Action given by principle of least action ($S$) and action variable given by action angle variable theory ($J$) are same?


1 Answer 1


In the ways these two notions are used, they are completely different things. However, I think their names share the common word action because conceptually they share a common underlying concept. But they are still different.

The common framework of both action and action-angle variables is the Hamiltonian formalism.

Actions. Good for deriving equations of motion. Let us have a Hamiltonain system with a Hamiltonian function $H= H(q, p)$, where $q = (q^1, ... ,q^n) \in \mathbb{R}^n$ and $p = (p_1, ... ,p_n) \in \mathbb{R}^n$. Then the dynamics of the system is determined by the solutions to the system of $2n$ differential equations \begin{align} &\frac{dq^k}{dt} \, = \, \frac{\partial H}{\partial p_k}\big( q, p\big)\\ &\frac{dp^k}{dt} \, = \, -\, \frac{\partial H}{\partial q^k}\big( q, p\big)\\ & \text{ for } k = 1, ..., n \end{align} Now, the action $S[\gamma]$ associated with the Hamiltonain $H(q, p)$ is a function, or rather a functional, whose

1. Input variables are functions $\gamma(t) = \big(q^1(t), ..., q^n(t), p_1(t), ...., p_n(t) \big)$ describing curves in $\mathbb{R}^{2n}$ with the property that $q(t_1) = q_1$ and $q(t_2) = q_2 \in \mathbb{R}^n$ are fixed ($t_1$ and $t_2$ are fixed and $q_1$ and $q_2$ are fixed);

2. The output is a number. The rule according to which the output is calculated is $$S_H[\gamma] \, = \, \int_{ \gamma } \, \left( \sum_{k=1}^{n}\, p_k \, dq^k \, -\, H(p, q)\, dt \right) = \int_{t_1}^{t_2} \left( \sum_{k=1}^{n}\, p_k(t) \, \frac{d q^k}{dt}(t) \, -\, H\big(p(t), q(t)\big)\, \right) dt$$

At the same time, the solutions of the system above coincide with the stationary curves (i.e. the critical curves) of the action $S_H[\gamma]$, a fact which is called the principle of stationary action $$\delta S_H[\gamma] = 0$$ It is well known that the solutions of the Hamilton's equations, which we said are the same as the stationary curves of the action, conserve the Hamiltonain function $H$, i.e. $$H\big(\gamma(t)\big) = H\big(q(t), p(t)\big) \equiv \text{constant}$$ An equivalent way of arriving at the principle of stationary action is by calculating the action $$S[\gamma] \, = \, \int_{ \gamma } \, \sum_{k=1}^{n}\, p_k \, dq^k = \int_{t_1}^{t_2} \left( \sum_{k=1}^{n}\, p_k(t) \, \frac{d q^k}{dt}(t) \right) dt$$ for all curves $\gamma(t) = \big(q^1(t), ..., q^n(t), p_1(t), ...., p_n(t) \big)$ in $\mathbb{R}^{2n}$ with the properties that

- $H\big(\gamma(t)\big) = H\big(q(t), p(t)\big) = c$ and

- $q(t_1) = q_1$ and $q(t_2) = q_2 \in \mathbb{R}^n$ are fixed;

Then, the Hamilton's equations are equivalent to the restricted principle of stationary action $$\delta S[\gamma] =0 \text{ under the restriction that } H(\gamma) = \text{const}$$

Action-angle variables. Good for simplifying Hamiltonian systems. Now in some cases, the Hamiltonain system of differential equations with function $H(q, p) = H_1(q, p)$ in $\mathbb{R}^{2n}$ possesses a full set of conservation laws, in the form of $n$ Hamiltonian functions $$H_1(q, p), \, H_2(q, p), \, ..., \, H_n(q, p)$$ with the properties that $$\text{the differentials } dH_1(q, p), ..., dH_n(q, p) \text{ are linearly independent for all points }(q,p)$$
$$\{H_i, H_j\} = 0 \text{ for all } i,j = 1, ..., n$$ The latter property includes the fact that the solutions $\gamma(t)$ of the Hamiltonian equations of $H = H_1$ conserve all the functions, i.e. $H_j\big(\gamma(r)\big) = c_j$ for all $j=1, ..., n$. A system with these properties is called integrable.

