In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundleY → 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 operatorsedit
A Lagrangian densityL (or, simply, a Lagrangian) of order r is defined as an n-form, n = dim X, on the r-order jet manifoldJrY of Y.
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
In a different way, Lagrangians, Euler–Lagrange operators and Euler–Lagrange equations are introduced in the framework of the calculus of variations.
Classical mechanicsedit
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]
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. 10 (5). World Scientific: 1350016. arXiv:1206.2508. doi:10.1142/S0219887813500163. ISSN 0219-8878.
External linksedit
Sardanashvily, G. (2009). "Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians". arXiv:0908.1886. Bibcode:2009arXiv0908.1886S. {{cite journal}}: Cite journal requires |journal= (help)