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Laws of Conservation of Energy, Momentum, and Angular Momentum

Work and power of the force $ \vec{F} $:

$$W = \int \vec{F}.d\vec{r} = \int F_s ds, P = \vec{F}.\vec{v}$$

Increment of the kinetic energy of a particle:

$$T_2 - T_1 = W,$$

where $ W $ is the work performed by the resultant of all the forces acting on the particle.

  • Work performed by the forces of a field is equal to the decrease of the potential energy of a particle in the given field:
    $$ W = U_1 - U_2.$$
  • Relationship between the force of a field and the potential energy of a particle in the field:
    $$\vec{F} = -\vec{\nabla} U,$$

    i.e. the force is equal to the anti-gradient of the potential energy.

  • Increment of the total mechanical energy of a particle in a given potential field:
    $$ E_2 - E_1 = W_{extr},$$

    where $ W_{extr} $ is the algebraic sum of works performed by all extraneous forces, that is, by the forces not belonging to those of the given field.

  • Increment of the total mechanical energy of a system:
    $$E_2 - E_1 = W_{ext} + W_{int}^{noncons},$$

    where $ E = T + U $, and $ U $ is the inherent potential energy of the system.

  • Law of momentum variation of a system:
    $$\frac{d\vec{p}}{dt} = \vec{F},$$

    where $ \vec{F} $ is the resultant os all external forces.

  • Equation of motion of the system's center of inertia:
    $$m \frac{d \vec{v_C}}{dt} = \vec{F},$$

    where $ \vec{F} $ is the resultant os all external forces.

  • Kinetic energy of a system:
    $$T = \widetilde{T} + \frac{m v_C^2}{2},$$

    where $ \widetilde{T} $ is its kinetic energy in the system of center of intertia.

  • Equation of dynamics of a body with variable mass:
    $$m \frac{d \vec{v}}{dt} = \vec{F} + \frac{d m}{dt} \vec{u},$$

    where $ \vec{u} $ is the velocity of the separated (gained) substance relative to the body considered.

  • Law of angular momentum variation of a system:
    $$\frac{d\vec{L}}{dt} = \vec{\tau},$$

    where $ \vec{L} $ is the angular momentum of the system, and $ \vec{\tau} $ is the total moment of all external forces.

  • Angular momentum of a system:
    $$\vec{L} = \widetilde{\vec{L}} + \vec{r_C}\times\vec{p},$$

    where $ \widetilde\vec{L} $ is its angular momentum in the system of the center of inertia, $ \vec{r_C} $ is the radius vector of the center of inertia, and $ \vec{p} $ is the momentum of the system.

 

Keywords: 
conservation, energy, momentum, angular momentum
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