Lagrangian system

Lagrangians and Euler–Lagrange operators

A Lagrangian density L (or, simply, a Lagrangian) of order r is defined as an n-form, , on the r-order jet manifold JrY 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

Given bundle coordinates x**λ, y**i on a fiber bundle Y and the adapted coordinates x**λ, y**i, y**iΛ, , ) on jet manifolds JrY, a Lagrangian L and its Euler–Lagrange operator read

: L=\mathcal{L}(x^\lambda,y^i,y^i_\Lambda) , d^nx,

: \delta L= \delta_i\mathcal{L} , dy^i\wedge d^nx,\qquad \delta_i\mathcal{L} =\partial_i\mathcal{L} + \sum_{|\Lambda|}(-1)^{|\Lambda|} , d_\Lambda , \partial_i^\Lambda\mathcal{L},

where

: d_\Lambda=d_{\lambda_1}\cdots d_{\lambda_k}, \qquad d_\lambda=\partial_\lambda + y^i_\lambda\partial_i +\cdots,

denote the total derivatives.

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

: L=\mathcal{L}(x^\lambda,y^i,y^i_\lambda) , d^nx,\qquad \delta_i L =\partial_i\mathcal{L} - d_\lambda \partial_i^\lambda\mathcal{L}.

Euler–Lagrange equations

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

Cohomology and Noether's theorems

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

: dL=\delta L + d_H \Theta_L,

where

: d_H\Theta_L=dx^\lambda\wedge d_\lambda\phi, \qquad \phi\in O^*_\infty(Y)

is the total differential and θL is a Lepage equivalent of L. 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 M or various fiber bundles Q over \mathbb{R}. A solution of the equations of motion is called a motion.

References

References

1. {{harvnb. Sardanashvily. 2013
2. {{harvnb. Arnold. 1989
3. {{harvnb. Giachetta. Mangiarotti. Sardanashvily. 2011