Prove principle of least action.

Let's prove the principle of least action, also known as Hamilton's Principle. This principle states that the path taken by a physical system between two states is the one for which the action is stationary (often a minimum). Step 1: Define the action (S) 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. Step 2: Apply the principle of stationary action 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 Step 3: Expand the variation of the Lagrangian 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 Step 4: Substitute δL into δS and integrate by parts 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 Step 5: Combine terms and derive the Euler-Lagrange equation 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. Step 6: Conclusion 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. What's next? 📸