- #1
Rasalhague
- 1,387
- 2
Wikipedia: Euler Lagrange Equation defines a function
[tex]L:[a,b] \times X \times TX \rightarrow \mathbb{R} \enspace\enspace\enspace (1)[/tex]
such that
[tex](t,q(t),q'(t)) \mapsto L(t,q(t),q'(t)) \enspace\enspace\enspace (2.)[/tex]
But (2) suggests that the domain of L is simply [a,b], thus:
[tex]L:[a,b] \rightarrow \mathbb{R},[/tex]
with [itex]L = G \circ F[/itex], and
[tex]F:[a,b] \rightarrow [a,b] \times X \times TX \; \bigg| \; F(t) = (t,q(t),q'(t)),[/tex]
[tex]G:[a,b] \times X \times TX \rightarrow \mathbb{R}.[/tex]
Is that what they mean by L?
[tex]L:[a,b] \times X \times TX \rightarrow \mathbb{R} \enspace\enspace\enspace (1)[/tex]
such that
[tex](t,q(t),q'(t)) \mapsto L(t,q(t),q'(t)) \enspace\enspace\enspace (2.)[/tex]
But (2) suggests that the domain of L is simply [a,b], thus:
[tex]L:[a,b] \rightarrow \mathbb{R},[/tex]
with [itex]L = G \circ F[/itex], and
[tex]F:[a,b] \rightarrow [a,b] \times X \times TX \; \bigg| \; F(t) = (t,q(t),q'(t)),[/tex]
[tex]G:[a,b] \times X \times TX \rightarrow \mathbb{R}.[/tex]
Is that what they mean by L?