\ldots ,u_{n}\}} are the n generalized coordinates that describe the system, and where L {\displaystyle L} is the
Pfaffian_constraint
Theorem on magnetism
{p} _{N};\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})} of these generalized coordinates is then ⟨ f ⟩ = ∫ f d P ∫ d P . {\displaystyle \langle f\rangle
Bohr–Van_Leeuwen_theorem
systems is mainly based on the Lagrangian which is a function of the generalized coordinates and the associated velocities. If all forces are derivable from
Electromechanical_modeling
Ensemble of states with an exactly specified total energy
qn) defined over the system's phase space. The phase space has n generalized coordinates called q1, ... qn, and n associated canonical momenta called p1
Microcanonical_ensemble
GENERALIZED COORDINATES
GENERALIZED COORDINATES
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
GENERALIZED COORDINATES
GENERALIZED COORDINATES
Boy/Male
Arabic, Muslim
Name of a Companion of the Prophet PBUH
Girl/Female
Hindu, Indian
Name of a Raga of Carnatic Music
Boy/Male
Muslim/Islamic
Of mercy
Girl/Female
African, American, Arabic, French, Indian, Iranian, Kannada, Muslim, Parsi, Punjabi, Sikh, Sindhi, Telugu
Faith; Belief; Faithful; Believer of Faith
Boy/Male
Arabic, Muslim
Opener; Untier; One who Opens
Girl/Female
British, Christian, English, Indian
Beautiful
Boy/Male
Hindu
Lord of Yoga
Girl/Female
Hebrew
Asked of God.
Girl/Female
Australian, German, Teutonic
Famous Defender
Girl/Female
Australian, French, Indian, Malayalam, Welsh
Idol; Image; A Small Bird
GENERALIZED COORDINATES
GENERALIZED COORDINATES
GENERALIZED COORDINATES
GENERALIZED COORDINATES
GENERALIZED COORDINATES
imp. & p. p.
of Mineralize
v. t.
To impregnate with a mineral; as, mineralized water.
imp. & p. p.
of Federalize
n.
The system by which power is centralized, as in a government.
p. pr. & vb. n.
of Generalize
v. t.
To derive or deduce (a general conception, or a general principle) from particulars.
n.
One who takes general or comprehensive views.
a.
Capable of being generalized, or reduced to a general form of statement, or brought under a general rule.
n.
A generalized concept of magnitude.
v. t.
To generalize or conclude as an inference from all the particulars; -- the opposite of deduce.
v. t.
To bring under a genus or under genera; to view in relation to a genus or to genera.
n.
A fishlike creature (Amphioxus lanceolatus), two or three inches long, found in temperature seas; -- also called the lancelet. Its body is pointed at both ends. It is the lowest and most generalized of the vertebrates, having neither brain, skull, vertebrae, nor red blood. It forms the type of the group Acrania, Leptocardia, etc.
a.
That can be passed over in a single course; -- said of a curve when the coordinates of the point on the curve can be expressed as rational algebraic functions of a single parameter /.
n.
The act or process of centralizing, or the state of being centralized; the act or process of combining or reducing several parts into a whole; as, the centralization of power in the general government; the centralization of commerce in a city.
a.
Comprising structural characters which are separated in more specialized forms; synthetic; as, a generalized type.
v. t.
To apply to other genera or classes; to use with a more extensive application; to extend so as to include all special cases; to make universal in application, as a formula or rule.
imp. & p. p.
of Generalize
v. i.
To form into a genus; to view objects in their relations to a genus or class; to take general or comprehensive views.
imp. & p. p.
of Centralize
v. t.
To make universal; to generalize.