A field guide, not a formula dump

Equations with meaning.

Use this page to recover the physical question behind a symbol. Every row names what the equation predicts, the assumptions hidden inside it, and where to derive or test it.

Time evolution

Relationship	Physical meaning and scope	Learn it
$\dot{x}=f(x,t)$	The current state and time determine its instantaneous rate under a chosen model.	Ch. 1
$v_{n+1}=v_n+h a_n$ $q_{n+1}=q_n+h v_{n+1}$	Semi-implicit Euler: update velocity before position. First order, often useful for mechanical stability.	Ch. 1
$E=T+V$	Mechanical energy used as a diagnostic when the modeled system should conserve it.	Ch. 1

Rigid-body motion

Relationship	Physical meaning and scope	Learn it
$p=mv$	Linear momentum for mass $m$ and COM velocity $v$ .	Ch. 3
$\tau=r\times F$	Torque about a reference point from an off-center force.	Ch. 2, 3
$L=I\omega$	Angular momentum when inertia and angular velocity are expressed in the same frame.	Ch. 3
$\dot{R}=[\omega]_{\times}R$	Orientation rate under the chapter's world-frame angular-velocity convention.	Ch. 3
$I_W=R I_B R^T$	Rotate a body-frame inertia tensor into world coordinates.	Ch. 2

Coordinates and articulated dynamics

Relationship	Physical meaning and scope	Learn it
$v_{site}=J(q)\dot q$	A kinematic Jacobian maps joint velocity into site velocity.	Ch. 4
$L(q,\dot q)=T-V$	The Lagrangian combines kinetic and potential energy.	Ch. 5
$M(q)\ddot q+c(q,\dot q)+g(q)=\tau$	Standard joint-space dynamics grouping inertia, velocity effects, gravity, and applied generalized force.	Ch. 5
$\tau=\operatorname{RNEA}(q,\dot q,\ddot q)$	Inverse dynamics: forces required for a requested acceleration.	Ch. 6
$\ddot q=\operatorname{ABA}(q,\dot q,\tau)$	Tree forward dynamics without constructing and factorizing the dense mass matrix.	Ch. 6

Architecture boundary: these generalized-coordinate equations explain common robotics algorithms. LavenderSim's current native engine instead stores Cartesian body state and enforces articulation with joint constraints and sequential impulses.

Constraint equations

Relationship	Physical meaning and scope	Learn it
$\phi(q)=0$	A holonomic configuration must remain on its constraint manifold.	Ch. 7
$J(q)\dot q=0$	Allowed velocity has no component that changes an ideal bilateral constraint.	Ch. 7
$v^+=v^*+M^{-1}J^T\lambda$	Constraint impulse changes unconstrained velocity through inverse mass.	Ch. 11
$A=JM^{-1}J^T$	Constraint-space effective mass, including coupling between rows.	Ch. 11

Collision and contact

Relationship	Physical meaning and scope	Learn it
$g_n\ge0,\;\lambda_n\ge0,\;g_n\lambda_n=0$	Ideal non-penetration: separation gap and pushing reaction cannot both be positive.	Ch. 9
$j_n=-\frac{(1+e)v_n^-}{m_{eff}^{-1}}$	One-dimensional normal bounce impulse for an approaching contact, before unilateral projection.	Ch. 9
$h_{rebound}\approx e^2h_{drop}$	Ideal gravity-only sphere bounce, useful as a scoped restitution check.	Ch. 9

Friction limits

Relationship	Physical meaning and scope	Learn it
$\lVert\lambda_t\rVert\le\mu\lambda_n$	Coulomb friction disk: legal tangent impulse grows with normal support.	Ch. 10
$\tan\theta_c=\mu_s$	Ideal block-on-incline threshold before static friction can no longer hold.	Ch. 10
$P=F\cdot v+\tau\cdot\omega$	Instantaneous mechanical power; dissipative friction should not add energy in its modeled direction.	Ch. 10

Complementarity and iteration

Relationship	Physical meaning and scope	Learn it
$w=A\lambda-b\ge0,\;\lambda\ge0,\;\lambda^Tw=0$	Linear complementarity form of a linearized unilateral solve.	Ch. 11
$\lambda_i\leftarrow\Pi_{\Omega_i}(\lambda_i+(b_i-A_{i:}\lambda)/A_{ii})$	Projected Gauss–Seidel row update onto the legal set $\Omega_i$ .	Ch. 11
$x_{k+1}=F^N(x_k,u_k)$	One control step as $N$ ordered physics substeps with held command.	Ch. 12

Symbol and unit conventions

Symbol	Meaning	SI unit
$q,v,a$	position, linear velocity, linear acceleration	m, m/s, m/s²
$R,\omega,\alpha$	rotation, angular velocity, angular acceleration	unitless, rad/s, rad/s²
$m,I$	mass and inertia tensor	kg, kg·m²
$F,\tau,j,\lambda$	force, torque, impulse, constraint force/impulse by context	N, N·m, N·s, declared per equation
$h,\Delta t$	timestep	s
$e,\mu$	restitution and friction coefficient	unitless

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