The first and second form of Euler-(Lagrange) equation with explicit time dependence

Question

I have learned the first and second form of Euler-(Lagrange) equation with no explicit time dependence (the time dependence only implicit on the function to be solved, say $y\left(t\right)$), from Thorton-Marion 5th Edition on Classical Dynamics. I will replace its functional $f\left(y\left(x\right), \frac{d}{dx}y\left(x\right); x\right)$ to this Lagrangian $L$ notation: $$L\left(y\left(t\right), \frac{d}{dt}y\left(t\right); t\right).$$

1. The first form of Euler-(Lagrange) equation is solved from the Lagrangian $$L\left(y\left(t\right),\dot{y}\left(t\right);t\right)$$ in Chap 6.3, but with $t$ dependence seems to play no role. Thorton-Marion obtains (6.18): $$ \boxed{\frac{\partial L}{\partial y}-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{y}}\right)=0} $$

2. The second form of Euler-(Lagrange) equation is solved from the Lagrangian $L\left(y\left(t\right), \frac{d}{dt}y\left(t\right); t\right)$ in Chap 6.4, but with explicit $t$ dependence does play a role. Thorton-Marion obtains (6.39): $$ \boxed{\frac{\partial L}{\partial t}-\frac{d}{dt}\left(L-\dot{y}\frac{\partial L}{\partial\dot{y}}\right)=0} $$ But this explicit $t$ dependence form does not give rise to the correct answer for Euler-(Lagrange) equation with explicit time dependence, because Thorton-Marion used (6.18) already to derive this (6.39). So some modification is necessarily required!!!

Question

• What are the first and second form of Euler-(Lagrange) equation with Lagrangian of explicit time dependence?

• How to modify and correct the derivations in Thorton-Marion (6.18) and (6.39) to get an Euler-(Lagrange) equation with Lagrangian of explicit time dependent system?

p.s. There is a related post on Lagrange equation with explicit time dependence: How to deal with explicit time dependence of the Lagrangian? But they do not work on the analogous first and second form of Euler-(Lagrange) equation. I hope you can provide some insights or explicit final form of equations.

Answer 1

The second form of the Euler-Lagrange equation can be rewritten as

$$ \frac{dh}{dt}~=~-\frac{\partial L}{\partial t},\tag{EL2}$$

where

$$ h(q,\dot{q},t)~:=~\left(\sum_{j=1}^n\dot{q}^j\frac{\partial }{\partial \dot{q}^j}-1 \right)L(q,\dot{q},t) \tag{h}$$

is the (Lagrangian) energy function. EL2 follows directly from the first form of the Euler-Lagrange equations $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}^j}-\frac{\partial L}{\partial q^j}~=~0, \qquad j~\in \{1,\ldots, n\},\tag{EL1} $$ for an arbitrary first-order Lagrangian $L(q,\dot{q},t)$ with possible explicit time-dependence.

Answer 2

What horrible notation! For an $L$ without expicit time dependence ($L=L(y,\dot y))$, the expression
$$ F= L-\dot y \frac{\partial L}{\partial \dot y} $$ obeys $$ \frac{d}{dt}F=0 $$ This first integral is a consequence of the E-L equation $$ \frac{\partial L}{\partial y}- \frac{d}{dt}\left(\frac{\partial L}{\partial \dot y}\right)=0. $$ It is not not a "second form of te E-L equation". In simple exmaples it is the energy, and it says that energy is conserved for time indepenedent sytems. For one-dimensional systems energy conservation is a useful way of solving the motion.

When there is more than one $y$ there is still a first integral $$ F= L-\sum_i \dot y_1 \frac{\partial L}{\partial \dot y_i}, $$ but there is only one, and one energy consrevation is not enough to solve the motion.

Similarly if $L(y,\dot y,t)$ then $$ \frac {d}{dt}F= \frac{\partial L}{\partial t} $$ is a consequence of the usual E-L equation. The E-L equation does not change if there is explicit time dependence.

I suggest you read a better book.