Lagrangian mechanics

LagrangianLagrange's equationsLagrangiansLagrangian dynamicscyclic coordinateEuler–Lagrange equationsLagrange equationsLagrange's equations of motionLagrangian formalismLagrangian formulation
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.wikipedia
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Classical mechanics

Newtonian mechanicsNewtonian physicsclassical
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton later developed Hamilton's principle that can be used to derive the Lagrange equation and was later recognized to be applicable to much of fundamental theoretical physics as well, particularly quantum mechanics and the theory of relativity.
Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics.

Noether's theorem

Noether currentNoether chargeNoether theorem
Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem.
The action of a physical system is the integral over time of a Lagrangian function (which may be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.

Lagrangian (field theory)

LagrangianLagrangian densityLagrangian field theory
Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density.
It is the field-theoretic analogue of Lagrangian mechanics.

Joseph-Louis Lagrange

LagrangeJoseph Louis LagrangeJoseph Lagrange
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.
But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and presented the so-called mechanical "principles" as simple results of the variational calculus.

Hamilton's principle

HamiltonianHamilton
Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton later developed Hamilton's principle that can be used to derive the Lagrange equation and was later recognized to be applicable to much of fundamental theoretical physics as well, particularly quantum mechanics and the theory of relativity.
(See that article for historical formulations.) It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it.

Euler–Lagrange equation

Euler–Lagrange equationsEuler-Lagrange equationLagrange's equation
Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange (EL) equations.
In Lagrangian mechanics, because of Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for the action of the system.

Relativistic Lagrangian mechanics

relativistic action
The above form of L does not hold in relativistic Lagrangian mechanics, and must be replaced by a function consistent with special or general relativity.
In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.

William Rowan Hamilton

HamiltonSir William Rowan HamiltonWilliam Hamilton
Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton later developed Hamilton's principle that can be used to derive the Lagrange equation and was later recognized to be applicable to much of fundamental theoretical physics as well, particularly quantum mechanics and the theory of relativity.
Paradoxically, the credit for discovering the quantity now called the Lagrangian and Lagrange's equations belongs to Hamilton.

Coordinate system

coordinatescoordinateaxis
Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system.

Generalized coordinates

generalized coordinategeneralized velocitiesGaussian coordinates
In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms: either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.
Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of Lagrange's equations of motion.

Energy

energy transferenergiestotal energy
Instead of forces, Lagrangian mechanics uses the energies in the system.
Another energy-related concept is called the Lagrangian, after Joseph-Louis Lagrange.

Configuration space (physics)

configuration spaceconfigurationconformation space
The vector q is a point in the configuration space of the system.
It is conventional to use the symbol q for a point in configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics, and in Lagrangian mechanics.

Quantum mechanics

quantum physicsquantum mechanicalquantum theory
Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton later developed Hamilton's principle that can be used to derive the Lagrange equation and was later recognized to be applicable to much of fundamental theoretical physics as well, particularly quantum mechanics and the theory of relativity.
Lagrangian mechanics applies to this.

Analytical mechanics

analyticalTheoretical MechanicsAnalytical method
A fundamental result in analytical mechanics is D'Alembert's principle, introduced in 1708 by Jacques Bernoulli to understand static equilibrium, and developed by D'Alembert in 1743 to solve dynamical problems.
Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space).

Generalized forces

generalized forcegeneralizedgeneralized "forces
The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N k along the virtual displacements δr k, and can without loss of generality be converted into the generalized analogues by the definition of generalized forces
Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates.

Angular momentum

conservation of angular momentumangular momentamomentum
Its dimensions are the same as [ angular momentum ], [energy]·[time], or [length]·[momentum].
Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate \phi expressed in the Lagrangian of the mechanical system.

Lagrange multiplier

Lagrange multipliersLagrangianLagrangian multiplier
In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms: either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.
As examples, in Lagrangian mechanics the equations of motion are derived by finding stationary points of the action, the time integral of the difference between kinetic and potential energy.

Canonical transformation

canonical transformationspoint transformationcanonical
Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates s according to a point transformation q = q(s, t), the new Lagrangian L′ is a function of the new coordinates
Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates

Equations of motion

equation of motionUniformly accelerated motionSUVAT equations
The velocity of each particle is how fast the particle moves along its path of motion, and is the time derivative of its position, thusIn Newtonian mechanics, the equations of motion are given by Newton's laws.
The Euler–Lagrange equations are

Action (physics)

actionaction principleaction integral
The time integral of the Lagrangian is another quantity called the action, defined as
:where the integrand L is called the Lagrangian.

Virtual displacement

The virtual displacements, δr k, are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system at an instant of time, i.e. in such a way that the constraint forces maintain the constrained motion.
This equation is used in Lagrangian mechanics to relate generalized coordinates, q j, to virtual work, δW, and generalized forces, Q j .

Total derivative

differentialtotaltotal differential
The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the total derivative of its position with respect to time, is
This expression is often used in physics for a gauge transformation of the Lagrangian, as two Lagrangians that differ only by the total time derivative of a function of time and the n generalized coordinates lead to the same equations of motion.

Conservation of energy

law of conservation of energyenergyenergy conservation law
is the total conserved energy of the system.
This focus on the vis viva by the continental physicists eventually led to the discovery of stationarity principles governing mechanics, such as the D'Alembert's principle, Lagrangian,

Hamiltonian mechanics

HamiltonianHamilton's equationsHamiltonian dynamics
A closely related formulation of classical mechanics is Hamiltonian mechanics.
Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.

Routhian mechanics

RouthianRouth's procedure
Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.
In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh.