## Lagrangian mechanics |

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**Lagrangian mechanics** is a reformulation of

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*,^{[1]} which treat ^{[2]}^{[3]} or the *Lagrange equations of the second kind*, which incorporate the constraints directly by judicious choice of ^{[1]}^{[4]} In each case, a **Lagrangian** is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.

No new physics are necessarily introduced in applying Lagrangian mechanics compared to ^{[5]} Generalized coordinates can be chosen for convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of

Lagrangian mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of * Mécanique analytique*,

Lagrangian mechanics is widely used to solve mechanical problems in physics and when

- introduction
- from newtonian to lagrangian mechanics
- properties of the euler–lagrange equation
- examples
- extensions to include non-conservative forces
- other contexts and formulations
- see also
- footnotes
- notes
- references
- further reading
- external links

Suppose we have a bead sliding around on a wire, or a swinging *independent*

For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a *N* point particles with *m*_{1}, *m*_{2}, ..., *m _{N}*, each particle has a

In Newtonian mechanics, the

applies to each particle. For an

Instead of forces, Lagrangian mechanics uses the **Lagrangian**, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The *non-relativistic* Lagrangian for a system of particles can be defined by^{[9]}

where

is the total ^{[10]} and *V* is the

Kinetic energy is the energy of the system's motion, and *v _{k}*

The *V* = *V*(**r**_{1}, **r**_{2}, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. *V* = *V*(**r**_{1}, **r**_{2}, ..., **v**_{1}, **v**_{2}, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally *V* = *V*(**r**_{1}, **r**_{2}, ..., **v**_{1}, **v**_{2}, ..., *t*).

The above form of *L* does not hold in *L*.

One or more of the particles may each be subject to one or more *f*(**r**, *t*) = 0. If the number of constraints in the system is *C*, then each constraint has an equation, *f*_{1}(**r**, *t*) = 0, *f*_{2}(**r**, *t*) = 0, ... *f _{C}*(

If *T* or *V* or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian *L*(**r**_{1}, **r**_{2}, ... **v**_{1}, **v**_{2}, ... *t*) is *explicitly time-dependent*. If neither the potential nor the kinetic energy depend on time, then the Lagrangian *L*(**r**_{1}, **r**_{2}, ... **v**_{1}, **v**_{2}, ...) is *explicitly independent of time*. In either case, the Lagrangian will always have implicit time-dependence through the generalized coordinates.

With these definitions, **Lagrange's equations of the first kind** are^{[12]}

where *k* = 1, 2, ..., *N* labels the particles, there is a *λ _{i}* for each constraint equation

are each shorthands for a vector of ^{[nb 1]} Each overdot is a shorthand for a *N* to 3*N* + *C*, because there are 3*N* coupled second order differential equations in the position coordinates and multipliers, plus *C* constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations.

In the Lagrangian, the position coordinates and velocity components are all *L* with respect to the *z*-velocity component of particle 2, *v*_{z2} = d*z*_{2}/d*t*, is just that; no awkward *z*_{2}).

In each constraint equation, one coordinate is redundant because it is determined from the other two. The number of *independent* coordinates is therefore *n* = 3*N* − *C*. We can transform each position vector to a common set of *n* *n*-tuple **q** = (*q*_{1}, *q*_{2}, ... *q _{n}*), by expressing each position vector, and hence the position coordinates, as

The vector **q** is a point in the

Given this **v**_{k}, the kinetic energy *in generalized coordinates* depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so *T* = *T*(**q**, d**q**/d*t*, *t*).

With these definitions we have the ** Euler–Lagrange equations**, or

are mathematical results from the *L*(**q**, d**q**/d*t*, *t*), gives the *N* to *n* = 3*N* − *C* coupled second order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for.

Although the equations of motion include *t* often involves

Other Languages

Afrikaans: Lagrange-meganika

العربية: ميكانيكا لاغرانج

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български: Механика на Лагранж

català: Formulació lagrangiana

Deutsch: Lagrange-Formalismus

eesti: Lagrange'i mehaanika

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español: Mecánica lagrangiana

Esperanto: Lagranĝa mekaniko

euskara: Lagrangeren mekanika

فارسی: مکانیک لاگرانژی

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galego: Mecánica lagranxiana

한국어: 라그랑주 역학

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Nederlands: Lagrangiaanse mechanica

日本語: ラグランジュ力学

norsk: Lagrange-mekanikk

norsk nynorsk: Lagrangemekanikk

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русский: Лагранжева механика

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中文: 拉格朗日力学