5
votes
Accepted
Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
The Lagrangian is a function, not a functional. The action is a functional, and is defined as
$$S[q; t_0,t_1] = \int_{t_0}^{t_1} L\big(q(t), \dot q(t), t\big) \mathrm dt$$
The partial derivatives ...
- 66k
4
votes
Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
All the problem, in my view, arises form the fact that the EL equations are introduced in a too sloppy way. (The variational approach makes even more obscure an obscure setup.)
Actually,
The ...
- 67.8k
2
votes
Accepted
Least action principle and uniform motion
Least action (or stationary action) means least along all those paths which begin and end at the given points, and that means the given times as well as locations. If a path sets out with a smaller ...
- 55.7k
1
vote
Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
On one hand, the Lagrangian $L(q,v,t)$ is a function of the position$^1$ $q$, the velocity $v$ and the time $t$.
On the other hand, the Lagrangian action
$$S[q] ~:=~ \int_{t_i}^{t_f}\mathrm{d}t \ L(q(...
- 193k
1
vote
Accepted
Does the linear combination of basis functions, need to use eigenfunctions as basis?
Continuing from our chat in $\hslash$, I have to concur with the other participants that you are conflating very many things together, and that that is the real source of all your confusions regarding ...
- 6,992
1
vote
What is the most general transformation between Lagrangians which give the same equation of motion?
Rather than answering the question point-by-point, I am going to cut to the essence.
Suppose that we are given $n$ independent variables $x=(x^i)=(x^1,\dots,x^n)$, $m$ "dependent variables" $...
- 10.5k
1
vote
Accepted
Variational Principle for Free Particle Motion (Relativistic)
I hope the notation of writing $L = \sqrt{\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}$ is clear as this makes the following calculations very compact.
\begin{equation}
\frac{\partial L}{\partial \dot{x}^{...
- 126
1
vote
Least action principle and uniform motion
The change in velocity would require an extra force which would change the Lagrangian. Even if the sudden force was not so sudden the Lagrangian would still change due to this interaction.
1
vote
Least action principle and uniform motion
Well, the problem is that the new path violates the boundary conditions (BCs) of the stationary action principle, usually Dirichlet BCs.
- 193k
1
vote
Why is it possible to neglect higher order terms in the variation of the action?
We want to fund the function $y(x)$ that will minimize
$$
I=\int_{a}^{b}dx\,F(x,y(x),y^{\prime }(x))\, , \tag{1}
$$
where the integrand
depends $x$, $y(x)$ and also on the derivative $y^{\prime }(x)$....
- 44.6k
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