Conservation Laws for One Particle in Three Dimensions

Basic Conservation Laws for One Particle in Three Dimensions 

As a direct consequence of Newton s laws one finds the elementary conservation laws for linear and angular momentum and energy. Consider again one particle with mass m in three-dimensional space. If the particle is moving freely, i.e., the resulting force acting on it vanishes, F = 0, the linear momentum p is conserved due to Eq. (2.2). 

Vectorial multiplication of Eq. (2.2) with r from the left yields 

For the discussion of energy conservation we first need the general concept of conservative force. A force F is said to be conservative if there exists a time-independent scalar potential energy U = U r) such that 

The most prominent example of such a velocity-dependent conservative force is the Lorentz force of electrodynamics, cf. Eq. (2.104) in section 2.4. After multiplication by f, Newton s second law in Eq. (2.2) yields for conservative forces 

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