When conservation rules are applied to classical mechanics, the process of elucidating the dynamics is substantially simplified. Isolated many-body systems in nature commonly preserve one or more first-order integrals for linear momentum, angular momentum, and mass/energy. Note that the conservation of mass and energy is related in the Theory of Relativity, but detached in non-relativistic mechanics. Other observables, including lepton and baryon numbers, are preserved, however these conservation rules can often be subsumed under the conservation of mass for the majority of issues in non-relativistic classical mechanics.

When addressing issues involving Newtonian mechanics, it is advantageous to combine the conservation laws with the initial integrals for linear momentum, angular momentum, and work-energy due to the effectiveness of conservation laws in calculating classical dynamics. These three conservation laws will be derived assuming Newton’s laws of motion; nonetheless, they are basic laws of nature that extend well beyond the scope of Newtonian mechanics.

conservation of linear momentum

In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant. This fact, known as the law of conservation of momentum, is implied by Newton’s laws of motion.

conservation of angular momentum

angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved)

Conservation of energy

 the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time

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