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Mathematics for Physics

242 System

A chosen object or collection of objects that is used to mark what is under study and what counts as its surroundings.

The chosen boundary. The observer chooses the system by drawing a boundary. Everything outside that boundary is the surroundings. This principle is used to decide which forces and energy transfers count as external.

Parameters versus state. Some properties, such as mass and charge, are fixed by how the system is made. Other properties make up the physical state and can change. This principle is used to separate parameters from variables.

The quantum wavefunction. In quantum mechanics the state is represented completely by a wavefunction. This principle is used to compute every measurement probability from one object.

Energy balance. The total energy of the system is conserved when no work is done on it by external forces and no heat crosses the boundary. This principle is used to write an energy balance after the boundary is chosen.

The isolated-system energy balance is

\[ W_{\mathrm{ext}} = 0 \quad\mathrm{and}\quad q = 0 \implies \Delta E_{\mathrm{sys}} = 0 \]

where

  • \(W_{\mathrm{ext}}\) is the work of external forces.
  • \(q\) is the heat absorbed by the system.
  • \(E_{\mathrm{sys}}\) is the total energy of the system.

Redrawing the boundary. The same collection can be redrawn so that interacting forces sit inside rather than outside. This principle is used to convert an external pair into an internal pair that cancels in the net force.

The Lagrangian description. A system can be described by a Lagrangian from which the equations of motion are derived. This principle is used to treat particles, fields, and constrained collections in one formalism.

Note: Also called a physical system. The boundary that separates the system from its surroundings is chosen by the observer.

242.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — system boundary; energy and work.
  2. Park, D. Introduction to the Quantum Theory. Dover, 2005. — properties and state; wavefunction.
  3. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — system parameters.
  4. Schwichtenberg, J. Physics from Symmetry. — Lagrangian description.
  5. Soare, M. V., Teodorescu, P. P., & Toma, I. Ordinary Differential Equations with Applications to Mechanics. Springer, 2007. — multiparticle constrained systems.