Prove principle of least action.

The action S for a system is defined as the time integral of the Lagrangian (L) between two fixed times t₁ and t₂:

S = ∫ from t₁ to t₂ of L(q, q̇, t) dt

Here, q represents the generalized coordinates, q̇ represents the generalized velocities (dq/dt), and L is the Lagrangian, which is typically the kinetic energy (T) minus the potential energy (V), i.e., L = T - V.

For the action to be stationary, its variation (δS) must be zero for any small variation in the path, provided the endpoints are fixed (δq(t₁) = 0 and δq(t₂) = 0).

δS = δ ∫ from t₁ to t₂ of L(q, q̇, t) dt = 0

We can move the variation inside the integral:

δS = ∫ from t₁ to t₂ of δL dt

The Lagrangian L is a function of q, q̇, and t. The variation δL can be written using the chain rule:

δL = (∂L/∂q)δq + (∂L/∂q̇)δq̇

Since q̇ = dq/dt, its variation δq̇ is d(δq)/dt. So,

δL = (∂L/∂q)δq + (∂L/∂q̇)d(δq)/dt

Substitute the expanded δL back into the expression for δS:

δS = ∫ from t₁ to t₂ of [(∂L/∂q)δq + (∂L/∂q̇)d(δq)/dt] dt

Now, we apply integration by parts to the second term:
∫ from t₁ to t₂ of (∂L/∂q̇)d(δq)/dt dt

Using ∫ u dv = uv - ∫ v du, let u = (∂L/∂q̇) and dv = d(δq)/dt dt.
Then du = d/dt(∂L/∂q̇) dt and v = δq.

So, the second term becomes:
[(∂L/∂q̇)δq] from t₁ to t₂ - ∫ from t₁ to t₂ of d/dt(∂L/∂q̇)δq dt

Since the endpoints are fixed, δq(t₁) = 0 and δq(t₂) = 0. Therefore, the boundary term [(∂L/∂q̇)δq] from t₁ to t₂ evaluates to zero.

The second term simplifies to:

• ∫ from t₁ to t₂ of d/dt(∂L/∂q̇)δq dt

Substitute this back into the expression for δS:

δS = ∫ from t₁ to t₂ of [(∂L/∂q)δq - d/dt(∂L/∂q̇)δq] dt

Factor out δq:

δS = ∫ from t₁ to t₂ of [(∂L/∂q) - d/dt(∂L/∂q̇)]δq dt

For δS to be zero for arbitrary variations δq (which are zero at the endpoints), the term in the square brackets must be zero:

(∂L/∂q) - d/dt(∂L/∂q̇) = 0

This is the Euler-Lagrange equation.

The Euler-Lagrange equation derived from the principle of least action provides the equations of motion for the system. For conservative systems, these equations are equivalent to Newton's second law (F = ma). Thus, the principle of least action successfully describes the dynamics of classical systems.