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 $yleft(tright)$), from Thorton-Marion 5th Edition on Classical Dynamics. I will replace its functional $fleft(yleft(xright), frac{d}{dx}yleft(xright); xright)$ to this Lagrangian $L$ notation: $$Lleft(yleft(tright), frac{d}{dt}yleft(tright); tright).$$

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

The second form of Euler-(Lagrange) equation is solved from the Lagrangian $Lleft(yleft(tright), frac{d}{dt}yleft(tright); tright)$ in Chap 6.4, but with explicit $t$ dependence does play a role. Thorton-Marion obtains (6.39):
$$
boxed{frac{partial L}{partial t}-frac{d}{dt}left(L-dot{y}frac{partial L}{partialdot{y}}right)=0}
$$
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.

Qmechanic · Answer

The second form of the Euler-Lagrange equation can be rewritten as
$$ frac{dh}{dt}~=~-frac{partial L}{partial t},tag{EL2}$$
where
$$ h(q,dot{q},t)~:=~left(sum_{j=1}^ndot{q}^jfrac{partial }{partial dot{q}^j}-1 right)L(q,dot{q},t) tag{h}$$
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.

mike stone · Answer

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.

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

Question

2 Answers

Add your own answers!

Ask a Question