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## Main Question or Discussion Point

In classical mechanics, if we consider the motion of a particle of mass [itex]m[/itex], then

[tex]m=constant[/tex][tex]\vec{v}=d\vec{r}/dt[/tex][tex]\vec{a}=d\vec{v}/dt[/tex][tex]\vec{j}=d\vec{a}/dt[/tex][tex]\ldots[/tex]

Definition of Momentum [itex](\vec{M})[/itex]

[tex]\vec{M} \; = \int_a^b m \, \vec{a} \, dt \; = \int_a^b m \,d\vec{v} \; = \Delta \; m \, \vec{v}[/tex]

[tex]If \quad \vec{a} = 0 \: \; \rightarrow \; \: \int_a^b m \, \vec{a} \, dt = 0[/tex]

[tex]\rightarrow \; \: \Delta \; m \, \vec{v} = 0[/tex]

[tex]\rightarrow \; \: m \, \vec{v} = constant[/tex]

[tex]\rightarrow \; \: \vec{P} = constant[/tex]

Definition of Momentum 2 [itex](\vec{M}_2)[/itex]

[tex]\vec{M}_2 \; = \int_a^b m \, \vec{j} \, dt \; = \int_a^b m \,d\vec{a} \; = \Delta \; m \, \vec{a}[/tex]

[tex]If \quad \vec{j} = 0 \: \; \rightarrow \; \: \int_a^b m \, \vec{j} \, dt = 0[/tex]

[tex]\rightarrow \; \: \Delta \; m \, \vec{a} = 0[/tex]

[tex]\rightarrow \; \: m \, \vec{a} = constant[/tex]

[tex]\rightarrow \; \: \vec{P}_2 = constant[/tex]

Definition of Work [itex](W)[/itex]

[tex]W \; = \int_a^b m \, \vec{a} \cdot d\vec{r} \; = \int_a^b m \,\frac{d\vec{v}}{dt} \cdot \vec{v} \, dt \; = \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{v}^{\: 2}[/tex]

[tex]If \quad \vec{a} = constant \: \; \rightarrow \; \: \int_a^b m \, \vec{a} \cdot d\vec{r} = \Delta \; m \, \vec{a} \cdot \vec{r}[/tex]

[tex]\rightarrow \; \: \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{v}^{\: 2} + \Delta \; \left( - \; m \, \vec{a} \cdot \vec{r} \right) = 0[/tex]

[tex]\rightarrow \; \: {\textstyle \frac{1}{2}} \, m \, \vec{v}^{\: 2} + \left( - \; m \, \vec{a} \cdot \vec{r} \right) = constant[/tex]

[tex]\rightarrow \; \: T + V = constant[/tex]

Definition of Work 2 [itex](W_2)[/itex]

[tex]W_2 \; = \int_a^b m \, \vec{j} \cdot d\vec{v} \; = \int_a^b m \,\frac{d\vec{a}}{dt} \cdot \vec{a} \, dt \; = \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{a}^{\: 2}[/tex]

[tex]If \quad \vec{j} = constant \: \; \rightarrow \; \: \int_a^b m \, \vec{j} \cdot d\vec{v} = \Delta \; m \, \vec{j} \cdot \vec{v}[/tex]

[tex]\rightarrow \; \: \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{a}^{\: 2} + \Delta \; \left( - \; m \, \vec{j} \cdot \vec{v} \right) = 0[/tex]

[tex]\rightarrow \; \: {\textstyle \frac{1}{2}} \, m \, \vec{a}^{\: 2} + \left( - \; m \, \vec{j} \cdot \vec{v} \right) = constant[/tex]

[tex]\rightarrow \; \: T_2 + V_2 = constant[/tex]

If [itex]\vec{a}[/itex], [itex]\vec{j}[/itex], [itex]\ldots[/itex] are not constant but [itex]\vec{a}[/itex], [itex]\vec{j}[/itex], [itex]\ldots[/itex] are functions of [itex]\vec{r}[/itex], [itex]\vec{v}[/itex], [itex]\ldots[/itex] respectively, then the same final result is obtained; even if Newton's second law were not valid (even if [itex]\sum \vec{F} \ne m\;\vec{a}[/itex])

