Lagrangian system

Template:More footnotes Template:CS1 config In mathematics, a Lagrangian system is a pair Template:Math, consisting of a smooth fiber bundle Template:Math and a Lagrangian density Template:Math, which yields the Euler–Lagrange differential operator acting on sections of Template:Math.

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

Lagrangians and Euler–Lagrange operators

A Lagrangian density Template:Math (or, simply, a Lagrangian) of order Template:Math is defined as an [[exterior form|Template:Math-form]], Template:Math, on the Template:Math-order jet manifold Template:Math of Template:Math.

A Lagrangian Template:Math can be introduced as an element of the variational bicomplex of the differential graded algebra Template:Math of exterior forms on jet manifolds of Template:Math. The coboundary operator of this bicomplex contains the variational operator Template:Math which, acting on Template:Math, defines the associated Euler–Lagrange operator Template:Math.

In coordinates

Given bundle coordinates Template:Math on a fiber bundle Template:Math and the adapted coordinates Template:Math, Template:Math, Template:Math) on jet manifolds Template:Math, a Lagrangian Template:Math and its Euler–Lagrange operator read

$L=ℒ(x^{\lambda },y^i,y_{\mathrm{\Lambda }}^i)d^{n}x,$

$\delta L={\delta }_iℒd{y}^i\wedge d^{n}x,{\delta }_iℒ={\partial }_iℒ+\sum _{|\mathrm{\Lambda }|}(-1{)}^{|\mathrm{\Lambda }|}{d}_{\mathrm{\Lambda }}\underset{i}{\overset{\mathrm{\Lambda }}{\partial }}ℒ,$

where

$d_{\mathrm{\Lambda }}=d_{{\lambda }_1}\cdots d_{{\lambda }_k},d_{\lambda }={\partial }_{\lambda }+y_{\lambda }^i{\partial }_i+\cdots ,$

denote the total derivatives.

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

$L=ℒ(x^{\lambda },y^i,y_{\lambda }^i)d^{n}x,{\delta }_{i}L={\partial }_iℒ-d_{\lambda }\underset{i}{\overset{\lambda }{\partial }}ℒ.$

Euler–Lagrange equations

The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations Template:Math.

Cohomology and Noether's theorems

Cohomology of the variational bicomplex leads to the so-called variational formula

$dL=\delta L+d_H{\mathrm{\Theta }}_L,$

where

$d_H{\mathrm{\Theta }}_L=d{x}^{\lambda }\wedge d_{\lambda }\varphi ,\varphi \in O_{\mathrm{\infty }}^{*}(Y)$

is the total differential and Template:Math is a Lepage equivalent of Template:Math. Noether's first theorem and Noether's second theorem are corollaries of this variational formula.

Graded manifolds

Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian systems of even and odd variables.

Alternative formulations

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

Classical mechanics

In classical mechanics equations of motion are first and second order differential equations on a manifold Template:Math or various fiber bundles Template:Math over $ℝ$. A solution of the equations of motion is called a motion. Template:Stub section

See also

• Lagrangian mechanics
• Calculus of variations
• Noether's theorem
• Noether identities
• Jet bundle
• Jet (mathematics)
• Variational bicomplex

References

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External links

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