diff --git a/slides/canonical-variables-definition.tex b/slides/canonical-variables-definition.tex index 12ccacc..0bf5b3a 100644 --- a/slides/canonical-variables-definition.tex +++ b/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 diff --git a/slides/constraints.tex b/slides/constraints.tex index 7af2aa3..89d720c 100644 --- a/slides/constraints.tex +++ b/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 @@ -84,7 +83,7 @@ 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) \\ @@ -92,7 +91,7 @@ \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 diff --git a/slides/first-order-vs-second-order-system.tex b/slides/first-order-vs-second-order-system.tex index 802f900..cb8dd5c 100644 --- a/slides/first-order-vs-second-order-system.tex +++ b/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} diff --git a/slides/introduction.tex b/slides/introduction.tex index e25e790..a424f53 100644 --- a/slides/introduction.tex +++ b/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. diff --git a/slides/lagrangian-and-hamiltonian-formalism.tex b/slides/lagrangian-and-hamiltonian-formalism.tex index 3fdf9fd..7471bb1 100644 --- a/slides/lagrangian-and-hamiltonian-formalism.tex +++ b/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} diff --git a/slides/linear-ostrogradskian-instability.tex b/slides/linear-ostrogradskian-instability.tex index 1efeec8..a7de1f7 100644 --- a/slides/linear-ostrogradskian-instability.tex +++ b/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}