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--- basic_tools:variational_calculus:the_variational_problem [2018/03/14 15:17]
iiqof created
+++ basic_tools:variational_calculus:the_variational_problem [2018/03/14 16:25] (current)
iiqof Clarified things
@@ Line 4: / Line 4: @@
 $$
-\delta S[q|\phi] =
+= \delta S[q|\phi] =
 \frac{\partial}{\partial \varepsilon} \int_a^b F\circ \Gamma(q+\varepsilon \phi) dt =
 \int_a^b  \frac{\partial}{\partial \varepsilon} F(q+\varepsilon \phi, \dot q+\varepsilon \dot\phi)dt =
-\int_a^b  \frac{\partial F}{\partial q}\phi + \frac{\partial F}{\partial \dot q}\dot\phi dt
 $$
+$$
+\int_a^b  \frac{\partial F}{\partial q}\phi + \frac{\partial F}{\partial \dot q}\dot\phi dt = \left[\frac{\partial F}{\partial \dot q}\phi\right]^b_a + \int_a^b  \left(\frac{\partial F}{\partial q} - \frac{d}{dt}\frac{\partial F}{\partial \dot q}\right)\phi dt
+$$
+We haven't made made any assumption of the form of $q$ and the space $\Omega$ where it resides. On the standard variational problem $\Omega$ is the space of the functions that $q(a)=q_a$ and $q(b)=q_b$, i.e, **fixed boundaries**, and are sufficiently smooth. Also, we asumed a standart lift: $\Gamma(q) = (q, \dot q)$.
+Saying that, now the [[basic_tools:variational_calculus:fundamental_lemma|fundamental lemma of variational calculus]] enters. The variation, $\phi$, is arbitrary but  with $\phi(a)=\phi(b)=0$. The later assertion, makes the term outside the integral zero. On the other hand, by the fundamental lemma, the interior parentheisi is zero, giving:
+$$
+\frac{\partial F}{\partial q} - \frac{d}{dt}\frac{\partial F}{\partial \dot q} = 0
+$$
+that is, the [[equations:euler_lagrange_equations|Euler-Lagrange equations]].
+__** Some variations worth studying**__
+. Change $\Omega$: the functions do not need to be continous at all points, this gives collisions on a variational setting.
+. Variable end points, i.e. $a$ and $b$ vary with $\varepsilon$
+. Change the interdependencies of the lift $\Gamma$. This will lead to Vakonomic Mechanics.

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