410 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
Definition 5.6. The dual jet bundle J^1 (Y)* is the vector bundle over Y whose
fiber at y E Y x is the set of affine maps from J1(Y)y to An+^1 (X)x, the bundle of
( n + 1 )-forms on X. A smooth section of J^1 (Y) * is therefore an affine bundle map
of J^1 (Y) to A n+l(X) covering n xy.
Fiber coordinates on J1(Y)* are (P,PAμ ), which correspond to the affine map
given in coordinates by
(5.37)where dn+lx = dx^1 /\ · · · /\ dxn /\ dx^0.
Analogous to the canonical one-and two-forms on a cotangent bundle, there arecanonical ( n + 1)-and ( n + 2)-forms on the dual jet bundle J1 (Y)*. In coordinates,
these forms are given by
(5.38)
and
f2 = dyA /\ dpAμ /\ dnXμ - dp /\ dn+lx. (5.39)A Lagrangian density .C: J1(Y)--+ An+^1 (X) is a smooth bundle map over
X. In coordinates, we write
(5.40)
The corresponding covariant Legendre transformation for .C is a fiber preserving
map over Y, IF .C : J^1 (Y) --+ J^1 (Y) *, expressed intrinsically as the first order vertical
Taylor approximation to .C:
IF.C(I) · 1' = .C(I) + d~ le:=O .C(I + i:;(!' - 1)) (5.41)
where 1, 1' E J1 (Y)y. A straightforward calculation shows that the covariant
Legendre transformation is given in coordinates by
PA μ -- ~' 8L and p -- L - ~v 8L A w (5.42)
uv μ uv μThe Cartan forms are the (n + 1)-form 8.c on J1(Y) given by
8.c = (IF.C)*8 , (5.43)and the (n + 2)-form D.c by
D.c = - d8.c = (IF.C)*D, (5.44)
with local coordinate expressions
(^8) .c - avAμ 8L dy A /\ d n Xμ + ( L - ovAμ 8L v A μ ) d n+l x,
(5.45)
,..., H.C - dy A /\ d ( ovAμ 8L ) /\ d n Xμ - d L [ - avAμ 8L v A μ ] /\ d n+l x.
This formalism can be used to express, in an intrinsic way, the Euler- Lagrange
equations, which in coordinates take the standard form
8L (.^1 ) 8 ( 8L. 1 )
f)yA J (¢) - OXμ OVAμ (J (¢)) = Q (5.46)