Linked Questions

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0answers
29 views

Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]

The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$. Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
33
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6answers
12k 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 ...
7
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1answer
5k 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 ...
5
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2answers
2k views

Variation of Action with time coordinate variations

I was trying to derive equation (65) in the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu23.html This slightly unusual then usual classical mechanics ...
4
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1answer
1k 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 ...
3
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1answer
393 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 ...
1
vote
1answer
478 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 ...
3
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1answer
464 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_{...
6
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1answer
328 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 ...
2
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1answer
189 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) = \...
2
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1answer
278 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{...
2
votes
1answer
141 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 $t_{initial}, \ t_{final}$. ...
0
votes
1answer
174 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 $...
2
votes
2answers
60 views

Is there a $n$-dimensional system such that the minimal action from a path from $x$ to $y$ is the scalar product?

Suppose we work (with a particle) in $\mathbb{R}^n$. Is there a Euler-Lagrange equation associated to the particle in question such that the minimal action of all path going from a position $x\in \...