You could define your own subtitle and subsubtitle commands with the help of the titling package as shown in the following example:
\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{titling}
\newcommand{\subtitle}[1]{%
\posttitle{%
\par\end{center}
\begin{center}\Large#1\end{center}
}%
}
\newcommand{\subsubtitle}[1]{%
\preauthor{%
\begin{center}
\large #1 \vskip0.5em
\begin{tabular}[t]{c}
}%
}
\title{Symplectic Geometry in Classical Mechanics }
\subtitle{Lecture 1}
\subsubtitle{Basics of Manifolds}
\author{}
\date{}
\begin{document}
\maketitle
It is a mathematical formalism underlying
\begin{itemize}
\item Geometrical Optics
\item Classical Mechanics
\item General Relativity
\item Quantum Mechanics
\end{itemize}
Such a unifying formalism help us to solve problems in one area of physics
by using ideas from another areas.
It doesn't work for systems with loss, friction or noise.
\end{document}

To remove the white space below the subtitle (that is 'reserved' for author and date, you could use the following:
\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{titling}
\newcommand{\subtitle}[1]{%
\posttitle{%
\par\end{center}
\begin{center}\Large#1\end{center}
}%
}
\newcommand{\subsubtitle}[1]{%
\preauthor{%
\begin{center}
\large #1 \vskip0.5em
%\begin{tabular}[t]{c}
}%
}
\postauthor{%
\end{center}
}%
\predate{}
\postdate{}
\title{Symplectic Geometry in Classical Mechanics }
\subtitle{Lecture 1}
\subsubtitle{Basics of Manifolds}
\author{}
\date{}
\begin{document}
\maketitle
It is a mathematical formalism underlying
\begin{itemize}
\item Geometrical Optics
\item Classical Mechanics
\item General Relativity
\item Quantum Mechanics
\end{itemize}
Such a unifying formalism help us to solve problems in one area of physics
by using ideas from another areas.
It doesn't work for systems with loss, friction or noise.
\end{document}

A more 'quick and dirty' approach would be the following example, where is used \author for the subtitle and \date for the subsubtitle:
\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\title{Symplectic Geometry in Classical Mechanics }
\author{Lecture 1}
\date{Basics of Manifolds}
\begin{document}
\maketitle
It is a mathematical formalism underlying
\begin{itemize}
\item Geometrical Optics
\item Classical Mechanics
\item General Relativity
\item Quantum Mechanics
\end{itemize}
Such a unifying formalism help us to solve problems in one area of physics
by using ideas from another areas.
It doesn't work for systems with loss, friction or noise.
\end{document}

\chapterin thearticleclass. It can't compile. – Bernard Jun 10 '18 at 09:43\\*seems quite strange. You should simply remove them. – TeXnician Jun 10 '18 at 09:44bookorreport(see this answer). You should rarely have to explicitly tell LaTeX to skip to the next line because it automatically begins things like chapter titles,itemizeitems etc. on separate lines. Note that paragraphs should be sepaparted by an empty line (or a\par). – Circumscribe Jun 10 '18 at 10:26