Principle of Stationary Action and Euler-Lagrange Equation
Principle of Stationary Action:
Given a mechanical system, there exists an action $S$ such that it is extremitized, or $\delta S=0$, for the actual motion of the system.
$$S = \int_{t_1}^{t_2}L(q, \dot{q}, t)dt$$
where $L$ is the Lagrangian of the system.
Euler-Lagrangian Equation: $$\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot{q}}\bigg) = \frac{\partial L}{\partial q}$$
My understanding that the extremum of S implies that the E-L Equation is satisfied.
My question is: Does it work the other way? i.e. Given a mechanical system, is demanding $\delta S = 0$ for its action equivalent to demanding $\frac{d}{dt}\big(\frac{\partial L}{\partial \dot{q}}\big) = \frac{\partial L}{\partial q}$ for its Lagrangian?
classical-mechanics lagrangian-formalism variational-principle action functional-derivatives
MatthewMatthew
$\begingroup$ Possible duplicates: physics.stackexchange.com/q/907/2451 , physics.stackexchange.com/q/69077/2451 , physics.stackexchange.com/q/122486/2451 , physics.stackexchange.com/q/209344/2451 and links therein. $\endgroup$ – Qmechanic♦ Mar 6 '16 at 11:36
$\begingroup$ The Euler- lagranges equations are only a necessary condition for action to be extremum -not sufficient condition - For other sufficient conditions, see Gelfand & Fomin 2000. Chapter 5: see Gelfand and Fomin (2000)Silverman, Richard A., ed. Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 3. ISBN 978-0486414485."The Second Variation.Chapter 6: "Fields. Sufficient Conditions for a Strong Extremum". Sufficient conditions for a strong minimum are given by the theorem on p. 148. $\endgroup$ – drvrm Mar 6 '16 at 17:16
The functional derivative of a functional $S[q]$ with respect to the function $q(t)$ is defined as $$ \frac{\delta S[q]}{\delta q(t)}\equiv \lim_{\alpha\to 0}\frac{S[q+\alpha\delta_t]-S[q]}{\alpha} $$ where $\delta_t$ is the Dirac delta function centered at $t$.
Your professor/book probably proved that the functional derivative coincides with the Euler-Lagrange derivative, $$ \frac{\delta S[q]}{\delta q(t)}=\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial L}{\partial\dot q}\right)-\frac{\partial L}{\partial q} $$ which means $\delta S=0$ iff E-L is satisfied. This means: as the functional derivative equals the E-L derivative, both are zero or neither is.
AccidentalFourierTransformAccidentalFourierTransform
$\begingroup$ Thank you very much for the concise and clear explanation! $\endgroup$ – Matthew Mar 6 '16 at 22:03
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