Action Principles

An action principle says that the physical trajectories of a system are stationary points of an action functional under a specified class of variations. For ordinary Lagrangian mechanics, the action is

$$S[q] = \int_{t_i}^{t_f} L(q,\dot q,t)\,dt,$$

and Hamilton’s principle is

$$\delta S=0$$

for all allowed variations with the chosen boundary data. This page collects the conceptual rules: what is varied, which endpoints are fixed, why boundary terms matter, how the phase-space action works, and why stationary action becomes stationary phase in semiclassical quantum mechanics.

Why Quantum Mechanics Needs This

Action principles appear throughout quantum mechanics:

• path integrals weight histories by $e^{iS/\hbar}$;
• stationary phase explains why classical paths dominate leading semiclassical approximations;
• WKB phases are action functions;
• variational approximations extremize functionals over trial states or trial paths;
• field theory and many-body theory are often organized by actions and source functionals;
• boundary terms and domains decide whether formal integrations by parts are legitimate.

The key discipline is to specify the functional and the allowed variations. Without that information, the phrase “use the action principle” is incomplete.

Stationary, Not Always Least

The traditional phrase “principle of least action” is often misleading. The classical path makes the action stationary:

$$\left. \frac{d}{d\epsilon} S[q+\epsilon\eta] \right\rvert_{\epsilon=0} = 0$$

for every allowed variation $\eta$. A stationary point can be a minimum, maximum, or saddle.

Real-time actions commonly have saddle behavior. This is especially important in path integrals, where the classical path is a stationary phase point of an oscillatory amplitude, not the most probable path of a positive probability measure.

Ingredients of an Action Principle

Every action principle needs four pieces:

• the space of histories, such as paths $q(t)$ or fields $\phi(x)$;
• the action functional $S$;
• the allowed variations, such as fixed-endpoint variations;
• the boundary conditions or endpoint terms.

For a particle path, an allowed fixed-endpoint variation is

$$q^i(t) \longmapsto q^i(t)+\epsilon\eta^i(t),$$

with

$$\eta^i(t_i)=\eta^i(t_f)=0.$$

Changing the allowed variations changes the variational problem. If endpoints are free, boundary terms that vanish in the fixed-endpoint problem can become physical conditions.

Configuration-Space Action

For a Lagrangian

$$L(q,\dot q,t),$$

the configuration-space action is

$$S[q] = \int_{t_i}^{t_f} L(q,\dot q,t)\,dt.$$

Its first variation has the form

$$\delta S = \int_{t_i}^{t_f} \sum_i \left[ \frac{\partial L}{\partial q^i} - \frac{d}{dt} \left( \frac{\partial L}{\partial\dot q^i} \right) \right] \eta^i(t)\,dt + \left[ \sum_i p_i\eta^i \right]_{t_i}^{t_f},$$

where

$$p_i = \frac{\partial L}{\partial\dot q^i}.$$

For fixed endpoints, the boundary term vanishes. Since the remaining variations are arbitrary in the interior, stationary action gives the Euler–Lagrange equations:

$$\frac{\partial L}{\partial q^i} - \frac{d}{dt} \left( \frac{\partial L}{\partial\dot q^i} \right) = 0.$$

The full mathematical derivation belongs to Calculus of Variations. The mechanics review is Lagrangian Mechanics Review.

Boundary Terms

Boundary terms are not bookkeeping debris. They encode what data are being held fixed and which additional conditions are required.

For fixed endpoints, the variation satisfies $\eta(t_i)=\eta(t_f)=0$, so a term such as

$$\left[ p_i\eta^i \right]_{t_i}^{t_f}$$

vanishes. If an endpoint is free, the same term may imply a natural boundary condition.

Adding a total time derivative to the Lagrangian,

$$L \longmapsto L+\frac{dF(q,t)}{dt},$$

changes the action by an endpoint term:

$$S \longmapsto S+F(q_f,t_f)-F(q_i,t_i).$$

The interior equations of motion are unchanged for fixed endpoints, but the endpoint phase in a path integral and the generating function for a canonical transformation can change.

Phase-Space Action

Hamiltonian mechanics has its own action principle. Treat $q^i(t)$ and $p_i(t)$ as independent paths and define

$$S[q,p] = \int_{t_i}^{t_f} \left[ \sum_i p_i\dot q^i - H(q,p,t) \right]dt.$$

Varying with respect to $p_i$ gives

$$\dot q^i = \frac{\partial H}{\partial p_i}.$$

Varying with respect to $q^i$ and using fixed endpoint variations for $q^i$ gives

$$\dot p_i = - \frac{\partial H}{\partial q^i}.$$

Thus the phase-space action produces Hamilton’s equations directly.

The term

$$\sum_i p_i\,dq^i$$

is the canonical one-form in local coordinates. Its exterior derivative gives the symplectic form used in Poisson brackets and canonical transformations.

