## Abstract

In introductory physics courses energy is usually introduced as a conserved quantity. In a closed system the energy is constant as the physical system moves in time. For any given physical system, each known conserved quantity provides an equation that can be very useful in the analysis of the system. For instance, conservation of energy for a ball of mass *m* thrown straight up into the air yields the equation \( E = \frac{1}{{2m}}{p^2} + mgr \), where E is the constant value of the energy, *g* is the constant strength of the downward gravitational force, *p* is the momentum of the ball and *r* is its height. From this equation alone, without using calculus, one can predict the maximum height of the ball from its initial position and momentum.

## Keywords

Phase Space Vector Field Hamiltonian System Symplectic Form Symplectic Manifold## Preview

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