[tex]m=constant[/tex][tex]\vec{v}=d\vec{r}/dt[/tex][tex]\vec{a}=d\vec{v}/dt[/tex][tex]\vec{j}=d\vec{a}/dt[/tex][tex]\ldots[/tex]

Definition of Momentum [itex](\vec{M})[/itex]

[tex]\vec{M} \; = \int_a^b m \, \vec{a} \, dt \; = \int_a^b m \,d\vec{v} \; = \Delta \; m \, \vec{v}[/tex]

[tex]If \quad \vec{a} = 0 \: \; \rightarrow \; \: \int_a^b m \, \vec{a} \, dt = 0[/tex]

[tex]\rightarrow \; \: \Delta \; m \, \vec{v} = 0[/tex]

[tex]\rightarrow \; \: m \, \vec{v} = constant[/tex]

[tex]\rightarrow \; \: \vec{P} = constant[/tex]

Definition of Momentum 2 [itex](\vec{M}_2)[/itex]

[tex]\vec{M}_2 \; = \int_a^b m \, \vec{j} \, dt \; = \int_a^b m \,d\vec{a} \; = \Delta \; m \, \vec{a}[/tex]

[tex]If \quad \vec{j} = 0 \: \; \rightarrow \; \: \int_a^b m \, \vec{j} \, dt = 0[/tex]

[tex]\rightarrow \; \: \Delta \; m \, \vec{a} = 0[/tex]

[tex]\rightarrow \; \: m \, \vec{a} = constant[/tex]

[tex]\rightarrow \; \: \vec{P}_2 = constant[/tex]

Definition of Work [itex](W)[/itex]

[tex]W \; = \int_a^b m \, \vec{a} \cdot d\vec{r} \; = \int_a^b m \,\frac{d\vec{v}}{dt} \cdot \vec{v} \, dt \; = \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{v}^{\: 2}[/tex]

[tex]If \quad \vec{a} = constant \: \; \rightarrow \; \: \int_a^b m \, \vec{a} \cdot d\vec{r} = \Delta \; m \, \vec{a} \cdot \vec{r}[/tex]

[tex]\rightarrow \; \: \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{v}^{\: 2} + \Delta \; \left( - \; m \, \vec{a} \cdot \vec{r} \right) = 0[/tex]

[tex]\rightarrow \; \: {\textstyle \frac{1}{2}} \, m \, \vec{v}^{\: 2} + \left( - \; m \, \vec{a} \cdot \vec{r} \right) = constant[/tex]

[tex]\rightarrow \; \: T + V = constant[/tex]

Definition of Work 2 [itex](W_2)[/itex]

[tex]W_2 \; = \int_a^b m \, \vec{j} \cdot d\vec{v} \; = \int_a^b m \,\frac{d\vec{a}}{dt} \cdot \vec{a} \, dt \; = \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{a}^{\: 2}[/tex]

[tex]If \quad \vec{j} = constant \: \; \rightarrow \; \: \int_a^b m \, \vec{j} \cdot d\vec{v} = \Delta \; m \, \vec{j} \cdot \vec{v}[/tex]

[tex]\rightarrow \; \: \Delta \; {\textstyle \frac{1}{2}} \, m \, \vec{a}^{\: 2} + \Delta \; \left( - \; m \, \vec{j} \cdot \vec{v} \right) = 0[/tex]

[tex]\rightarrow \; \: {\textstyle \frac{1}{2}} \, m \, \vec{a}^{\: 2} + \left( - \; m \, \vec{j} \cdot \vec{v} \right) = constant[/tex]

[tex]\rightarrow \; \: T_2 + V_2 = constant[/tex]

If [itex]\vec{a}[/itex], [itex]\vec{j}[/itex], [itex]\ldots[/itex] are not constant but [itex]\vec{a}[/itex], [itex]\vec{j}[/itex], [itex]\ldots[/itex] are functions of [itex]\vec{r}[/itex], [itex]\vec{v}[/itex], [itex]\ldots[/itex] respectively, then the same final result is obtained; even if Newton's second law were not valid (even if [itex]\sum \vec{F} \ne m\;\vec{a}[/itex])