Classical Mechanics 4
Angular Momentum and Hamilton’s Equations.
1 Angular Momentum
We begin our discussion of angular momentum in \(\mathbb{R}^2\), which serves as the simplest non-trivial case for understanding rotational dynamics.
In geometric terms, this can be expressed as \(J = |\mathbf{x}||\mathbf{p}| \sin \phi\), where \(\phi\) is the angle between the position vector \(\mathbf{x}\) and the momentum vector \(\mathbf{p}\). If we employ polar coordinates \((r, \theta)\) such that \(x_1 = r\cos\theta\) and \(x_2 = r\sin\theta\), a direct calculation using the operator \(\mathrm{D}\) reveals: \[J = m(x_1 \mathrm{D}_t x_2 - x_2 \mathrm{D}_t x_1) = mr^2 \mathrm{D}_t \theta.\] Furthermore, we can relate \(J\) to the rate at which the position vector sweeps out area. Let \(A\) be the area function defined by the integral \(A = \frac{1}{2}\mathrm{I}(r^2, \theta_0, \theta)\). It follows that: \[J = 2m \mathrm{D}_t A.\]
A pivotal property of angular momentum is its conservation under central force fields.
To prove this, we compute the time derivative of \(J\) along a trajectory: \[\mathrm{D}_t J = \mathrm{D}_t (x_1 p_2 - x_2 p_1) = (\mathrm{D}_t x_1 p_2 + x_1 \mathrm{D}_t p_2) - (\mathrm{D}_t x_2 p_1 + x_2 \mathrm{D}_t p_1).\] Since \(\mathrm{D}_t x_i p_j = \mathrm{D}_t x_i (m \mathrm{D}_t x_j)\), terms of the form \(\mathrm{D}_t x_1 p_2\) and \(\mathrm{D}_t x_2 p_1\) are equal and thus cancel. Substituting the force components \(\mathrm{D}_t p_i = F_i = -\mathrm{D}_{x_i} V\), we obtain: \[\mathrm{D}_t J = x_1 F_2 - x_2 F_1 = -x_1 \mathrm{D}_{x_2} V + x_2 \mathrm{D}_{x_1} V.\] The condition \(\mathrm{D}_t J = 0\) requires the gradient \(\nabla V\) to be everywhere parallel to the position vector \(\mathbf{x}\). This alignment occurs if and only if \(V\) depends exclusively on the radial distance \(r\), confirming that \(V\) is centrally symmetric.
The conservation of angular momentum directly leads to Kepler’s Second Law.
In \(\mathbb{R}^3\), the angular momentum is typically represented as a vector \(\mathbf{J} = \mathbf{x} \times \mathbf{p}\). However, this vector representation is a low-dimensional artifact. In general \(\mathbb{R}^n\), angular momentum is properly characterized as a skew-symmetric matrix.
The conservation of this total angular momentum is linked to the rotational invariance of the potential energy.
Consider an infinitesimal rotation \(R(s) = \exp(s\Omega)\) where \(\Omega\) is a skew-symmetric matrix. If \(V\) is invariant, the derivative with respect to the rotation parameter vanishes: \(\mathrm{D}_s |_{s=0} V(e^{s\Omega}\boldsymbol{X}) = 0\). By the multivariable chain rule: \[\sum_{l=1}^N \langle \nabla^l V, \Omega \mathbf{x}_l \rangle = 0 \implies \sum_{l=1}^N \sum_{j,k} \mathrm{D}_{x_l^j} V \Omega_{jk} x_l^k = 0.\] Using the force relation \(\mathbf{F}_l = -\nabla^l V\) and rearranging the summation indices, we find: \[\sum_{j,k} \Omega_{jk} \left( \sum_{l=1}^N F_l^j x_l^k \right) = 0.\] Since this holds for any skew-symmetric matrix \(\Omega\), the term in parentheses must be symmetric in \(j\) and \(k\), which implies the vanishing of the skew-symmetric part \(\sum_l (F_l^j x_l^k - F_l^k x_l^j)\). This sum is precisely the time derivative \(\mathrm{D}_t J_{jk}\), establishing the conservation of total angular momentum.
As momentum conservation stems from translational symmetry, angular momentum conservation reflects the isotropy of space—the fact that the laws of physics are independent of the system’s orientation.
2 Poisson Brackets and Hamiltonian Mechanics
We now transition to the Hamiltonian formulation of mechanics, which provides a profound framework for understanding conserved quantities. The hallmark of this approach is the transition from the configuration space \((\mathbf{x}, \mathbf{v})\) to the phase space \((\mathbf{x}, \mathbf{p})\), where energy is expressed as a function of position and momentum, termed the Hamiltonian.
