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2014 | OriginalPaper | Buchkapitel

2. The Geometry of Hamiltonian Mechanics

verfasst von : Stavros C. Farantos

Erschienen in: Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics

Verlag: Springer International Publishing

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Abstract

In this chapter an introduction to Lagrangian and Hamiltonian mechanics is given. An effort is made to present Hamiltonian theory from the analytical mechanics point of view, which unveils the geometrical characteristics of the theory, such as its symplectic symmetry. The relations among the tangent bundle, cotangent bundle and the mixed tangent-cotangent bundle of the configuration manifold are discussed. The Euler–Lagrange equations and Hamilton’s equations of motion are extracted from the principle of least action. The canonical equations are also formulated by the symplectic $2-$form and the symplectic transformations are explained. Poisson brackets, which provide the bridge to pass from classical to quantum mechanics are introduced.

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Vorheriges Kapitel Introduction and Overview

Nächstes Kapitel Dynamical Systems

Holonomic constraints may contain the velocities $\phi ^i(r^1,\dots ,r^n, \dot{r}^1,\dots ,\dot{r}^N) = c^i$, which however, can be integrated to equations without the velocities.

The letter superscript ($T$) denotes a column vector and generally the transpose of a matrix.

The components of Kronecker delta tensor, $\delta _i^j$, are equal to 1 for $i=j$ and 0 for $i\ne j$.

For time dependent Hamiltonians, $H(q, p, t)$, we can apply the same formalism of conservative Hamiltonians by introducing time as a new variable, $q^0=t$, with conjugate momentum, $p_0$, and new Hamiltonian, $H_t = p_0 + H(q, p, t) = 0$. Thus, the extended phase space $M_t=T^*Q_t$ of the extended configuration manifold, $Q_t=(t, q^1, \dots , q^n)^T$, is of $2(n+1)-$dimension and in its cotangent bundle we define the Canonical Poincaré $1-$Form

$$\begin{aligned} \hat{\theta _t} = \sum _{i=0}^n p_idq^i = p_0dq^0 + \sum _{i=1}^n p_idq^i = -H(q, p, t)dt + \hat{\theta }, \end{aligned}$$

(2.72)

and symplectic $2-$form

$$\begin{aligned} \hat{\omega _t} = -d\hat{\theta _t} = dH\wedge dt - d\hat{\theta } = -dt\wedge dH + \sum _{i=1}^n dq^i\wedge dp_i. \end{aligned}$$

(2.73)

The new Hamiltonian vector field $(X_{H_t})$ is defined by the equation

$$\begin{aligned} \hat{\omega _t}(X_{H_t}, \bullet ) = dH_t, \end{aligned}$$

(2.74)

$$\begin{aligned} \left( \begin{array}{r} (X_{H_t})^0 \\ \overline{(X_{H_t})}_0 \end{array} \right) = \left( \begin{array}{c} 1 \\ -{\partial H}/{\partial t} \end{array} \right) , \left( \begin{array}{r} (X_{H_t})^i \\ \overline{(X_{H_t})}_i \end{array} \right) = \left( \begin{array}{r} {\partial H}/{\partial p_i} \\ - {\partial H}/{\partial q^i} \end{array} \right) ,\;\; i=1,\dots , n . \end{aligned}$$

(2.75)

The Hamiltonian vector field lives in the tangent bundle of the extended phase space, $T(T^*Q_t)$, the base vector fields of which are

$$\begin{aligned} \left( \frac{\partial }{\partial t}, \frac{\partial }{\partial p_0}\right) , \;\; \left( \frac{\partial }{\partial q^i}, \frac{\partial }{\partial p_i}\right) , \;\; i=1,\dots , n. \end{aligned}$$

(2.76)

$(M_t, \hat{\omega }_t, X_{H_t})$ is a Hamiltonian system and the Canonical Poincaré $1-$Form, Eq. 2.72, is related to the total differential of action (Eq. 2.47). We can see, that with this formulation of time dependent systems the trajectories are projected at each time $t$ in the physical phase space of the system of $2n-$dimension, $x=(q^1, \dots , q^n, p_1, \dots , p_n)^T$, and they are given by Hamilton’s equations of motion with the time dependent Hamiltonian

$$\begin{aligned} \dot{x}(t) = J \partial H(x, t). \end{aligned}$$

(2.77)

Arnold VI (1980) Mathematical methods of classical mechanics., Graduate text in mathematics Springer, Berlin

Ezra GS, Waalkens H, Wiggins S (2009) Microcanonical rates, gap times, and phase space dividing surfaces. J Chem Phys 130(164):118

Frankel T (2004) The geometry of physics: an introduction. Cambridge University Press, Cambridge

Ginoux JM (2009) Differential geometry applied to dynamical systems., Nonlinear science, World Scientific Publishing Co., Pte. Ltd, Singapore

Meyer KR, Hall GR, Offin D (2009) Introduction to Hamiltonian dynamical systems and the n-body problem, 2nd edn., Applied mathematical sciences Springer, Heidelberg

Scheck F (1990) Mechanics, 1st edn. Springer, BerlinCrossRef

Spivak M (1965) Calculus on manifolds: a modern approach to classical theorems of advanced calculus. Addison-Wisley, Massachusetts

Titel: The Geometry of Hamiltonian Mechanics
verfasst von: Stavros C. Farantos
Verlag: Springer International Publishing
Buch: Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics
Print ISBN: 978-3-319-09987-3

Electronic ISBN: 978-3-319-09988-0

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-09988-0_2

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