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Laws of Conservation of Energy, Momentum, and Angular Momentum

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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.

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