For a particle moving in \(\mathbb{R}^n\) with a standard potential, the Hamiltonian \(H\) is defined as: \[H(\mathbf{x}, \mathbf{p}) = \frac{1}{2m} \sum_{j=1}^{n} p_j^2 + V(\mathbf{x}),\] where \(p_j = m \mathrm{D}_t x_j\). Under this representation, Newton’s second law is equivalent to the following system of first-order equations, known as Hamilton’s equations: \[\begin{cases} \mathrm{D}_t x_j = \mathrm{D}_{p_j} H \\ \mathrm{D}_t p_j = -\mathrm{D}_{x_j} H \end{cases}\] To understand the structural elegance of these equations, we introduce the formal machinery of the Poisson bracket.
The Poisson bracket satisfies several fundamental algebraic properties that characterize the geometry of phase space.
1 These properties imply that the Poisson bracket defines a Lie bracket on the space of smooth functions, thereby endowing \(C^\infty(\mathbb{R}^{2n})\) with the structure of a Lie algebra.
The linearity and anti-symmetry follow directly from the algebraic structure of the definition. The Leibniz rule is a consequence of the bracket acting as a first-order differential operator satisfying the product rule. The Jacobi identity, while requiring more extensive calculation, can be verified by expanding the nested derivatives; the symmetry of mixed second-order derivatives \(\mathrm{D}_{x_j}\mathrm{D}_{p_k} = \mathrm{D}_{p_k}\mathrm{D}_{x_j}\) ensures that all higher-order terms cancel, consistent with the bracket’s interpretation as a Lie derivative along a Hamiltonian vector field.
Consider the basis elements \(x_i x_j\), \(p_i p_j\), and \(x_i p_j\). By evaluating their mutual Poisson brackets, one observes that they satisfy the commutation relations of \(\mathfrak{sp}(2n, \mathbb{R})\). For example, a direct computation shows \(\{x_i p_j, x_k p_l\} = \delta_{jk} x_i p_l - \delta_{il} x_k p_j\). This preservation of the Lie structure under the mapping establishes the isomorphism.
The canonical relationship between position and momentum functions is captured by the fundamental brackets, which serve as the classical correspondence to the canonical commutation relations in quantum mechanics.
Applying the definition to the third case, we have \(\{x_j, p_k\} = \sum_l ( \mathrm{D}_{x_l} x_j \mathrm{D}_{p_l} p_k - \mathrm{D}_{p_l} x_j \mathrm{D}_{x_l} p_k )\). Since \(\mathrm{D}_{x_l} x_j = \delta_{jl}\) and \(\mathrm{D}_{p_l} p_k = \delta_{kl}\), and noting that the cross-derivatives \(\mathrm{D}_p x\) and \(\mathrm{D}_x p\) vanish identically, the expression reduces to \(\sum_l \delta_{jl} \delta_{kl} = \delta_{jk}\). The proofs for the remaining identities proceed in an analogous fashion.
The evolution of any physical observable \(f\) along a trajectory is elegantly described by its bracket with the Hamiltonian.
2 This is often referred to as the Heisenberg picture of classical mechanics.
By the multivariable chain rule, the time derivative is expressed as \(\mathrm{D}_t f = \sum_j ( \mathrm{D}_{x_j} f \mathrm{D}_t x_j + \mathrm{D}_{p_j} f \mathrm{D}_t p_j )\). Substituting the Hamiltonian equations \(\mathrm{D}_t x_j = \mathrm{D}_{p_j} H\) and \(\mathrm{D}_t p_j = -\mathrm{D}_{x_j} H\) into this sum, we obtain \(\mathrm{D}_t f = \sum_j ( \mathrm{D}_{x_j} f \mathrm{D}_{p_j} H - \mathrm{D}_{p_j} f \mathrm{D}_{x_j} H )\), which is precisely the definition of \(\{f, H\}\).
The Poisson bracket thus reveals that conservation laws are deeply embedded in the Lie algebraic structure of the phase space.
3 Symplectic Geometry and Liouville’s Theorem
The significance of conserved quantities lies in their ability to restrict the dynamics of a system to specific submanifolds of the phase space. If \(H\) and \(f\) are two independent conserved quantities, the motion is confined to the intersection of their level sets. This reduction of dimensionality is a cornerstone for integrating complex dynamical systems.
