Lagrangian system
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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[]
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 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 xλ, yi on a fiber bundle Y and the adapted coordinates xλ, yi, yiΛ, (Λ = (λ1, ...,λk), |Λ| = k ≤ r) on jet manifolds JrY, 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[]
The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations δL = 0.
Cohomology and Noether's theorems[]
Cohomology of the variational bicomplex leads to the so-called variational formula
where
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.[1]
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 ℝ. A solution of the equations of motion is called a motion.[2][3]
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See also[]
- Lagrangian mechanics
- Calculus of variations
- Noether's theorem
- Noether identities
- Jet bundle
- Jet (mathematics)
- Variational bicomplex
References[]
- Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 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. 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[]
- Sardanashvily, G. (2009). "Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians". arXiv:0908.1886. Bibcode:2009arXiv0908.1886S. Cite journal requires
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- Differential operators
- Calculus of variations
- Dynamical systems
- Lagrangian mechanics