**Equations of Motion**

The Hamiltonian for a particle of mass *m* moving freely without friction on a surface is:

where is a potential designed to be zero inside the region in which the particle can move, and infinity otherwise:

This form of the potential guarantees a specular reflection on the boundary. The kinetic term guarantees that the particle moves in a straight line, without any change in energy. If the particle is to move on a non-Euclidean manifold, then the Hamiltonian is replaced by:

where is the metric tensor at point . Because of the very simple structure of this Hamiltonian, the equations of motion for the particle, the Hamiltonâ€“Jacobi equations, are nothing other than the geodesic equations on the manifold: the particle moves along geodesics.

Read more about this topic: Dynamical Billiards

### Other articles related to "equations of motion, equations, equation, motion, motions":

**Equations Of Motion**- Analogues For Waves and Fields

... Field

**equations**Equations that describe the spatial dependence and time evolution of fields are called field

**equations**... These include the Navierâ€“Stokes

**equations**for the velocity field of a fluid, Maxwell's

**equations**for the electromagnetic field, the Einstein field

**equation**for gravitation (Newton's law of gravity is a ... Wave

**equations**Equations of wave

**motion**are called wave

**equations**...

**Equations of Motion**

... rocket configurations, because these exhibit greater symmetry than aeroplanes, and the

**equations of motion**are correspondingly simpler ... that the vehicle is roll-controlled, the pitch and yaw

**motions**may be treated in isolation ... practice to consider the yaw plane, so that only 2D

**motion**need be considered ...

**Equations of Motion**

... The

**equations of motion**of an idealized bike, consisting of a rigid frame, a rigid fork, two knife-edged, rigid wheels, all connected with frictionless bearings and ... These

**equations**have been verified by comparison with multiple numeric models derived completely independently ... The

**equations**show that the bicycle is like an inverted pendulum with the lateral position of its support controlled by terms representing roll acceleration, roll velocity ...

**Equations of Motion**

... to use the usual procedures from Lagrangian mechanics to derive "

**equations of motion**" that describe the time evolution of both the metric and its conjugate momentum ... The result and is a non-linear set of partial differential

**equations**... variations with respect to the lapse and shift provide constraint

**equations**and and the lapse and shift themselves can be freely specified, reflecting the fact that ...

### Famous quotes containing the word motion:

“A movement is only composed of people moving. To feel its warmth and *motion* around us is the end as well as the means.”

—Gloria Steinem (b. 1934)