The solution to the Hamilton’s equations on \(\mathbb{R}^{2n}\) can be viewed as a flow, denoted by a family of diffeomorphisms \(\Phi_t: \mathbb{R}^{2n} \to \mathbb{R}^{2n}\), where \(\Phi_t(\mathbf{x}_0, \mathbf{p}_0)\) represents the state of the system at time \(t\) given the initial condition at \(t=0\). Depending on the potential \(V\), a flow is called complete if \(\Phi_t\) is defined for all \(t \in \mathbb{R}\) on the entire phase space.
A fundamental property of these flows is the preservation of the phase space volume.
To prove Liouville’s Theorem, it suffices to show that the divergence of the Hamiltonian vector field \(X_H = (\mathrm{D}_t \mathbf{x}, \mathrm{D}_t \mathbf{p})\) vanishes. The divergence in phase space is defined as: \[\text{div}(X_H) = \sum_{j=1}^n \left( \mathrm{D}_{x_j} (\mathrm{D}_t x_j) + \mathrm{D}_{p_j} (\mathrm{D}_t p_j) \right).\] Substituting Hamilton’s equations \(\mathrm{D}_t x_j = \mathrm{D}_{p_j} H\) and \(\mathrm{D}_t p_j = -\mathrm{D}_{x_j} H\), we obtain: \[\text{div}(X_H) = \sum_{j=1}^n \left( \mathrm{D}_{x_j} \mathrm{D}_{p_j} H - \mathrm{D}_{p_j} \mathrm{D}_{x_j} H \right) = 0,\] where the equality follows from the symmetry of mixed partial derivatives. By the divergence theorem, a flow generated by a divergence-free vector field preserves the volume measure.
Beyond volume preservation, Hamiltonian flows satisfy a much more rigid geometric constraint: they preserve the Symplectic Form \(\omega = \sum dx_j \wedge dp_j\). This implies that the Jacobian matrix of \(\Phi_t\) is a symplectic matrix at every point.
Algebraically, the preservation of the symplectic structure is equivalent to the preservation of Poisson brackets. For any smooth functions \(f, g\), we have: \[\{f \circ \Phi_t, g \circ \Phi_t\} = \{f, g\} \circ \Phi_t.\] In mathematics, such a mapping \(\Psi\) is called a Symplectomorphism, while in physics, it is referred to as a Canonical Transformation.
Every smooth function \(f\) on \(\mathbb{R}^{2n}\) can be viewed as a Hamiltonian generator of a flow. While the flow generated by \(H\) represents physical time evolution \(\mathrm{D}_t\), flows generated by other functions represent geometric symmetries of the system.
For \(f_{\mathbf{a}} = \sum a_j p_j\), the equations of the flow using the parameter \(s\) are: \[\mathrm{D}_s x_j = \mathrm{D}_{p_j} f_{\mathbf{a}} = a_j, \quad \mathrm{D}_s p_j = -\mathrm{D}_{x_j} f_{\mathbf{a}} = 0.\] Integrating these directly with respect to \(s\) yields the translation \(\mathbf{x}(s) = \mathbf{x}_0 + s\mathbf{a}\) and constant momentum \(\mathbf{p}(s) = \mathbf{p}_0\). Similarly, for \(g_{\mathbf{b}} = \sum b_j x_j\): \[\mathrm{D}_s x_j = \mathrm{D}_{p_j} g_{\mathbf{b}} = 0, \quad \mathrm{D}_s p_j = -\mathrm{D}_{x_j} g_{\mathbf{b}} = -b_j,\] which results in constant position \(\mathbf{x}(s) = \mathbf{x}_0\) and the momentum shift \(\mathbf{p}(s) = \mathbf{p}_0 - s\mathbf{b}\).
These results illustrate that the momentum function generates spatial translations, whereas the position function generates translations in momentum space. This deep connection between conserved quantities and physical symmetries is generalized by the principle that a quantity \(f\) is conserved (\(\mathrm{D}_t f = 0\)) if and only if the Hamiltonian \(H\) is invariant under the flow generated by \(f\).
4 Angular Momentum as a Generator of Rotation
The geometric interpretation of angular momentum in the Hamiltonian framework reveals that while \(H\) generates the physical evolution \(\mathrm{D}_t\), other functions generate different geometric transformations.
