# 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 meaning of action.

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The action has no immediate physical interpretation, but may be understood as the generating function for a canonical transformation; see e.g., http://en.wikipedia.org/wiki/Hamilton-Jacobi_equation

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The Hamiltonian H and Lagrangian L which are rather abstract constructions in classical mechanics get a very simple interpretation in relativistic quantum mechanics. Both are proportional to the number of phase changes per unit of time. The Hamiltonian runs over the time axis (the vertical axis in the drawing) while the Lagrangian runs over the trajectory of the moving particle, the t’-axis.

The Illustration shows the relativistic de Broglie wave in a Minkowski diagram. The triangle represents the relation between the Lagrangian an the Hamiltonian, which holds in both relativistic and non-relativistic physics.

$$L ~=~pv-H$$

The Hamiltonian counts the phase-changes per unit of time on the vertical axis while the term pv counts the phase-changes per unit on the horizontal axis representing distance: v is the distance traveled per unit of time while p is proportional with the phase-changes per unit of distance, hence the term pv.

The Action can now be seen as being proportional to the total number of phase changes over the trajectory of the particle. The principle of least action is thus equivalent to the principle of least phase change. In the theory of special relativity the latter is equivalent to the principle of least proper time since the 'proper time' as experienced by the particle is proportional to the number of phase changes over the trajectory.

Hans

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+1 This is why I love this place, Perhaps you could point me to some elaboration on this? –  Prathyush Oct 18 '12 at 18:27
Do you have a Natural extension of this way of thought for fields? –  Prathyush Oct 18 '12 at 18:38

I) At least three different quantities in physics are customary called an action and denoted with the letter $S$:

1. The off-shell action $S[q;t_i,t_f]$,

2. The (Dirichlet) on-shell action $S(q_f,t_f;q_i,t_i)$, and

3. Hamilton's principal function $S(q,\alpha, t).$

For their definitions and how they are interrelated, see e.g. this Phys.SE answer. (Here the words on-shell and off-shell refer to whether the equations of motion (eom) are satisfied or not, see the general definition)

II) OP is apparently thinking of the first option: The off-shell action

$$S[q;t_i,t_f]~=~\int_{t_i}^{t_f}\! dt ~L,$$

which may be evaluated along (possibly virtual) paths $q:[t_i,t_f]\to\mathbb{R}$, which do not necessarily satisfy Euler-Lagrange equations (=eom). The Lagrangian $L$ is typically the difference between the kinetic and potential energy, but we warn that this need not be the case, cf. e.g. this Phys.SE post and links therein.

III) One may ask: Why do we consider virtual/unphysical paths that do not necessarily satisfy eom?

Answer: For at least two reasons:

1. One cannot derive Euler-Lagrange equations without allowing virtual paths, cf. the principle of stationary action.

2. In quantum mechanics, the virtual paths contribute to the path integral as quantum fluctuations, and have physical consequences. (They are e.g. responsible for the Van Vleck determinant in the semiclassical approximation via Gaussian integration.)

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I agree with Arnold, more or less, confining our attention to classical dynamics. In quantum mechanics (QM) and field theory (QFT), however, the action is the natural logarithm of the probability amplitude to propagate a system from an initial configuration of particles in QM or fields in QFT. Feynman exploited a comment by Dirac in his QM book that, paraphrasing, the exponential of $-i \hbar S$ is related to the propagation probability amplitude.
The action is a functional of not-yet-defined functions $q(t)$ and $\dot q(t)$ such that its minimum (or a stationary condition $\delta S =0$) determines a family of possible real motions of a physical system as differential equation general solutions. The final choice of one real motion out of this family is determined with giving some endpoints (or more often - initial conditions). It fixes the arbitrary constants and yields a unique curve.