Action and Action integral: Different kinds of variational principles

Question

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.

Answer 1

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.

Answer 2

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.