Linked Questions
17 questions linked to/from Hamilton-Jacobi Equation
42
votes
6
answers
20k
views
What is the physical meaning of the action in Lagrangian mechanics?
The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian.
I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical ...
- 573
9
votes
2
answers
3k
views
Variation of Action with time coordinate variations
I was trying to derive equation (65) in the review by László B. Szabados in Living Reviews in Relativity (2002, Article 4)
This slightly unusual then usual classical mechanics because it includes a ...
- 93
9
votes
1
answer
10k
views
Hamilton's characteristic and principal functions and separability
Just hoping for some clarity regarding Hamilton's characteristic function $W$. When we take a time independent Hamiltonian we can separate the Principal function $S$ up into the characteristic ...
- 2,551
7
votes
1
answer
2k
views
What variables does the action $S$ depend on?
Action is defined as,
$$S ~=~ \int L(q, q', t) dt,$$
but my question is what variables does $S$ depend on?
Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?
In Wikipedia I've ...
- 173
7
votes
1
answer
1k
views
Intuition for Hamilton-Jacobi equation derived from least action
I am trying to understand the Hamilton-Jacobi equation without the framework of the canonical transformations. Even on the case of a 1D free particle I'm getting stuck.
The system starts at fixed ...
- 876
3
votes
1
answer
692
views
Hamilton-Jacobi formalism and on-shell actions
My question is essentially how to extract the canonical momentum out of an on-shell action.
The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
- 659
2
votes
1
answer
318
views
Geometry of Hamilton-Jacobi Equation
I'm trying to understand the geometry of the Hamilton-Jacobi equation (working from Gelfand + Fomin), but I'm stuck. I know that:
If we define the function $S(t,y;t_0, y_0)$ as:
$$S(t,y;t_0,y_0) = \...
- 1,423
2
votes
1
answer
874
views
Is this action a Hamilton's principal function?
In Ref. 1, in section 3, they wrote:
\begin{equation}
L'(q,\dot{q},\ddot{q},t)~=~L(q,\dot{q},\ddot{q},t)-\frac{dS(q,\dot{q},t)}{dt}.\tag{14}
\end{equation}
Then the Hamilton-Jacobi equation is
\begin{...
- 446
3
votes
1
answer
838
views
A problem in deriving the Hamilton-Jacobi equation from a variational principle
As I've already said, I have a problem in understanding a reasoning from which we derive the Hamilton–Jacobi equation from a variational principle.
Let's take the Hamilton functional:
$$ S = \int_{...
- 105
1
vote
1
answer
699
views
Hamilton-Jacobi theory and initial value problem?
Having read through some recent posts regarding the Lagrangian formulation being interpreted into an initial value problem rather than the familiar boundary condition problem we are familiar with, I ...
- 2,551
1
vote
0
answers
705
views
Why I find two possible $S$ of a free particle by solving the Hamilton-Jacobi Equation?
For a free particle, the Hamiltonian is
$$H(p)=\frac{p^2}{2m}.$$
The corresponding H-J equation thus can be written as
$$\frac{1}{2m} \left(\frac{\partial S(q,t)}{\partial q}\right)^2=- \frac{\...
- 59
2
votes
1
answer
304
views
Conceptual problem with action considered as function of endpoints
I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
- 2,256
1
vote
1
answer
541
views
Hamilton-Jacobi equation and Action Functional
Let the action functional $S[q]$ given by
\begin{equation}\label{eq16}
S[q]=\int\limits^{t_2}_{t_1}L\left(q^i(t),\dot{q}^i(t)\right)dt.\tag{1}
\end{equation}
Also, we know that using Legendre ...
- 403
2
votes
1
answer
268
views
Partial time derivative of the on-shell action
I have a few questions about differentiating the on-shell action.
Here is what I currently understand (or think I do!):
Given that a system with Lagrangian $\mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, ...
- 2,168
0
votes
1
answer
301
views
Hamilton-Jacobi Equation and uniform circular motion
I tried to apply Hamilton-Jacobi equation to uniform circular motion and it doesn't come out right.
Lets say a particle of mass $m$ undergoes uniform circular motion around another particle of mass $...
- 761