final correction on slides

slides/canonical-variables-definition.tex

@@ -32,7 +32,7 @@
 
 \begin{frame}{Motivation behind the definition of canonical variables}
   \begin{alertblock}{}
-    In order to translating Lagrangian formalism in the Hamiltonian one, we
+    In order to translate Lagrangian formalism into the Hamiltonian one, we
     need a \emph{map}~\footnote{
       There is no need to map $q^{(6)}$ because it appears in Euler-Lagrange
       equation as a simple derivation of terms depending on $q^{(5)}$. When

slides/constraints.tex

@@ -20,8 +20,7 @@
 \begin{frame}{Constraints classification}
   \begin{alertblock}{Second classification}
     \begin{itemize}
-      \item \emph{Primary constraints}: holds independently from the equations
-        of motion.
+      \item \emph{Primary constraints}: are the original constraints.
       \begin{equation*} \label{eq:constraint}
         \phi_1(Q, P) = 0
       \end{equation*}
@@ -47,7 +46,7 @@
 \begin{frame}{Constrained Hamiltonians}
   \begin{alertblock}{Introducing m-constraints in n-th order Hamiltonian}
   \end{alertblock}
-  Constraints can be imposed in the Hamiltonian with the usage of
+  Constraints can be imposed on the Hamiltonian using
   auxiliary variables $\lambda_i$ in the corresponding Lagrangian
   \begin{equation*}
     L = L(q, \dot{q}, \ddot{q}, \ldots, q^{(n)},
@@ -56,7 +55,7 @@
     \frac{\partial L}{\partial \lambda_i} = 0 \quad i=1, 2, \ldots, m
   \end{equation*}
 
-  Canonical coordinates has to be chosen also for $\lambda_i$
+  Canonical coordinates have to be chosen also for $\lambda_i$
   \begin{equation*} \label{eq:def_canonical_coordinates_lambda}
     \Lambda_{i}:= \lambda_{i}
     \quad \leftrightarrow \quad
@@ -84,15 +83,15 @@
   different number of secondary constraints can be found using the consistency
   relations.
 
-  Finally, constraints relation can be organize as follows
+  Finally, constraints relations can be organized as follows
   \begin{equation*}
     \begin{cases}
       \Lambda_i =\ f_i(Q_1, \ldots, Q_n, P_n) \\
       \Pi_i =\ 0
     \end{cases}
     \qquad i = 1, 2, \ldots, m
   \end{equation*}
-  and substitute in $\tilde{H}$ obtaining the \emph{constrained
+  and substituted into $\tilde{H}$ obtaining the \emph{constrained
   Hamiltonian}
   \begin{equation*}
     H =\ P_n h + P_{n-1} Q_n + \cdots + P_1 Q_2

slides/first-order-vs-second-order-system.tex

@@ -1,6 +1,6 @@
 \begin{frame}{First order vs Second order Lagrangian}
   We would like to find the differences between systems that can be
-  describe by these two Lagrangians:
+  described by these two Lagrangians:
   \vspace{0.5em}
   \begin{columns}
     \begin{column}{0.4\textwidth}

slides/introduction.tex

@@ -23,7 +23,7 @@
 \begin{frame}{Why study HD systems?}
   \begin{alertblock}{QFT + GR = Theory of everything}
       \vspace{0.5em}
-      This is what string theory try to achieve but at the moment there are
+      This is what string theory tries to achieve but at the moment there are
       no known doable experiments to corroborate it.
   \end{alertblock}
   So other paths may be explored in order to formulate a Theory of everything.

slides/lagrangian-and-hamiltonian-formalism.tex

@@ -82,7 +82,7 @@
 
 \begin{enumerate}
 \begin{frame}{Hamiltonian formalism}
-  \begin{alertblock}{From Lagrange to Hamilton in 3 step}
+  \begin{alertblock}{From Lagrange to Hamilton in 3 steps}
     \vspace{0.5em}
     \item Define i-th \emph{canonical coordinates as} $(i = 1, \ldots, n)$
       \begin{equation*} \label{eq:def_canonical_coordinates}

slides/linear-ostrogradskian-instability.tex

@@ -1,13 +1,13 @@
 \begin{frame}{Linear Ostrogradskian instability}
   \begin{alertblock}{$H_2$ system isolated}
     \vspace{0.5em}
-    If the energy is conserve even though the spectra is not bounded the
-    energy stay constant.
+    If the energy is conserved even though the spectra is not bounded the
+    energy stays constant.
   \end{alertblock}
   \vspace{2.0em}
   \begin{alertblock}{$H_2$ interacting with $H_1$}
     \vspace{0.5em}
-    $H_2$ system try to reach the minimum of the Hamiltonian $H_2$ by giving
+    $H_2$ system tries to reach the minimum of the Hamiltonian $H_2$ by giving
     energy to $H_1$ system. This is behaviour goes on endlessly. This is the
     so called \alert{Linear Ostrogradskian instability}~\cite{Kallosh08,
     Eliezer89}.
@@ -23,12 +23,12 @@
 \end{frame}
 
 \begin{frame}{Curing linear Ostrogradskian instability with constraints}
-  As shown in~\cite{Chen13} linear Ostrogradskian instability can be cured by
-  imposition of constrains on the system. This can happen when the constrained
+  As shown in~\cite{Chen13} the linear Ostrogradskian instability can be cured
+  by imposition of constrains on the system. This means that the constrained
   Hamiltonian lives in phase space with lower dimensionality than the original
   phase space.\\\vspace{0.3em}
-  Constraints have been introduced in the Lagrangian using auxiliary variables
-  $\lambda$, then they have been studied using the Dirac formalism for
+  Constraints may be introduced in the Lagrangian using auxiliary variables
+  $\lambda$, then they can be studied using the Dirac formalism for
   constraints.\vspace{0.7em}
 
   \begin{block}{Example: classical Pais-Uhlenbeck oscillator}