# Lagrangian system – Wikipedia

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In mathematics, a **Lagrangian system** is a pair (*Y*, *L*), consisting of a smooth fiber bundle *Y* → *X* and a Lagrangian density *L*, which yields the Euler–Lagrange differential operator acting on sections of *Y* → *X*.

In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle *Q* → ℝ over the time axis ℝ. In particular, *Q* = ℝ × *M* if a reference frame is fixed. In classical field theory, all field systems are the Lagrangian ones.

## Lagrangians and Euler–Lagrange operators[edit]

A **Lagrangian density** *L* (or, simply, a Lagrangian) of order *r* is defined as an *n*-form, *n* = dim *X*, on the *r*-order jet manifold *J*^{r}*Y* of *Y*.

A Lagrangian *L* can be introduced as an element of the variational bicomplex of the differential graded algebra *O*^{∗}_{∞}(*Y*) of exterior forms on jet manifolds of *Y* → *X*. The coboundary operator of this bicomplex contains the variational operator *δ* which, acting on *L*, defines the associated Euler–Lagrange operator *δL*.

### In coordinates[edit]

Given bundle coordinates *x*^{λ}, *y*^{i} on a fiber bundle *Y* and the adapted coordinates *x*^{λ}, *y*^{i}, *y*^{i}_{Λ}, (Λ = (*λ*_{1}, …,*λ*_{k}), |Λ| = *k* ≤ *r*) on jet manifolds *J*^{r}*Y*, a Lagrangian *L* and its Euler–Lagrange operator read

where

denote the total derivatives.

For instance, a first-order Lagrangian and its second-order Euler–Lagrange operator take the form

### Euler–Lagrange equations[edit]

The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations *δL* = 0.

## Cohomology and Noether’s theorems[edit]

Cohomology of the variational bicomplex leads to the so-called

variational formula

where

is the total differential and θ_{L} is a Lepage equivalent of *L*. Noether’s first theorem and Noether’s second theorem are corollaries of this variational formula.

## Graded manifolds[edit]

Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian systems of even and odd variables.^{[1]}

## Alternative formulations[edit]

In a different way, Lagrangians, Euler–Lagrange operators and Euler–Lagrange equations are introduced in the framework of the calculus of variations.

## Classical mechanics[edit]

In classical mechanics equations of motion are first and second order differential equations on a manifold *M* or various fiber bundles *Q* over ℝ. A solution of the equations of motion is called a *motion*.^{[2]}^{[3]}

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## See also[edit]

## References[edit]

- Arnold, V. I. (1989),
*Mathematical Methods of Classical Mechanics*, Graduate Texts in Mathematics, vol. 60 (second ed.), Springer-Verlag, ISBN 0-387-96890-3 - Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (1997).
*New Lagrangian and Hamiltonian Methods in Field Theory*. World Scientific. ISBN 981-02-1587-8. - Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (2011).
*Geometric formulation of classical and quantum mechanics*. World Scientific. doi:10.1142/7816. hdl:11581/203967. ISBN 978-981-4313-72-8. - Olver, P. (1993).
*Applications of Lie Groups to Differential Equations*(2 ed.). Springer-Verlag. ISBN 0-387-94007-3. - Sardanashvily, G. (2013). “Graded Lagrangian formalism”.
*Int. J. Geom. Methods Mod. Phys*. World Scientific.**10**(5): 1350016. arXiv:1206.2508. doi:10.1142/S0219887813500163. ISSN 0219-8878.

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