HamiltonianFormOfLagrangesEquation 2026-08-19 23:07:46 2026-08-19 23:07:46 Definition % this is the default PhysicsLibrary preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \subsection*{Modern conventions} We use generalized coordinates $q_i$, generalized velocities $\dot q_i$, and for a natural mechanical system the Lagrangian \[ L(q,\dot q,t)=T(q,\dot q,t)-V(q,t). \] The canonical momenta are \[ p_i=\frac{\partial L}{\partial \dot q_i}. \] When the Legendre transform is regular, the Hamiltonian is \[ H(q,p,t)=\sum_{i=1}^{n} p_i\dot q_i-L, \] with the velocities expressed in terms of $(q,p,t)$. If generalized nonconservative forces are present, they are denoted by $Q_i^{\mathrm{nc}}$. Then Hamilton's equations take the form \[ \dot q_i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q_i}+Q_i^{\mathrm{nc}}. \] In the conservative case $Q_i^{\mathrm{nc}}=0$. \subsection*{Relation with Lagrange's equations} Start from the Euler--Lagrange equations with nonconservative generalized forces, \[ \frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i} =Q_i^{\mathrm{nc}}. \] Because $p_i=\partial L/\partial \dot q_i$, these become \[ \dot p_i-\frac{\partial L}{\partial q_i}=Q_i^{\mathrm{nc}}. \tag{1} \] Now differentiate the Hamiltonian, \[ dH=\sum_i \dot q_i\,dp_i-\sum_i \frac{\partial L}{\partial q_i}\,dq_i- \frac{\partial L}{\partial t}\,dt. \] Therefore \[ \frac{\partial H}{\partial p_i}=\dot q_i, \qquad \frac{\partial H}{\partial q_i}=-\frac{\partial L}{\partial q_i}. \tag{2} \] Combining (1) and (2) yields Hamilton's equations \[ \dot q_i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q_i}+Q_i^{\mathrm{nc}}. \tag{3} \] \subsection*{Autonomous natural systems} If the coordinate transformation is time-independent and \[ L=T(q,\dot q)-V(q), \] then the Hamiltonian coincides with the total energy, \[ H=T+V. \] This is the standard case behind many of Byerly's examples, even though he originally formulated the discussion in terms of a momentum-version of the kinetic energy alone. \subsection*{Interpretation} Hamilton's equations replace $n$ second-order Euler--Lagrange equations by $2n$ first-order equations in phase-space variables $(q_i,p_i)$. They are not usually computationally shorter for elementary problems, but they are the natural entrance to canonical transformations, phase space, Poisson brackets, and modern analytical mechanics. \section*{Modern notation references} The notation and terminology in this modernized transcription follow standard analytical mechanics conventions, especially: \begin{enumerate} \item H. Goldstein, C. Poole, and J. Safko, \emph{Classical Mechanics}, 3rd ed., Addison--Wesley, 2002. \item C. Lanczos, \emph{The Variational Principles of Mechanics}, 4th ed., Dover, 1986. \item J. R. Taylor, \emph{Classical Mechanics}, University Science Books, 2005. \end{enumerate} \section*{Source} This article is a modernized restatement of the corresponding portion of William Elwood Byerly, \emph{An Introduction to the Use of Generalized Co\"ordinates in Mechanics and Physics}, Ginn and Company, 1916, Chapter II. The 1916 source work is in the public domain in the United States.