Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

What are the difference between:

  1. the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the principle of least action, and

  2. the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in Hamilton's variational principle?

References:

  1. H. Goldstein, Classical Mechanics, Section 8.6.

  2. H. Goldstein, Classical Mechanics, Chapter 2.

share|improve this question
2  
I've never seen anything defined as $\int L+H\,dt$, could you cite a source that uses this form? –  Kyle Kanos Jul 20 at 2:02

2 Answers 2

The more common names for what you are talking about are the abbreviated action

$$S_0[q] := \int p \mathrm{d}q$$

versus the action

$$ S[q] := \int_{t_1}^{t_2}L(q,\dot q,t)\mathrm{d}t$$

Both are used in different formulations of classical mechanics, and deliver a different "flavor" of solutions. On both one can do variations calculus and obtains the classical trajectory $q_{cl}$ as the extremum of the action.

The abbreviated action obeys Maupertuis' principle and gives you the classical path provided you know the conserved energy along the trajectory and the start- and endpoints.

The action's extrema are found by the usual Euler-Lagrange equations and gives you the classical path provided you know the start- and endtimes as well as points.

Now, what has the abbreviated action to do with $L + H$? Observe that

$$\int p \mathrm{d}q =\int p \dot q \mathrm{d}t = \int (L + H) \mathrm{d}t$$

since $L$ and $H$ are Legrendre transforms w.r.t. $\dot q$ resp. $p$ of each other.

share|improve this answer

As user ACuriousMind correctly writes:

  1. What Goldstein calls the principle of least action $\int p~\mathrm{d}q$ is usually called Maupertuis' principle or the principle of abbreviated action.

  2. What Goldstein calls the Hamilton's variational principle is often also called the the principle of least/extremal/stationary action $\int L~\mathrm{d}t$.

This is also explained in a footnote in Goldstein, Section 8.6. At the physical level, besides the different appearances, the important thing is to realize that different quantities are kept fixed in the two variational principles 1 and 2.

Also note that confusingly the Hamilton's principle is a variational principle in the Lagrangian formulation (as opposed to the Hamiltonian formulation).

Finally, let us remark that the word action has several different meanings in physics and mathematics.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.