Classical Paths in Path Integrals

The real-time path-integral weight is

$$\exp\left( \frac{i}{\hbar}S[q] \right).$$

When the action scale is large compared with $\hbar$, phases vary rapidly as the path is varied. Contributions away from stationary paths cancel by destructive interference, while neighborhoods of stationary paths contribute coherently.

The leading semiclassical approximation is therefore organized around solutions of

$$\delta S=0.$$

Those are classical paths, but the approximation remains quantum. The amplitude also contains fluctuation determinants, Maslov phases, tunneling exponents, and interference between multiple stationary paths.

Relation to Hamilton–Jacobi Theory

If the action is evaluated on a classical path connecting endpoints, it becomes Hamilton’s principal function:

$$S(q_f,t_f;q_i,t_i) = \int_{t_i}^{t_f} L(q_{\mathrm{cl}},\dot q_{\mathrm{cl}},t)\,dt.$$

Endpoint derivatives of this function give endpoint momenta, and the function satisfies the Hamilton–Jacobi equation. Thus action principles define the classical path, while Hamilton–Jacobi theory studies the endpoint-dependent action function built from that path.

Quantum Variational Principles

Not every action principle in quantum mechanics is a path integral. Variational methods also extremize functionals over trial states. For example, the stationary condition of

$$\langle\psi\rvert H\lvert\psi\rangle - E \left( \langle\psi\rvert\psi\rangle-1 \right)$$

with respect to $\langle\psi\rvert$ gives

$$H\lvert\psi\rangle=E\lvert\psi\rangle.$$

Time-dependent variational principles restrict Schrödinger evolution to a trial manifold. These methods are different from Hamilton’s classical action principle, but they share the same variational discipline: specify the functional, constraints, allowed variations, and error interpretation.

Common Mistakes

• Saying “least action” when the result is only stationary.
• Forgetting to specify endpoint conditions.
• Dropping boundary terms automatically.
• Varying $q$ and $\dot q$ as if they were independent in the configuration-space action.
• Forgetting that $q$ and $p$ are independent variables in the phase-space action.
• Treating stationary phase in a path integral as a positive-probability maximum.
• Assuming adding a total derivative changes nothing in every context.
• Confusing a classical action principle with a quantum variational approximation over states.

Cross-Links

• Calculus of Variations
• Lagrangian Mechanics Review
• Hamiltonian Mechanics Review
• Phase Space
• Poisson Brackets
• Canonical Transformations
• Symplectic Vector Spaces
• Symplectic Manifolds, First Look
• Hamilton–Jacobi Theory
• Functional Derivatives
• Semiclassical Limit
• Why Path Integrals?
• From Propagators to Path Integrals
• Action and Phase
• Semiclassical Limit Overview
• Path Integrals

References

• H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002.
• L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Butterworth-Heinemann, 1976.
• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989.
• I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 2000.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.

Exercises

1. Why does fixed-endpoint variation make the boundary term $\left[p_i\eta^i\right]_{t_i}^{t_f}$ vanish?

Solution

Fixed endpoints mean the varied path agrees with the original path at $t_i$ and $t_f$. Therefore the variation satisfies

$$\eta^i(t_i)=\eta^i(t_f)=0.$$

Substituting into the boundary term gives zero at both endpoints:

$$\left[ p_i\eta^i \right]_{t_i}^{t_f} = p_i(t_f)\eta^i(t_f) - p_i(t_i)\eta^i(t_i) = 0.$$

2. Derive Hamilton’s equations from the phase-space action

$$S[q,p] = \int \left( p\dot q-H(q,p,t) \right)dt$$

for one degree of freedom.

Solution

Varying $p$ gives

$$\delta_p S = \int \left( \dot q-\frac{\partial H}{\partial p} \right)\delta p\,dt.$$

Since $\delta p$ is arbitrary,

$$\dot q=\frac{\partial H}{\partial p}.$$

Varying $q$ gives

$$\delta_q S = \int \left( p\,\delta\dot q - \frac{\partial H}{\partial q}\delta q \right)dt.$$

Integrating $p\,\delta\dot q$ by parts and using fixed endpoint variations,

$$\delta_q S = \int \left( - \dot p - \frac{\partial H}{\partial q} \right)\delta q\,dt.$$

Thus

$$\dot p=-\frac{\partial H}{\partial q}.$$

3. Explain why adding $dF/dt$ to a Lagrangian does not change the Euler–Lagrange equations for fixed endpoints.

Solution

The action changes by

$$\int_{t_i}^{t_f}\frac{dF}{dt}\,dt = F(q_f,t_f)-F(q_i,t_i).$$

For fixed endpoints, this endpoint contribution is fixed under allowed variations, so its first variation vanishes. Therefore the interior stationary-action equations are unchanged.

4. Why does stationary action become stationary phase in a real-time path integral?

Solution

The path-integral weight is

$$\exp\left( \frac{i}{\hbar}S[q] \right).$$

When $S/\hbar$ is large, small changes in the path usually produce rapidly varying phases that cancel. Near a stationary path, $\delta S=0$, so the phase is stable to first order and nearby paths add coherently. This is stationary phase, not probability maximization.