By William Elwood Byerly
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Additional info for An introduction to generalized coordinates in mechanics and physics
In a general setting, one also considers other Lepage equivalents of L [91, 92]. 12) takes its values into a subbundle n−1 J 1 Y ×(T ∗ Y ∧( ∧ T ∗ X )) Y Y n of ∧ T ∗ J 1 Y . 14) is an imbedded subbundle i L : Z L → Z Y of the fibre bundle Z Y → Y . This morphism is called the homogeneous Legendre map. 13) is said to be the homogeneous Legendre bundle. 15) ∂y j ∂ y i μ p . 13) reads ( piμ , p) ◦ HL = (πiμ , L − yμi πiμ ). 8) modelled over the pull-back vector bundle n Π × ∧ T ∗ X → Π. 5) is exactly the composition of morphisms L = π Z Π ◦ HL : J 1 Y → Π.
64) restricted to N L . 79) r ∗ (u d H | N L ) = 0, where r is a section of N L → X and u is an arbitrary vertical vector field on Π → X . 78). 23). Proof It is readily observed that L = π Z Π ◦ HL . 16) yields a surjection of Z L onto N L . 75), we have a morphism HL ◦ Ψ : N L → Z L . 7), this is a surjection such that π Z Π ◦ HL ◦ Ψ = Id N L . Hence, HL ◦Ψ is a bundle isomorphism over Y which is independent of the choice of a global section Ψ . 76) results in HN = ( HL ◦ Ψ )∗ Ξ L that leads to a desired equivalence.
35) termed the energy-momentum tensor relative to a connection Γ [42, 100, 119]. 34): 0 ≈ −dλ [πiλ τ μ (yμi − Γμi ) − δμλ τ μ L ]. 34 3 Lagrangian and Hamiltonian Field Theories For instance, let a fibre bundle Y → X admit a flat connection Γ . 35) takes a form J0 λ μ = πiλ yμi − δμλ L of the familiar canonical energy–momentum tensor. 41) with respect to a reference frame Γ (Sect. 3). 4. It is parameterized by vector fields τ as sections of the vector bundle T X → X . 7). 6) depends on derivatives of gauge parameters τ , and can become a gauge symmetry.
An introduction to generalized coordinates in mechanics and physics by William Elwood Byerly