\documentclass{beamer}
\usepackage[english,italian]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{listings}
\usepackage{calligra}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{cancel}
\graphicspath{ {./figure/} }
\usetheme{default}
\usecolortheme{beaver}
\usepackage{tikz}
\usetikzlibrary{shapes,arrows}

\title{vibrazioni}
\author{\calligra Fulvio Ciriaco}
\subject{}
\begin{document}
  \pgfdeclareimage[width=1.5cm]{logo}{uniba.png}
  \logo{\pgfuseimage{logo}}
  \begin{frame}{\titlepage}
  \tableofcontents
  \end{frame}

\begin{frame} 
  \frametitle{piccole oscillazioni}
  \begin{equation*}
    V=V_{x_e}+\left(\frac {\partial V}{\partial x}\right)_{x_e}(x-x_e)+
    \frac 1 2 \left(\frac {\partial^2 V}{\partial x^2}\right)_{x_e}(x-x_e)^2+...
  \end{equation*}
\end{frame}

\begin{frame} 
  \frametitle{meccanica classica}
  \begin{align*}
    m_1 x''_1 &= -k (x_1-x_2) \\
    m_2 x''_2 &= -k (x_2-x_1)
  \end{align*}
  \begin{equation} \begin{bmatrix} x_1 \\ x_2\end{bmatrix}''=
  \begin{bmatrix} x''_1 \\ x''_2\end{bmatrix}=
  \begin{bmatrix} -\frac k {m_1} & \frac k {m_1} \\
  \frac k {m_2} & -\frac k {m_2}\end{bmatrix}
  \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
\end{equation}
\end{frame}

\begin{frame} 
  \frametitle{disaccoppiamento}
  so risolvere
  \begin{equation} 
    \begin{bmatrix} x_1 \\ x_2\end{bmatrix}''=
    -\begin{bmatrix} \omega_1^2 & 0 \\ 0 & \omega_2^2 \end{bmatrix}
    \begin{bmatrix} x_1 \\ x_2\end{bmatrix}
  \end{equation}
  \begin{align*}
    x_1 &= A_1 e^{i \omega_1 t} + B_1 e^{-i \omega_1 t} \\
    x_2 &= A_2 e^{i \omega_2 t} + B_2 e^{-i \omega_2 t} 
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{diagonalizzazione della matrice K/M}
  \begin{equation*}
  \det \begin{bmatrix} -\frac k {m_1} -\lambda & \frac k {m_1} \\
  \frac k {m_2} & -\frac k {m_2} -\lambda\end{bmatrix}\end{equation*}

  \begin{align*}
    \lambda^2 + \lambda (\frac  k {m_1} + \frac k {m_2}) &= 0 \implies \\
    \lambda_1 &= 0 \\
    \lambda_2 &= -\frac k \mu
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{soluzione 1}
  $\lambda=0$

  \begin{align*}
  \begin{bmatrix} -\frac k {m_1} & \frac k {m_1} \\
    \frac k {m_2} & -\frac k {m_2} \end{bmatrix} 
    \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &= 0 \\
    x &= A \begin{bmatrix} 1 \\ 1 \end{bmatrix}
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{soluzione 2}
  $\lambda=-\frac k \mu= \frac k {m_1} + \frac k {m_2}$
  
  \begin{align*}
    \det \begin{bmatrix} -\frac k {m_1} +\frac k {m_1} + \frac k {m_2}& \frac k {m_1} \\
  \frac k {m_2} & -\frac k {m_2} +\frac k {m_1} + \frac k {m_2}\end{bmatrix}
      \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &= 0 \\
    x&=A\begin{bmatrix} m_2 \\ -m_1 \end{bmatrix}
    \end{align*}

\end{frame}

\end{document}