- #1

jostpuur

- 2,116

- 19

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S(x) = \int\limits_{t_A}^{t_B} L\big(t,x(t),\dot{x}(t)\big) dt

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For a small real parameter [itex]\delta[/itex] and some path [itex]\eta:[t_A,t_B]\to\mathbb{R}^N[/itex] such that [itex]\eta(t_A)=0[/itex] and [itex]\eta(t_B)=0[/itex] the action for [itex]x+\delta\eta[/itex] can be approximated as follows:

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S(x+\delta\eta) = \int\limits_{t_A}^{t_B}\Big( L\big(t,x(t),\dot{x}(t)\big)

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\quad+ \delta\sum_{n=1}^N \Big(\eta_n(t) \frac{\partial L(t,x(t),\dot{x}(t))}{\partial x_n}

+ \dot{\eta}_n(t) \frac{\partial L(t,x(t),\dot{x}(t))}{\partial \dot{x}_n}\Big)

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\quad+\frac{1}{2}\delta^2 \sum_{n,n'=1}^N\Big(\eta_n(t)\eta_{n'}(t)

\frac{\partial^2 L(t,x(t),\dot{x}(t))}{\partial x_n\partial x_{n'}}

+ 2\eta_n(t)\dot{\eta}_n(t) \frac{\partial^2 L(t,x(t),\dot{x}(t))}{\partial x_n\partial \dot{x}_{n'}}

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[tex]

\quad\quad + \dot{\eta}_n(t)\dot{\eta}_{n'}(t)\frac{\partial^2 L(t,x(t),\dot{x}(t))}{\partial\dot{x}_n\partial \dot{x}_{n'}}\Big) + O(\delta^3)\Big)dt

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The Euler-Lagrange equations deal with condition that the first order term is zero. Has the positivity of the second order term been studied at all? It is quite common, that in classical mechanics it is believed that the action is minimized, but it is not proven in any ordinary books. What if the zero of the gradient was a saddle point in some case? Would that be a surprise? Saddle points can exist in infinite dimensional spaces too.