Lagrangian system – Wikipedia

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^rY 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, yⁱ on a fiber bundle Y and the adapted coordinates x^λ, yⁱ, yⁱ_Λ, (Λ = (λ₁, …,λ_k), |Λ| = k ≤ r) on jet manifolds J^rY, a Lagrangian L and its Euler–Lagrange operator read

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

${displaystyle delta L=delta _{i}{mathcal {L}},dy^{i}wedge d^{n}x,qquad delta _{i}{mathcal {L}}=partial _{i}{mathcal {L}}+sum _{|Lambda |}(-1)^{|Lambda |},d_{Lambda },partial _{i}^{Lambda }{mathcal {L}},}$

where

${displaystyle d_{Lambda }=d_{lambda _{1}}cdots d_{lambda _{k}},qquad d_{lambda }=partial _{lambda }+y_{lambda }^{i}partial _{i}+cdots ,}$

denote the total derivatives.

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

${displaystyle L={mathcal {L}}(x^{lambda },y^{i},y_{lambda }^{i}),d^{n}x,qquad delta _{i}L=partial _{i}{mathcal {L}}-d_{lambda }partial _{i}^{lambda }{mathcal {L}}.}$

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

${displaystyle dL=delta L+d_{H}Theta _{L},}$

where

${displaystyle d_{H}Theta _{L}=dx^{lambda }wedge d_{lambda }phi ,qquad phi in O_{infty }^{*}(Y)}$

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.

External links[edit]