Symmetry in Mechanics pp 19-45 | Cite as

# Phase Spaces of Mechanical Systems are Symplectic Manifolds

## Abstract

The concept, if not the name, of a symplectic manifold is familiar to any student of mechanics. Each symplectic manifold we study in this chapter is a *phase space* of a mechanical system. In other words, each is a set of distinguishable *states* of a particular mechanical system. For instance, to specify the state of a particle in space, it is enough to specify its position and momentum. Because Newton’s second law \( F = ma \) is second-order (acceleration **a** is the second derivative of position with respect to time, and the force **F** can depend on position and velocity but not on acceleration), knowing the position and velocity at any one time allows one to predict the particle’s motion at all future times. To put it another way, if we think of Newton’s law as \( F = \frac{d}{{dt}}p \) and remember that the force never depends on derivatives of the momentum **p**, we see that knowing the position and momentum allows one to predict the particle’s motion at all future times. So we can think of the phase space as made up of position-momentum pairs. (Some authors use the phrase “state space” synony-mously with “phase space.”) Note that the *configuration space* is something different: it is the set of possible positions.

## Keywords

Phase Space Vector Field Bilinear Form Jacobian Matrix Differential Form## Preview

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