Integrability allows us to find a set of new variables, such that when we change the variables $(q,p)$ to the new ones, the original Hamilton's equations simplify drastically. These variables are called the action-angle variables.

Action variables. First, for a set of $n$ constants $c = \big(c_1, ..., c_n \big) \in \mathbb{R}^n$ define the set $$M_c = \big\{ (q, p) \in \mathbb{R}^{2n} \, | \, H_1(q, p) = c_1, ..., H_n(q, p) = c_n\big\}$$ Then, $M_c$ is a smooth embedded Lagrangian submanifold of $\mathbb{R}^n $ and if it is compact, it is an $n-$dimensional torus. On this torus, one can choose a set of $n$ simple closed loops $\sigma_1, ..., \sigma_n$, generating its fundamental group. The exact shape and choice of the loops is irrelevant, all that matters is the that the loops are chosen to cover all $n$ transverse directions of the torus. Thanks to that, we can derive the new quantities $$I_j = I_j(q, p) = \int_{\sigma_j} \sum_{k=1}^{n} p_k dq^k$$ By varying the points $(q,p)$ we select different tori $M_c$ for different $c_k = H_k(q,p)$ and thus the set of all these new quantities $$I_j = I_j(q, p)$$ becomes a set of $n$ new variables smoothly depending on the old variables $(q, p)$.

The name action variables comes from the fact that we are constructing the variables $I_j$ by applying the action $$S[\sigma_j] =\int_{\sigma_j} \sum_{k=1}^{n} p_k dq^k$$ under the conditions that $\sigma_j$ are loops that satisfy the stronger restrictions $H_1(\sigma_j) = c_1, ..., H_n(\sigma_j) = c_n$. This is a special case of the action from the first part of the discussion. Hence the name.

Angle variables. To change the variables completely and at the same time keep the Hamiltonain (so called canonical) structure of the equations of motion, we need a set of $n$ extra variables $\theta^1, ..., \theta^n$ which completes the set of action variables so that the new extra variables, called angle variables, are the conjugate variables to the action variables, i.e. $$\{I_j, \theta^k\} = \delta_{j}^{k}, \,\, \{I_j, I_k\} = \{\theta^j, \theta^k\} = 0$$ This can be done.

Now we have a new set of variables $I_j = I_j(q, p)$ and $\theta^k = \theta^k(q, p)$ for $j, k = 1,..., n$ written as functions of the old variables $(q,p)$. The formulas can be inverted and $H(q, p) = \tilde{H}(I) = \tilde{H}(I_1,...,I_n)$. Consequently, the old system \begin{align} &\frac{dq^k}{dt} \, = \, \frac{\partial H}{\partial p_k}\big( q, p\big)\\ &\frac{dp^k}{dt} \, = \, -\, \frac{\partial H}{\partial q^k}\big( q, p\big)\\ & \text{ for } k = 1, ..., n \end{align} can be written in the new action-angle variables as the simplified system \begin{align} &\frac{dI_k}{dt} \, = \, 0\\ &\frac{d\theta^k}{dt} \, = \, -\, \frac{\partial \tilde{H}}{\partial I_k}\big( I_1, ..., I_n\big)\\ & \text{ for } k = 1, ..., n \end{align} with Hamiltonain function $\tilde{H}$. The latter system is easy to solve, so the solutions look like \begin{align} &I_k(t) = I_k^{0} \\ &\theta^k(t) = \theta_0^k - \, \frac{\partial \tilde{H}}{\partial I_k}\big( I_1^0, ..., I_n^0\big)\,t & \text{ for } k=1, ..., n \end{align}

  • $\begingroup$ What's your conclusion ? @ Futurologist $\endgroup$
    – user172266
    Commented Nov 13, 2018 at 15:50
  • 2
    $\begingroup$ @robin What do you think? What did you gather form reading my post? $\endgroup$ Commented Nov 13, 2018 at 19:45

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