Spring Mass Systems: Simulation in Phase Space

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If a system is self-contained, i.e., does not get influenced by additional external forces,
then this state space contains enough information to calculate the next state in time of the physical system.
For our spring-mass systems, the positions of all the particles defines the stretching of the springs and thus the forces between the particles.
The accumulated forces for each particle, divided by its mass, define the change in velocity for that particle;
and since the velocity of each particle is known at each point in phase space, the new velocities are also readily known.
Thus this 6N-dimensional phase space can again be seen as filled with a huge vector field
that predicts the evolution of the system as we trace the "current-state point" along these field lines.
Of course, all the issues with selecting the right time-step size h still apply!

More details in Discussion Sections

To Learn More:

Andy Witkin and David Baraff: "Physically Based Modeling: Principles and Practice"
http://www-2.cs.cmu.edu/~baraff/sigcourse/index.html

Grinspun, Hirani, Desbrun, and Peter Schroder: "Discrete Shells," SCA 2003

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