To prove this, we treat \(J\) as the Hamiltonian and define the flow via the operator \(\mathrm{D}_s\). The corresponding Hamilton’s equations are: \[ \begin{aligned} \mathrm{D}_s x_1 &= \mathrm{D}_{p_1} J = -x_2, & \mathrm{D}_s x_2 &= \mathrm{D}_{p_2} J = x_1 \\ \mathrm{D}_s p_1 &= -\mathrm{D}_{x_1} J = -p_2, & \mathrm{D}_s p_2 &= -\mathrm{D}_{x_2} J = p_1 \end{aligned} \] These constitute two decoupled systems of linear equations of the form \(\mathrm{D}_s \mathbf{u} = A\mathbf{u}\) with \(A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\). The solution is given by the matrix exponential \(e^{sA}\), which corresponds to the rotation matrix \(R(s)\).
In \(\mathbb{R}^{2n}\), the angular momentum in the \((j, k)\) plane is defined as a smooth function that generates a simultaneous rotation of the corresponding position and momentum coordinates.
To prove this, we treat \(L_{jk}\) as the Hamiltonian generator and define the flow through the differential operator \(\mathrm{D}_s\). The evolution of the phase space coordinates is governed by Hamilton’s equations: \[\mathrm{D}_s x_i = \mathrm{D}_{p_i} L_{jk}, \quad \mathrm{D}_s p_i = -\mathrm{D}_{x_i} L_{jk}.\] Evaluating the non-zero partial derivatives of \(L_{jk} = x_j p_k - x_k p_j\), we observe that for the \(j\) and \(k\) components, the system of equations becomes: \[ \begin{aligned} \mathrm{D}_s x_j &= \mathrm{D}_{p_j} L_{jk} = -x_k, & \mathrm{D}_s x_k &= \mathrm{D}_{p_k} L_{jk} = x_j, \\ \mathrm{D}_s p_j &= -\mathrm{D}_{x_j} L_{jk} = p_k, & \mathrm{D}_s p_k &= -\mathrm{D}_{x_k} L_{jk} = -p_j. \end{aligned} \] These relations constitute two decoupled systems of linear ordinary differential equations. For the position coordinates, the system can be expressed in matrix form as \(\mathrm{D}_s \mathbf{u} = A\mathbf{u}\) with \(A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\). The solution is given by the matrix exponential \(\exp(sA)\), which is exactly the rotation matrix \(R(s) = \begin{pmatrix} \cos s & -\sin s \\ \sin s & \cos s \end{pmatrix}\). An identical structure governs the momentum components \(p_j\) and \(p_k\).
For any index \(l\) such that \(l \neq j\) and \(l \neq k\), the derivatives \(\mathrm{D}_{p_l} L_{jk}\) and \(\mathrm{D}_{x_l} L_{jk}\) vanish identically. Consequently, \(\mathrm{D}_s x_l = 0\) and \(\mathrm{D}_s p_l = 0\), ensuring that all coordinates outside the \((j, k)\)-plane are conserved under the flow.
The parameter \(s\) represents the angle of rotation rather than physical time. Thus, \(J\) is characterized as the Hamiltonian generator of rotations. In this context, any function \(g\) satisfies \(\mathrm{D}_s g = \{g, J\}\) along the rotation flow.
3 This illustrates that physical conservation laws are reflections of the underlying symmetries of the Hamiltonian \(H\).
The Hamiltonian framework extends to \(N\) particles in \(\mathbb{R}^n\), where the phase space \(\mathbb{R}^{2nN}\) is spanned by the tuples \((\boldsymbol{X}, \mathbf{p})\) with \(\mathbf{x}^j, \mathbf{p}^j \in \mathbb{R}^n\).
We verify the equivalence by direct computation. From the first equation: \[\mathrm{D}_t x_k^j = \mathrm{D}_{p_k^j} \left( \sum_i \frac{|\mathbf{p}^i|^2}{2m_i} \right) = \frac{p_k^j}{m_j} \implies p_k^j = m_j \mathrm{D}_t x_k^j.\] Applying the operator \(\mathrm{D}_t\) to both sides and substituting the second equation: \[m_j \mathrm{D}_t^2 x_k^j = \mathrm{D}_t p_k^j = -\mathrm{D}_{x_k^j} H = -\mathrm{D}_{x_k^j} V.\] This matches the \(k\)-th component of Newton’s law for the \(j\)-th particle. The identity \(\mathrm{D}_t f = \{f, H\}\) follows from the chain rule: \[\mathrm{D}_t f = \sum_{j,k} \left( \mathrm{D}_{x_k^j} f \mathrm{D}_t x_k^j + \mathrm{D}_{p_k^j} f \mathrm{D}_t p_k^j \right),\] which, upon substituting Hamilton’s equations, becomes the definition of the Poisson bracket \(\{f, H\}\).