Suppose that a particle with mass m is moving in \mathbb{R} ^ 3 under the influence of a force field vec F (x,y,z) . Assume that the force field has the following special property : vec F; vec F =- nabla V , where V(x, y, z) is some smooth scalar-valued function (here, V / (\mathbb{R} ^ 3) -> \mathbb{R} and VV is the gradient of V). (Later in the course, we will call V a "potential function " for ) vec F . Denote by () the position of the particle at time t. Assume that the usual Second Newton's Law is applicable to our situation: m vec a = vec F where stands for the acceleration vector of the particle (remember vec a the formula : vec a =( vec r )^ prime prime ) . Define the total energy function E(t) of the particle at time t as: E(t)=(V vec r )+ (m||( vec r )^ prime ||^ 2)/2 , where (V vec r ) is the composition of V(x, y, z) and vec r (t) . (Note that vec r depends on t, so that the right hand side is a function of t. We suppress t in our writing of (t) and simply write to make the formulas look vec r simpler.) Prove that E(t) is constant for all t.

To prove that the total energy function \( E(t) \) of the particle is constant for all time \( t \), we need to show that the derivative of \( E(t) \) with respect to \( t \) is zero, i.e., \( \frac{dE(t)}{dt} = 0 \).

### Step 1: Define the energy function \( E(t) \)

The total energy function \( E(t) \) is given by:
\[
E(t) = V(\vec{r}(t)) + \frac{m}{2} || \vec{r}'(t) ||^2
\]
Here:
- \( V(\vec{r}(t)) \) is the potential energy, which depends on the position vector \( \vec{r}(t) = (x(t), y(t), z(t)) \).
- \( \frac{m}{2} || \vec{r}'(t) ||^2 \) is the kinetic energy, where \( \vec{r}'(t) \) is the velocity vector and \( m \) is the mass of the particle.

### Step 2: Differentiate \( E(t) \) with respect to \( t \)

We need to differentiate the total energy function with respect to time \( t \). Using the chain rule for differentiation, we get:
\[
\frac{dE(t)}{dt} = \frac{d}{dt} \left( V(\vec{r}(t)) \right) + \frac{d}{dt} \left( \frac{m}{2} || \vec{r}'(t) ||^2 \right)
\]

#### 2.1: Differentiate the potential energy term \( V(\vec{r}(t)) \)

By the chain rule:
\[
\frac{d}{dt} \left( V(\vec{r}(t)) \right) = \nabla V(\vec{r}(t)) \cdot \vec{r}'(t)
\]
where \( \nabla V(\vec{r}(t)) \) is the gradient of the potential function \( V \), and \( \vec{r}'(t) \) is the velocity vector.

#### 2.2: Differentiate the kinetic energy term \( \frac{m}{2} || \vec{r}'(t) ||^2 \)

The kinetic energy is \( \frac{m}{2} || \vec{r}'(t) ||^2 \), where \( || \vec{r}'(t) ||^2 = \vec{r}'(t) \cdot \vec{r}'(t) \). Differentiating this term with respect to \( t \) gives:
\[
\frac{d}{dt} \left( \frac{m}{2} || \vec{r}'(t) ||^2 \right) = m \vec{r}'(t) \cdot \vec{r}''(t)
\]
where \( \vec{r}''(t) \) is the acceleration vector.

### Step 3: Use Newton's Second Law

According to Newton's Second Law, the force acting on the particle is:
\[
m \vec{r}''(t) = \vec{F}(\vec{r}(t))
\]
where \( \vec{F}(\vec{r}(t)) = -\nabla V(\vec{r}(t)) \). Substituting this into the equation for the kinetic energy derivative, we get:
\[
m \vec{r}'(t) \cdot \vec{r}''(t) = \vec{r}'(t) \cdot \vec{F}(\vec{r}(t)) = \vec{r}'(t) \cdot \left( -\nabla V(\vec{r}(t)) \right)
\]
Thus, we have:
\[
m \vec{r}'(t) \cdot \vec{r}''(t) = - \vec{r}'(t) \cdot \nabla V(\vec{r}(t))
\]

### Step 4: Combine the results

Now, substitute the results from Steps 2.1 and 2.2 into the expression for \( \frac{dE(t)}{dt} \):
\[
\frac{dE(t)}{dt} = \nabla V(\vec{r}(t)) \cdot \vec{r}'(t) + m \vec{r}'(t) \cdot \vec{r}''(t)
\]
Using the fact that \( m \vec{r}''(t) = - \nabla V(\vec{r}(t)) \), this becomes:
\[
\frac{dE(t)}{dt} = \nabla V(\vec{r}(t)) \cdot \vec{r}'(t) - \nabla V(\vec{r}(t)) \cdot \vec{r}'(t) = 0
\]

### Step 5: Conclusion

Since \( \frac{dE(t)}{dt} = 0 \), the total energy \( E(t) \) is constant for all \( t \).

Thus, we have proven that the total energy of the particle remains constant over time.

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Would you like more details or further explanation?

Here are five related questions:
1. What is the physical interpretation of a potential function in a force field?
2. How can you determine the force from a given potential function?
3. What is the significance of energy conservation in a closed system?
4. How does the gradient of a scalar field relate to the direction of the force?
5. What are the conditions under which Newton's laws hold in this scenario?

**Tip**: The potential energy function \( V(\vec{r}(t)) \) often simplifies the process of finding the force in conservative systems.

Suppose that a particle with mass m is moving in ℝ^3 under the influence of a force field F(x, y, z). Assume that the force field has the property: F = -∇V, where V(x, y, z) is a smooth scalar-valued function (potential function). Denote the position of the particle at time t by r(t). According to Newton's Second Law, m * a = F, where a is the acceleration vector (a = r''(t)). Define the total energy function E(t) as E(t) = V(r(t)) + (m||r'(t)||²)/2. Prove that E(t) is constant for all t.

This problem explores the conservation of total energy for a particle moving in a 3D force field governed by Newton's Second Law. Learn how to derive that the energy remains constant over time using the potential function, kinetic energy, and vector calculus. Ideal for university-level physics or calculus students.

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