Derivations of the equations of motion

Status of these derivations

Optional deeper understanding — not required by the official syllabus. The syllabus excludes full derivations of Lagrange's and Hamilton's equations. Their statements, meanings, assumptions, and applications are taught in the main chapters. This appendix supplies the derivations for readers who want to follow the logical connection from D’Alembert's principle.

The conservation-law proofs, kinetic-energy identities, central-force proofs, Binet equation, Kepler laws, and Runge–Lenz proofs in the main course remain part of the relevant core development. The optional status here does not extend to those results.

Lagrange’s equations

Assume constant-mass particles in an inertial frame, smooth ideal holonomic constraints, independent generalized coordinates, and conservative forces described by a velocity-independent potential $V(q,t)$ . Keep any remaining applied generalized forces as $Q_i$ .

D’Alembert's principle is

\sum_a(\mathbf F_a- m_a\ddot{\mathbf r}_a)\cdot\delta\mathbf r_a=0.

(P.1)

At fixed time, $\delta\mathbf r_a=\sum_i\partial_{q_i}\mathbf r_a\delta q_i$ . The conservative-force contribution is

\sum_a\mathbf F_a^{(c)}\cdot\frac{\partial\mathbf r_a}{\partial q_i} =-\sum_a\nabla_aV\cdot\frac{\partial\mathbf r_a}{\partial q_i} =-\frac{\partial V}{\partial q_i},

(P.2)

by the chain rule. The remaining contribution is $Q_i$ by its virtual-work definition. The kinetic-energy identity proved in Chapter 10 gives

\sum_am_a\ddot{\mathbf r}_a\cdot\frac{\partial\mathbf r_a}{\partial q_i} =\frac d{dt}\frac{\partial T}{\partial\dot q_i}-\frac{\partial T}{\partial q_i}.

(P.3)

Therefore

\sum_i\left[-\frac{\partial V}{\partial q_i}+Q_i -\frac d{dt}\frac{\partial T}{\partial\dot q_i}+\frac{\partial T}{\partial q_i}\right]\delta q_i=0.

(P.4)

The variations are independent, so every bracket is zero. Rearranging and using $V_{\dot q_i}=0$ yields

\frac d{dt}\frac{\partial(T-V)}{\partial\dot q_i}-\frac{\partial(T-V)}{\partial q_i}=Q_i.

(P.5)

With $L=T-V$ this is Lagrange's equation. The minus sign between energies follows from the conservative force being the negative gradient of $V$ . The derivation also identifies its limit: independent virtual variations were used, so arbitrary non-holonomic velocity constraints cannot be eliminated by the same naive substitution.

Hamilton’s equations

Assume a regular Lagrangian so that $p_i=L_{\dot q_i}$ can be inverted for $\dot q(q,p,t)$ . Define $H=\sum_ip_i\dot q_i-L$ . Take its differential before collecting variables:

dH=\sum_i(\dot q_i\,dp_i+p_i\,d\dot q_i) -\sum_i\left(L_{q_i}\,dq_i+L_{\dot q_i}\,d\dot q_i\right)-L_t\,dt.

(P.6)

The $d\dot q_i$ coefficients cancel because $p_i=L_{\dot q_i}$ . Thus

dH=\sum_i\dot q_i\,dp_i-\sum_iL_{q_i}\,dq_i-L_t\,dt.

(P.7)

Since the independent arguments of $H$ are $q,p,t$ , comparison with $dH=\sum_iH_{q_i}dq_i+\sum_iH_{p_i}dp_i+H_tdt$ gives

H_{p_i}=\dot q_i,\qquad H_{q_i}=-L_{q_i},\qquad H_t=-L_t.

(P.8)

Lagrange's equation is $\dot p_i-L_{q_i}=Q_i$ . Substituting the second identity gives

\boxed{\dot q_i=H_{p_i},\qquad\dot p_i=-H_{q_i}+Q_i.}

(P.9)

For $Q_i=0$ these are the canonical Hamilton equations. The same differential calculation also proves the correspondence of cyclic coordinates and the opposite explicit-time derivatives of $H$ and $L$ . All derivatives are evaluated at the corresponding state, but the variables held fixed differ between the two descriptions.