Ex 1.33 Properties of E

Understanding the Euler-Lagrange Operator

Let $\displaystyle{F}$ and $\displaystyle{G}$ be two Lagrangian-like functions of a local tuple, $\displaystyle{C}$ be a local-tuple transformation function, and $\displaystyle{c}$ a constant.

Demonstrate the following properties:

a. $\displaystyle{E[F + G] = E[F] + E[G]}$

…

d. $\displaystyle{\mathcal{E}[F \circ C] = D_t (DF \circ C) \partial_2 C + DF \circ C \mathcal{E}[C]}$

~~~

Eq. (1.167):

$\displaystyle{\bar \Gamma (\bar f) (t, q, v, \dots) = \bar f [\mathcal{O} (t,q,v, \dots)](t)}$

Eq. (1.174):

$\displaystyle{E[L] = D_t \partial_2 L - \partial_1 L}$

$\displaystyle{ \begin{aligned} E[L] &= D_t \partial_2 L - \partial_1 L \\ E[F+G] &= D_t \left( \partial_2 (F+G) \right) - \partial_1 (F+G) \\ &= D_t \left( \partial_2 F+\partial_2 G \right) - (\partial_1 F+ \partial_1 G) \\ &= D_t \partial_2 F+D_t \partial_2 G - (\partial_1 F+ \partial_1 G) \\ &= D_t \partial_2 F- \partial_1 F +D_t \partial_2 G - \partial_1 G \\ &= E[F] + E[G] \\ \end{aligned} }$

— Me@2025-05-30 03:40:41 PM

The Problem

$\displaystyle{ \begin{aligned} &E[F \circ C] (t,q,v,\dots) \\ &= (D_t \partial_2 (F \circ C) - \partial_1 (F \circ C)) (t,q,v,\dots)\\ &= (D_t \partial_2 (F (C(t,q,v,\dots))) - \partial_1 (F(C(t,q,v,\dots)))) \\ \end{aligned} }$

Prove that

$\displaystyle{\mathcal{E}[F \circ C] = D_t (DF \circ C) \partial_2 C + DF \circ C \mathcal{E}[C]}$

Key Terms Explained

Local Tuple: Think of this as a snapshot of a system’s state along a path. It includes:
- $\displaystyle{ t }$ : time,
- $\displaystyle{ q }$ : generalized coordinate (e.g., position),
- $\displaystyle{ v = \frac{dq}{dt} }$ : velocity,
- and possibly higher derivatives like acceleration. We’ll use $\displaystyle{ \eta = (t, q, v) }$ for simplicity.
Lagrangian-like Function $\displaystyle{ F }$ : A scalar function of the local tuple, such as $\displaystyle{ F(t, q, v) }$ , akin to a Lagrangian in mechanics.
Local-Tuple Transformation $\displaystyle{ C }$ : A function that maps one local tuple to another. For example, $\displaystyle{ C(\eta) = (t, C_q(t, q, v), C_v(t, q, v)) }$ , where $\displaystyle{ C_q }$ and $\displaystyle{ C_v }$ transform the coordinate and velocity.
Composition $\displaystyle{ F \circ C }$ : This is $\displaystyle{ F }$ evaluated at the transformed tuple: $\displaystyle{ (F \circ C)(\eta) = F(t, C_q(t, q, v), C_v(t, q, v)) }$ .
Euler-Lagrange Operator $\displaystyle{ E }$ : For a function $\displaystyle{ G(t, q, v) }$ , it’s defined as:
$\displaystyle{ E[G] = \frac{\partial G}{\partial q} - D_t \left( \frac{\partial G}{\partial v} \right) }$
This operator extracts the equations of motion when applied to a Lagrangian.
Total Time Derivative $\displaystyle{ D_t }$ : This accounts for how a function changes over time, considering all variables. For $\displaystyle{ h(t, q, v) }$ :
$\displaystyle{ D_t h = \frac{\partial h}{\partial t} + v \frac{\partial h}{\partial q} + a \frac{\partial h}{\partial v} }$
where $\displaystyle{ a = \frac{dv}{dt} }$ is acceleration.
Derivative $\displaystyle{ DF }$ : The derivative of $\displaystyle{ F }$ with respect to its spatial arguments, typically $\displaystyle{ DF = \left( \frac{\partial F}{\partial q}, \frac{\partial F}{\partial v} \right) }$ .